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Page 1: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

Game TheoryRepeated Games

Jordi Massó

International Doctorate in Economic Analysis (IDEA)Universitat Autònoma de Barcelona (UAB)

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 1 / 64

Page 2: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 3: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 4: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 5: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 6: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 7: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 8: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 9: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 10: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

R. Aumann and L. Shapley. �Long Term Competition �A GameTheoretic Analysis,�mimeo, The Hebrew University, 1976.

R. Aumann. �Survey of Repeated Games,� in Essays in Game Theoryand Mathematical Economics in Honor of Oskar Morgenstern, 1981.

Relationships between players last over time: long-term strategicinteraction.

We observe non-equilibrium behavior; for instance, cooperation ininteractions like the Prisoners�Dilemma.

Diamonds market.

Cartels (like the OPEC).

Reputation phenomena.

Con�icts.

Etc.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 2 / 64

Page 11: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Is it possible to sustain non-equilibrium behavior (for instance,cooperation in the Prisoners�Dilemma) as equilibrium of a largergame through out repetition?

The goal is to introduce dynamic aspects in the strategic interaction.

Two basic hypothesis:

Perfect monitoring.

Only pure strategies.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 3 / 64

Page 12: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Is it possible to sustain non-equilibrium behavior (for instance,cooperation in the Prisoners�Dilemma) as equilibrium of a largergame through out repetition?

The goal is to introduce dynamic aspects in the strategic interaction.

Two basic hypothesis:

Perfect monitoring.

Only pure strategies.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 3 / 64

Page 13: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Is it possible to sustain non-equilibrium behavior (for instance,cooperation in the Prisoners�Dilemma) as equilibrium of a largergame through out repetition?

The goal is to introduce dynamic aspects in the strategic interaction.

Two basic hypothesis:

Perfect monitoring.

Only pure strategies.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 3 / 64

Page 14: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Is it possible to sustain non-equilibrium behavior (for instance,cooperation in the Prisoners�Dilemma) as equilibrium of a largergame through out repetition?

The goal is to introduce dynamic aspects in the strategic interaction.

Two basic hypothesis:

Perfect monitoring.

Only pure strategies.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 3 / 64

Page 15: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Is it possible to sustain non-equilibrium behavior (for instance,cooperation in the Prisoners�Dilemma) as equilibrium of a largergame through out repetition?

The goal is to introduce dynamic aspects in the strategic interaction.

Two basic hypothesis:

Perfect monitoring.

Only pure strategies.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 3 / 64

Page 16: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form. Ai isthe set of player i�s actions and A = ∏

i2IAi is the set of action pro�les.

The game G is repeated over time: t = 1, 2, ...

The duple (I , (Ai )i2I ) is a game form.

De�ne, for every t � 1, At = A� � � � � A| {z }t�times

.

That is, (a1, ..., at ) 2 At , where for every 1 � s � t,as = (as1, ..., a

sn) 2 A.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 4 / 64

Page 17: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form. Ai isthe set of player i�s actions and A = ∏

i2IAi is the set of action pro�les.

The game G is repeated over time: t = 1, 2, ...

The duple (I , (Ai )i2I ) is a game form.

De�ne, for every t � 1, At = A� � � � � A| {z }t�times

.

That is, (a1, ..., at ) 2 At , where for every 1 � s � t,as = (as1, ..., a

sn) 2 A.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 4 / 64

Page 18: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form. Ai isthe set of player i�s actions and A = ∏

i2IAi is the set of action pro�les.

The game G is repeated over time: t = 1, 2, ...

The duple (I , (Ai )i2I ) is a game form.

De�ne, for every t � 1, At = A� � � � � A| {z }t�times

.

That is, (a1, ..., at ) 2 At , where for every 1 � s � t,as = (as1, ..., a

sn) 2 A.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 4 / 64

Page 19: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form. Ai isthe set of player i�s actions and A = ∏

i2IAi is the set of action pro�les.

The game G is repeated over time: t = 1, 2, ...

The duple (I , (Ai )i2I ) is a game form.

De�ne, for every t � 1, At = A� � � � � A| {z }t�times

.

That is, (a1, ..., at ) 2 At , where for every 1 � s � t,as = (as1, ..., a

sn) 2 A.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 4 / 64

Page 20: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.1.- Introduction

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form. Ai isthe set of player i�s actions and A = ∏

i2IAi is the set of action pro�les.

The game G is repeated over time: t = 1, 2, ...

The duple (I , (Ai )i2I ) is a game form.

De�ne, for every t � 1, At = A� � � � � A| {z }t�times

.

That is, (a1, ..., at ) 2 At , where for every 1 � s � t,as = (as1, ..., a

sn) 2 A.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 4 / 64

Page 21: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Given the game in normal form G = (I , (Ai )i2I , (hi )i2I ), de�ne thesuper-game form as the game form (I , (Fi )i2I ), where for every i 2 I ,

Fi =�fi = ff ti g∞

t=1 j f 1i 2 Ai and 8t � 1, f t+1i : At �! Ai.

Perfect monitoring: the domain of f t+1i is At .

Pure actions: the range of f t+1i is a subset of Ai .

Notation: F = ∏i2IFi .

Given f = (fi )i2I 2 F we represent the sequence of actions inducedby f as

a(f ) = fat (f )g∞t=1,

which is de�ned recursively as follows:

a1(f ) 2 A is given by a1i (f ) = f 1i for all i 2 I ,andfor all t � 1, at+1(f ) 2 A is given by at+1i (f ) = f t+1i (a1(f ), ..., at (f ))for all i 2 I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 5 / 64

Page 22: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Given the game in normal form G = (I , (Ai )i2I , (hi )i2I ), de�ne thesuper-game form as the game form (I , (Fi )i2I ), where for every i 2 I ,

Fi =�fi = ff ti g∞

t=1 j f 1i 2 Ai and 8t � 1, f t+1i : At �! Ai.

Perfect monitoring: the domain of f t+1i is At .

Pure actions: the range of f t+1i is a subset of Ai .

Notation: F = ∏i2IFi .

Given f = (fi )i2I 2 F we represent the sequence of actions inducedby f as

a(f ) = fat (f )g∞t=1,

which is de�ned recursively as follows:

a1(f ) 2 A is given by a1i (f ) = f 1i for all i 2 I ,andfor all t � 1, at+1(f ) 2 A is given by at+1i (f ) = f t+1i (a1(f ), ..., at (f ))for all i 2 I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 5 / 64

Page 23: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Given the game in normal form G = (I , (Ai )i2I , (hi )i2I ), de�ne thesuper-game form as the game form (I , (Fi )i2I ), where for every i 2 I ,

Fi =�fi = ff ti g∞

t=1 j f 1i 2 Ai and 8t � 1, f t+1i : At �! Ai.

Perfect monitoring: the domain of f t+1i is At .

Pure actions: the range of f t+1i is a subset of Ai .

Notation: F = ∏i2IFi .

Given f = (fi )i2I 2 F we represent the sequence of actions inducedby f as

a(f ) = fat (f )g∞t=1,

which is de�ned recursively as follows:

a1(f ) 2 A is given by a1i (f ) = f 1i for all i 2 I ,andfor all t � 1, at+1(f ) 2 A is given by at+1i (f ) = f t+1i (a1(f ), ..., at (f ))for all i 2 I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 5 / 64

Page 24: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Given the game in normal form G = (I , (Ai )i2I , (hi )i2I ), de�ne thesuper-game form as the game form (I , (Fi )i2I ), where for every i 2 I ,

Fi =�fi = ff ti g∞

t=1 j f 1i 2 Ai and 8t � 1, f t+1i : At �! Ai.

Perfect monitoring: the domain of f t+1i is At .

Pure actions: the range of f t+1i is a subset of Ai .

Notation: F = ∏i2IFi .

Given f = (fi )i2I 2 F we represent the sequence of actions inducedby f as

a(f ) = fat (f )g∞t=1,

which is de�ned recursively as follows:

a1(f ) 2 A is given by a1i (f ) = f 1i for all i 2 I ,andfor all t � 1, at+1(f ) 2 A is given by at+1i (f ) = f t+1i (a1(f ), ..., at (f ))for all i 2 I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 5 / 64

Page 25: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Given the game in normal form G = (I , (Ai )i2I , (hi )i2I ), de�ne thesuper-game form as the game form (I , (Fi )i2I ), where for every i 2 I ,

Fi =�fi = ff ti g∞

t=1 j f 1i 2 Ai and 8t � 1, f t+1i : At �! Ai.

Perfect monitoring: the domain of f t+1i is At .

Pure actions: the range of f t+1i is a subset of Ai .

Notation: F = ∏i2IFi .

Given f = (fi )i2I 2 F we represent the sequence of actions inducedby f as

a(f ) = fat (f )g∞t=1,

which is de�ned recursively as follows:

a1(f ) 2 A is given by a1i (f ) = f 1i for all i 2 I ,andfor all t � 1, at+1(f ) 2 A is given by at+1i (f ) = f t+1i (a1(f ), ..., at (f ))for all i 2 I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 5 / 64

Page 26: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Given the game in normal form G = (I , (Ai )i2I , (hi )i2I ), de�ne thesuper-game form as the game form (I , (Fi )i2I ), where for every i 2 I ,

Fi =�fi = ff ti g∞

t=1 j f 1i 2 Ai and 8t � 1, f t+1i : At �! Ai.

Perfect monitoring: the domain of f t+1i is At .

Pure actions: the range of f t+1i is a subset of Ai .

Notation: F = ∏i2IFi .

Given f = (fi )i2I 2 F we represent the sequence of actions inducedby f as

a(f ) = fat (f )g∞t=1,

which is de�ned recursively as follows:

a1(f ) 2 A is given by a1i (f ) = f 1i for all i 2 I ,and

for all t � 1, at+1(f ) 2 A is given by at+1i (f ) = f t+1i (a1(f ), ..., at (f ))for all i 2 I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 5 / 64

Page 27: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Given the game in normal form G = (I , (Ai )i2I , (hi )i2I ), de�ne thesuper-game form as the game form (I , (Fi )i2I ), where for every i 2 I ,

Fi =�fi = ff ti g∞

t=1 j f 1i 2 Ai and 8t � 1, f t+1i : At �! Ai.

Perfect monitoring: the domain of f t+1i is At .

Pure actions: the range of f t+1i is a subset of Ai .

Notation: F = ∏i2IFi .

Given f = (fi )i2I 2 F we represent the sequence of actions inducedby f as

a(f ) = fat (f )g∞t=1,

which is de�ned recursively as follows:

a1(f ) 2 A is given by a1i (f ) = f 1i for all i 2 I ,andfor all t � 1, at+1(f ) 2 A is given by at+1i (f ) = f t+1i (a1(f ), ..., at (f ))for all i 2 I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 5 / 64

Page 28: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Examples of strategies in the Prisoners�Dilemma.

�Play always C�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,f t+1i (a1, ..., at ) = C .

�Play C during 5 periods and D thereafter�: f 1i = C , for all1 � t < 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = C and for allt � 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = D.

Trigger strategy. �Start playing C and play C as long as the otherplayer has played always C , once the other player has played D playD always�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,

f t+1i (a1, ..., at ) =�C if for all 1 � s � t, as3�i = CD if there exists 1 � s � t such that as3�i = D.

Tit-for-tat. �Start playing C and then play the action taken by theother player last period�: f 1i = C and for all t � 1 and all(a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = at3�i .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 6 / 64

Page 29: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Examples of strategies in the Prisoners�Dilemma.

�Play always C�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,f t+1i (a1, ..., at ) = C .

�Play C during 5 periods and D thereafter�: f 1i = C , for all1 � t < 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = C and for allt � 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = D.

Trigger strategy. �Start playing C and play C as long as the otherplayer has played always C , once the other player has played D playD always�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,

f t+1i (a1, ..., at ) =�C if for all 1 � s � t, as3�i = CD if there exists 1 � s � t such that as3�i = D.

Tit-for-tat. �Start playing C and then play the action taken by theother player last period�: f 1i = C and for all t � 1 and all(a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = at3�i .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 6 / 64

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5.2.- Strategies

Examples of strategies in the Prisoners�Dilemma.

�Play always C�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,f t+1i (a1, ..., at ) = C .

�Play C during 5 periods and D thereafter�: f 1i = C , for all1 � t < 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = C and for allt � 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = D.

Trigger strategy. �Start playing C and play C as long as the otherplayer has played always C , once the other player has played D playD always�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,

f t+1i (a1, ..., at ) =�C if for all 1 � s � t, as3�i = CD if there exists 1 � s � t such that as3�i = D.

Tit-for-tat. �Start playing C and then play the action taken by theother player last period�: f 1i = C and for all t � 1 and all(a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = at3�i .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 6 / 64

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5.2.- Strategies

Examples of strategies in the Prisoners�Dilemma.

�Play always C�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,f t+1i (a1, ..., at ) = C .

�Play C during 5 periods and D thereafter�: f 1i = C , for all1 � t < 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = C and for allt � 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = D.

Trigger strategy. �Start playing C and play C as long as the otherplayer has played always C , once the other player has played D playD always�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,

f t+1i (a1, ..., at ) =�C if for all 1 � s � t, as3�i = CD if there exists 1 � s � t such that as3�i = D.

Tit-for-tat. �Start playing C and then play the action taken by theother player last period�: f 1i = C and for all t � 1 and all(a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = at3�i .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 6 / 64

Page 32: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Examples of strategies in the Prisoners�Dilemma.

�Play always C�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,f t+1i (a1, ..., at ) = C .

�Play C during 5 periods and D thereafter�: f 1i = C , for all1 � t < 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = C and for allt � 5 and all (a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = D.

Trigger strategy. �Start playing C and play C as long as the otherplayer has played always C , once the other player has played D playD always�: f 1i = C and for all t � 1 and all (a1, ..., at ) 2 At ,

f t+1i (a1, ..., at ) =�C if for all 1 � s � t, as3�i = CD if there exists 1 � s � t such that as3�i = D.

Tit-for-tat. �Start playing C and then play the action taken by theother player last period�: f 1i = C and for all t � 1 and all(a1, ..., at ) 2 At , f t+1i (a1, ..., at ) = at3�i .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 6 / 64

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5.2.- Strategies

Sequences of actions generated by some strategy pro�les.

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

(f1, f2) : For all 1 � s � 5, as (f1, f2) = (C ,C ), a6(f1, f2) = (D,C )and for all t � 7, a7(f1, f2) = (D,D).

a(f1, f2) =((C ,C ), (C ,C ), (C ,C ), (C ,C ), (C ,C ), (D,C ), (D,D), ...|{z}

(D ,D ) always

).

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 7 / 64

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5.2.- Strategies

Sequences of actions generated by some strategy pro�les.

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

(f1, f2) : For all 1 � s � 5, as (f1, f2) = (C ,C ), a6(f1, f2) = (D,C )and for all t � 7, a7(f1, f2) = (D,D).

a(f1, f2) =((C ,C ), (C ,C ), (C ,C ), (C ,C ), (C ,C ), (D,C ), (D,D), ...|{z}

(D ,D ) always

).

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 7 / 64

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5.2.- Strategies

Sequences of actions generated by some strategy pro�les.

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}

(C ,C ) always

).

(f1, f2) : For all 1 � s � 5, as (f1, f2) = (C ,C ), a6(f1, f2) = (D,C )and for all t � 7, a7(f1, f2) = (D,D).

a(f1, f2) =((C ,C ), (C ,C ), (C ,C ), (C ,C ), (C ,C ), (D,C ), (D,D), ...|{z}

(D ,D ) always

).

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 7 / 64

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5.2.- Strategies

Sequences of actions generated by some strategy pro�les.

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}

(C ,C ) always

).

(f1, f2) : For all 1 � s � 5, as (f1, f2) = (C ,C ), a6(f1, f2) = (D,C )and for all t � 7, a7(f1, f2) = (D,D).

a(f1, f2) =((C ,C ), (C ,C ), (C ,C ), (C ,C ), (C ,C ), (D,C ), (D,D), ...|{z}

(D ,D ) always

).

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 7 / 64

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5.2.- Strategies

Sequences of actions generated by some strategy pro�les.

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}

(C ,C ) always

).

(f1, f2) : For all 1 � s � 5, as (f1, f2) = (C ,C ), a6(f1, f2) = (D,C )and for all t � 7, a7(f1, f2) = (D,D).

a(f1, f2) =((C ,C ), (C ,C ), (C ,C ), (C ,C ), (C ,C ), (D,C ), (D,D), ...|{z}

(D ,D ) always

).

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 7 / 64

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5.2.- Strategies

Sequences of actions generated by some strategy pro�les.

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}

(C ,C ) always

).

(f1, f2) : For all 1 � s � 5, as (f1, f2) = (C ,C ), a6(f1, f2) = (D,C )and for all t � 7, a7(f1, f2) = (D,D).

a(f1, f2) =((C ,C ), (C ,C ), (C ,C ), (C ,C ), (C ,C ), (D,C ), (D,D), ...|{z}

(D ,D ) always

).

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).

a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}(C ,C ) always

).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 7 / 64

Page 39: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.2.- Strategies

Sequences of actions generated by some strategy pro�les.

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}

(C ,C ) always

).

(f1, f2) : For all 1 � s � 5, as (f1, f2) = (C ,C ), a6(f1, f2) = (D,C )and for all t � 7, a7(f1, f2) = (D,D).

a(f1, f2) =((C ,C ), (C ,C ), (C ,C ), (C ,C ), (C ,C ), (D,C ), (D,D), ...|{z}

(D ,D ) always

).

(f1, f2) : For all t � 1, at (f1, f2) = (C ,C ).a(f1, f2) = ((C ,C ), (C ,C ), ...|{z}

(C ,C ) always

).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 7 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form and letT 2 N. The �nitely T�times repeated game is the game in normalform GT = (I , (Fi )i2I , (HTi )i2I , where (I , (Fi )i2I ) is the super-gameform and for each i 2 I , HTi : F �! R is de�ned as follows: for allf 2 F ,

HTi (f ) =1T

T

∑t=1hi (at (f )).

Remark: Since GT is a game in normal form, we can de�ne F �T as theset of Nash equilibria of GT .

Examples:

T = 10, H101 (f1, f2) =110 (5 � 3+ 4+ 4 � 1) =

2310 .

T = 6, H62 (f1, f2) =16 (5 � 3+ 4 � 1) =

196 .

For any T � 1, HTi (f1, f2) =1T (3 � T ) = 3.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 8 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form and letT 2 N. The �nitely T�times repeated game is the game in normalform GT = (I , (Fi )i2I , (HTi )i2I , where (I , (Fi )i2I ) is the super-gameform and for each i 2 I , HTi : F �! R is de�ned as follows: for allf 2 F ,

HTi (f ) =1T

T

∑t=1hi (at (f )).

Remark: Since GT is a game in normal form, we can de�ne F �T as theset of Nash equilibria of GT .

Examples:

T = 10, H101 (f1, f2) =110 (5 � 3+ 4+ 4 � 1) =

2310 .

T = 6, H62 (f1, f2) =16 (5 � 3+ 4 � 1) =

196 .

For any T � 1, HTi (f1, f2) =1T (3 � T ) = 3.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 8 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form and letT 2 N. The �nitely T�times repeated game is the game in normalform GT = (I , (Fi )i2I , (HTi )i2I , where (I , (Fi )i2I ) is the super-gameform and for each i 2 I , HTi : F �! R is de�ned as follows: for allf 2 F ,

HTi (f ) =1T

T

∑t=1hi (at (f )).

Remark: Since GT is a game in normal form, we can de�ne F �T as theset of Nash equilibria of GT .

Examples:

T = 10, H101 (f1, f2) =110 (5 � 3+ 4+ 4 � 1) =

2310 .

T = 6, H62 (f1, f2) =16 (5 � 3+ 4 � 1) =

196 .

For any T � 1, HTi (f1, f2) =1T (3 � T ) = 3.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 8 / 64

Page 43: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form and letT 2 N. The �nitely T�times repeated game is the game in normalform GT = (I , (Fi )i2I , (HTi )i2I , where (I , (Fi )i2I ) is the super-gameform and for each i 2 I , HTi : F �! R is de�ned as follows: for allf 2 F ,

HTi (f ) =1T

T

∑t=1hi (at (f )).

Remark: Since GT is a game in normal form, we can de�ne F �T as theset of Nash equilibria of GT .

Examples:

T = 10, H101 (f1, f2) =110 (5 � 3+ 4+ 4 � 1) =

2310 .

T = 6, H62 (f1, f2) =16 (5 � 3+ 4 � 1) =

196 .

For any T � 1, HTi (f1, f2) =1T (3 � T ) = 3.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 8 / 64

Page 44: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form and letT 2 N. The �nitely T�times repeated game is the game in normalform GT = (I , (Fi )i2I , (HTi )i2I , where (I , (Fi )i2I ) is the super-gameform and for each i 2 I , HTi : F �! R is de�ned as follows: for allf 2 F ,

HTi (f ) =1T

T

∑t=1hi (at (f )).

Remark: Since GT is a game in normal form, we can de�ne F �T as theset of Nash equilibria of GT .

Examples:

T = 10, H101 (f1, f2) =110 (5 � 3+ 4+ 4 � 1) =

2310 .

T = 6, H62 (f1, f2) =16 (5 � 3+ 4 � 1) =

196 .

For any T � 1, HTi (f1, f2) =1T (3 � T ) = 3.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 8 / 64

Page 45: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form and letT 2 N. The �nitely T�times repeated game is the game in normalform GT = (I , (Fi )i2I , (HTi )i2I , where (I , (Fi )i2I ) is the super-gameform and for each i 2 I , HTi : F �! R is de�ned as follows: for allf 2 F ,

HTi (f ) =1T

T

∑t=1hi (at (f )).

Remark: Since GT is a game in normal form, we can de�ne F �T as theset of Nash equilibria of GT .

Examples:

T = 10, H101 (f1, f2) =110 (5 � 3+ 4+ 4 � 1) =

2310 .

T = 6, H62 (f1, f2) =16 (5 � 3+ 4 � 1) =

196 .

For any T � 1, HTi (f1, f2) =1T (3 � T ) = 3.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 8 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form. We say thatG is bounded if

sup fhi (a) j i 2 I and a 2 Ag < ∞.

Note that if G is �nite then G is bounded.

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form andlet λ 2 (0, 1). The λ�discounted repeated game is the game innormal form Gλ = (I , (Fi )i2I , (Hλ

i )i2I , where (I , (Fi )i2I ) is thesuper-game form and for each i 2 I , Hλ

i : F �! R is de�ned asfollows: for all f 2 F ,

Hλi (f ) = (1� λ)

∑t=1

λt�1hi (at (f )).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 9 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form. We say thatG is bounded if

sup fhi (a) j i 2 I and a 2 Ag < ∞.

Note that if G is �nite then G is bounded.

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form andlet λ 2 (0, 1). The λ�discounted repeated game is the game innormal form Gλ = (I , (Fi )i2I , (Hλ

i )i2I , where (I , (Fi )i2I ) is thesuper-game form and for each i 2 I , Hλ

i : F �! R is de�ned asfollows: for all f 2 F ,

Hλi (f ) = (1� λ)

∑t=1

λt�1hi (at (f )).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 9 / 64

Page 48: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form. We say thatG is bounded if

sup fhi (a) j i 2 I and a 2 Ag < ∞.

Note that if G is �nite then G is bounded.

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form andlet λ 2 (0, 1). The λ�discounted repeated game is the game innormal form Gλ = (I , (Fi )i2I , (Hλ

i )i2I , where (I , (Fi )i2I ) is thesuper-game form and for each i 2 I , Hλ

i : F �! R is de�ned asfollows: for all f 2 F ,

Hλi (f ) = (1� λ)

∑t=1

λt�1hi (at (f )).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 9 / 64

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5.3.- Payo¤s

Example:

Hλ1 (f1, f2) = (1� λ)(3+ 3λ+ 3λ2 + 3λ3 + 3λ4 + 4λ5 + λ6 + λ7 + ...

= (1� λ)

31� λ5

1� λ+ 4λ5 +

λ6

1� λ

!

= 3(1� λ5) + 4(1� λ)λ5 + λ6

= 3+ λ5 � 3λ6.

Since Gλ is a game in normal form, we can de�ne F �λ as the set ofNash equilibria of Gλ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 10 / 64

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5.3.- Payo¤s

Example:

Hλ1 (f1, f2) = (1� λ)(3+ 3λ+ 3λ2 + 3λ3 + 3λ4 + 4λ5 + λ6 + λ7 + ...

= (1� λ)

31� λ5

1� λ+ 4λ5 +

λ6

1� λ

!

= 3(1� λ5) + 4(1� λ)λ5 + λ6

= 3+ λ5 � 3λ6.

Since Gλ is a game in normal form, we can de�ne F �λ as the set ofNash equilibria of Gλ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 10 / 64

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5.3.- Payo¤s

Example:

Hλ1 (f1, f2) = (1� λ)(3+ 3λ+ 3λ2 + 3λ3 + 3λ4 + 4λ5 + λ6 + λ7 + ...

= (1� λ)

31� λ5

1� λ+ 4λ5 +

λ6

1� λ

!

= 3(1� λ5) + 4(1� λ)λ5 + λ6

= 3+ λ5 � 3λ6.

Since Gλ is a game in normal form, we can de�ne F �λ as the set ofNash equilibria of Gλ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 10 / 64

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5.3.- Payo¤s

Example:

Hλ1 (f1, f2) = (1� λ)(3+ 3λ+ 3λ2 + 3λ3 + 3λ4 + 4λ5 + λ6 + λ7 + ...

= (1� λ)

31� λ5

1� λ+ 4λ5 +

λ6

1� λ

!

= 3(1� λ5) + 4(1� λ)λ5 + λ6

= 3+ λ5 � 3λ6.

Since Gλ is a game in normal form, we can de�ne F �λ as the set ofNash equilibria of Gλ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 10 / 64

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5.3.- Payo¤s

Remarks on Hλi (f ) = (1� λ)

∑t=1

λt�1hi (at (f )):

(1� λ) is a very useful normalization (remember that hi is a vNMutility function and (1� λ)hi is a positive a¢ ne transformation); forinstance, it assigns x to the constant sequence fx t = xg∞

t=1, since

(1� λ)∞

∑t=1

λt�1x = (1� λ) 11�λx = x .

If G is not bounded, the series may be divergent, and therefore Hλi

would not necessarily be well-de�ned.

The payo¤ Hλi (f ) can be interpreted as player i�s expected payo¤ of

playing f when at t, the probability of playing the game at t + 1 isequal to λ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 11 / 64

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5.3.- Payo¤s

Remarks on Hλi (f ) = (1� λ)

∑t=1

λt�1hi (at (f )):

(1� λ) is a very useful normalization (remember that hi is a vNMutility function and (1� λ)hi is a positive a¢ ne transformation); forinstance, it assigns x to the constant sequence fx t = xg∞

t=1, since

(1� λ)∞

∑t=1

λt�1x = (1� λ) 11�λx = x .

If G is not bounded, the series may be divergent, and therefore Hλi

would not necessarily be well-de�ned.

The payo¤ Hλi (f ) can be interpreted as player i�s expected payo¤ of

playing f when at t, the probability of playing the game at t + 1 isequal to λ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 11 / 64

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5.3.- Payo¤s

Remarks on Hλi (f ) = (1� λ)

∑t=1

λt�1hi (at (f )):

(1� λ) is a very useful normalization (remember that hi is a vNMutility function and (1� λ)hi is a positive a¢ ne transformation); forinstance, it assigns x to the constant sequence fx t = xg∞

t=1, since

(1� λ)∞

∑t=1

λt�1x = (1� λ) 11�λx = x .

If G is not bounded, the series may be divergent, and therefore Hλi

would not necessarily be well-de�ned.

The payo¤ Hλi (f ) can be interpreted as player i�s expected payo¤ of

playing f when at t, the probability of playing the game at t + 1 isequal to λ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 11 / 64

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5.3.- Payo¤s

Remarks on Hλi (f ) = (1� λ)

∑t=1

λt�1hi (at (f )):

(1� λ) is a very useful normalization (remember that hi is a vNMutility function and (1� λ)hi is a positive a¢ ne transformation); forinstance, it assigns x to the constant sequence fx t = xg∞

t=1, since

(1� λ)∞

∑t=1

λt�1x = (1� λ) 11�λx = x .

If G is not bounded, the series may be divergent, and therefore Hλi

would not necessarily be well-de�ned.

The payo¤ Hλi (f ) can be interpreted as player i�s expected payo¤ of

playing f when at t, the probability of playing the game at t + 1 isequal to λ.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 11 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form. Thein�nitely repeated game is the game in normal formG∞ = (I , (Fi )i2I , (H∞

i )i2I , where (I , (Fi )i2I ) is the super-game formand for each i 2 I , H∞

i : F �! R that will be de�ned later.

The �natural�payo¤ function would be: for all f 2 F ,

limT!∞

1T

T

∑t=1hi (at (f )) = lim

T!∞HTi (f ).

Problem: This limit may not exist (its existence depends on theparticular strategies used by players).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 12 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form. Thein�nitely repeated game is the game in normal formG∞ = (I , (Fi )i2I , (H∞

i )i2I , where (I , (Fi )i2I ) is the super-game formand for each i 2 I , H∞

i : F �! R that will be de�ned later.

The �natural�payo¤ function would be: for all f 2 F ,

limT!∞

1T

T

∑t=1hi (at (f )) = lim

T!∞HTi (f ).

Problem: This limit may not exist (its existence depends on theparticular strategies used by players).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 12 / 64

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5.3.- Payo¤s

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form. Thein�nitely repeated game is the game in normal formG∞ = (I , (Fi )i2I , (H∞

i )i2I , where (I , (Fi )i2I ) is the super-game formand for each i 2 I , H∞

i : F �! R that will be de�ned later.

The �natural�payo¤ function would be: for all f 2 F ,

limT!∞

1T

T

∑t=1hi (at (f )) = lim

T!∞HTi (f ).

Problem: This limit may not exist (its existence depends on theparticular strategies used by players).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 12 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Let fxng∞n=1 be a bounded sequence of real numbers (i.e., fxng 2 l∞).

We say that x 2 R is the limit superior of fxng, lim supn!∞

fxng, if x isthe highest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn < x + ε(from N on, the sequence is never above x + ε).

for all ε > 0 and all m 2 N there exists n > m such that xn > x � ε(the sequence always goes back to be close to x).

We say that x 2 R is the limit inferior of fxng, lim infn!∞

fxng, if x is thesmallest accumulation point of fxng; that is,

for all ε > 0 there exists N 2 N such that for all n > N, xn > x � ε(from N on, the sequence is never below x � ε).

for all ε > 0 and all m 2 N there exists n > m such that xn < x + ε(the sequence always goes back to be close to x).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 13 / 64

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5.3.- Payo¤s [Parenthesis]

Remark: for all fxng 2 l∞, lim infn!∞

fxng = �lim supn!∞

fyng, where for alln � 1, yn = �xn.

Example: xn =�1 if n is odd�1 if n is even

. Then, lim supn!∞

fxng = 1 and

lim infn!∞

fxng = �1.

Properties: for all fxng, fyng 2 l∞,

if limn!∞

fxng exists then lim infn!∞fxng = lim

n!∞fxng = lim sup

n!∞fxng.

lim infn!∞

fxng+ lim infn!∞fyng � lim inf

n!∞fxn + yng

� lim infn!∞

fxng+ lim supn!∞

fyng

� lim supn!∞

fxn + yng

� lim supn!∞

fxng+ lim supn!∞

fyng.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 14 / 64

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5.3.- Payo¤s [Parenthesis]

Remark: for all fxng 2 l∞, lim infn!∞

fxng = �lim supn!∞

fyng, where for alln � 1, yn = �xn.

Example: xn =�1 if n is odd�1 if n is even

. Then, lim supn!∞

fxng = 1 and

lim infn!∞

fxng = �1.

Properties: for all fxng, fyng 2 l∞,

if limn!∞

fxng exists then lim infn!∞fxng = lim

n!∞fxng = lim sup

n!∞fxng.

lim infn!∞

fxng+ lim infn!∞fyng � lim inf

n!∞fxn + yng

� lim infn!∞

fxng+ lim supn!∞

fyng

� lim supn!∞

fxn + yng

� lim supn!∞

fxng+ lim supn!∞

fyng.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 14 / 64

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5.3.- Payo¤s [Parenthesis]

Remark: for all fxng 2 l∞, lim infn!∞

fxng = �lim supn!∞

fyng, where for alln � 1, yn = �xn.

Example: xn =�1 if n is odd�1 if n is even

. Then, lim supn!∞

fxng = 1 and

lim infn!∞

fxng = �1.

Properties: for all fxng, fyng 2 l∞,

if limn!∞

fxng exists then lim infn!∞fxng = lim

n!∞fxng = lim sup

n!∞fxng.

lim infn!∞

fxng+ lim infn!∞fyng � lim inf

n!∞fxn + yng

� lim infn!∞

fxng+ lim supn!∞

fyng

� lim supn!∞

fxn + yng

� lim supn!∞

fxng+ lim supn!∞

fyng.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 14 / 64

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5.3.- Payo¤s [Parenthesis]

Remark: for all fxng 2 l∞, lim infn!∞

fxng = �lim supn!∞

fyng, where for alln � 1, yn = �xn.

Example: xn =�1 if n is odd�1 if n is even

. Then, lim supn!∞

fxng = 1 and

lim infn!∞

fxng = �1.

Properties: for all fxng, fyng 2 l∞,if limn!∞

fxng exists then lim infn!∞fxng = lim

n!∞fxng = lim sup

n!∞fxng.

lim infn!∞

fxng+ lim infn!∞fyng � lim inf

n!∞fxn + yng

� lim infn!∞

fxng+ lim supn!∞

fyng

� lim supn!∞

fxn + yng

� lim supn!∞

fxng+ lim supn!∞

fyng.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 14 / 64

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5.3.- Payo¤s [Parenthesis]

Remark: for all fxng 2 l∞, lim infn!∞

fxng = �lim supn!∞

fyng, where for alln � 1, yn = �xn.

Example: xn =�1 if n is odd�1 if n is even

. Then, lim supn!∞

fxng = 1 and

lim infn!∞

fxng = �1.

Properties: for all fxng, fyng 2 l∞,if limn!∞

fxng exists then lim infn!∞fxng = lim

n!∞fxng = lim sup

n!∞fxng.

lim infn!∞

fxng+ lim infn!∞fyng � lim inf

n!∞fxn + yng

� lim infn!∞

fxng+ lim supn!∞

fyng

� lim supn!∞

fxn + yng

� lim supn!∞

fxng+ lim supn!∞

fyng.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 14 / 64

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5.3.- Payo¤s [Parenthesis]

If G is bounded then, for all f 2 F ,(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

Desirable properties for H∞i (f ).

If limn!∞

1T

T

∑t=1

hi (at (f )) exists then H∞i (f ) should be equal to it.

lim infn!∞

1T

T

∑t=1

hi (at (f )) � H∞i (f ) � lim supn!∞

1T

T

∑t=1

hi (at (f )).

Note that the later implies the former.

Since we will have to check (equilibrium condition) whetherH∞i (f )�H∞

i (gi , f�i ) � 0, we would like that H∞i (f ) be linear.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 15 / 64

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5.3.- Payo¤s [Parenthesis]

If G is bounded then, for all f 2 F ,(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

Desirable properties for H∞i (f ).

If limn!∞

1T

T

∑t=1

hi (at (f )) exists then H∞i (f ) should be equal to it.

lim infn!∞

1T

T

∑t=1

hi (at (f )) � H∞i (f ) � lim supn!∞

1T

T

∑t=1

hi (at (f )).

Note that the later implies the former.

Since we will have to check (equilibrium condition) whetherH∞i (f )�H∞

i (gi , f�i ) � 0, we would like that H∞i (f ) be linear.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 15 / 64

Page 75: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

If G is bounded then, for all f 2 F ,(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

Desirable properties for H∞i (f ).

If limn!∞

1T

T

∑t=1

hi (at (f )) exists then H∞i (f ) should be equal to it.

lim infn!∞

1T

T

∑t=1

hi (at (f )) � H∞i (f ) � lim supn!∞

1T

T

∑t=1

hi (at (f )).

Note that the later implies the former.

Since we will have to check (equilibrium condition) whetherH∞i (f )�H∞

i (gi , f�i ) � 0, we would like that H∞i (f ) be linear.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 15 / 64

Page 76: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

If G is bounded then, for all f 2 F ,(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

Desirable properties for H∞i (f ).

If limn!∞

1T

T

∑t=1

hi (at (f )) exists then H∞i (f ) should be equal to it.

lim infn!∞

1T

T

∑t=1

hi (at (f )) � H∞i (f ) � lim supn!∞

1T

T

∑t=1

hi (at (f )).

Note that the later implies the former.

Since we will have to check (equilibrium condition) whetherH∞i (f )�H∞

i (gi , f�i ) � 0, we would like that H∞i (f ) be linear.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 15 / 64

Page 77: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

If G is bounded then, for all f 2 F ,(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

Desirable properties for H∞i (f ).

If limn!∞

1T

T

∑t=1

hi (at (f )) exists then H∞i (f ) should be equal to it.

lim infn!∞

1T

T

∑t=1

hi (at (f )) � H∞i (f ) � lim supn!∞

1T

T

∑t=1

hi (at (f )).

Note that the later implies the former.

Since we will have to check (equilibrium condition) whetherH∞i (f )�H∞

i (gi , f�i ) � 0, we would like that H∞i (f ) be linear.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 15 / 64

Page 78: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

If G is bounded then, for all f 2 F ,(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

Desirable properties for H∞i (f ).

If limn!∞

1T

T

∑t=1

hi (at (f )) exists then H∞i (f ) should be equal to it.

lim infn!∞

1T

T

∑t=1

hi (at (f )) � H∞i (f ) � lim supn!∞

1T

T

∑t=1

hi (at (f )).

Note that the later implies the former.

Since we will have to check (equilibrium condition) whetherH∞i (f )�H∞

i (gi , f�i ) � 0, we would like that H∞i (f ) be linear.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 15 / 64

Page 79: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

Proposition There exists a linear function H : l∞ �! R (called aBanach limit) such that for all fxng 2 l∞,

lim infn!∞

fxng � H(fxng) � lim supn!∞

fxng.

It follows from the Hahn-Banach Theorem.

Remarks:

There are many Banach limits.

Results will be invariant with respect to which one we will use.

It is not known a functional form of a Banach limit.

End of Parenthesis.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 16 / 64

Page 80: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

Proposition There exists a linear function H : l∞ �! R (called aBanach limit) such that for all fxng 2 l∞,

lim infn!∞

fxng � H(fxng) � lim supn!∞

fxng.

It follows from the Hahn-Banach Theorem.

Remarks:

There are many Banach limits.

Results will be invariant with respect to which one we will use.

It is not known a functional form of a Banach limit.

End of Parenthesis.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 16 / 64

Page 81: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

Proposition There exists a linear function H : l∞ �! R (called aBanach limit) such that for all fxng 2 l∞,

lim infn!∞

fxng � H(fxng) � lim supn!∞

fxng.

It follows from the Hahn-Banach Theorem.

Remarks:

There are many Banach limits.

Results will be invariant with respect to which one we will use.

It is not known a functional form of a Banach limit.

End of Parenthesis.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 16 / 64

Page 82: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

Proposition There exists a linear function H : l∞ �! R (called aBanach limit) such that for all fxng 2 l∞,

lim infn!∞

fxng � H(fxng) � lim supn!∞

fxng.

It follows from the Hahn-Banach Theorem.

Remarks:

There are many Banach limits.

Results will be invariant with respect to which one we will use.

It is not known a functional form of a Banach limit.

End of Parenthesis.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 16 / 64

Page 83: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

Proposition There exists a linear function H : l∞ �! R (called aBanach limit) such that for all fxng 2 l∞,

lim infn!∞

fxng � H(fxng) � lim supn!∞

fxng.

It follows from the Hahn-Banach Theorem.

Remarks:

There are many Banach limits.

Results will be invariant with respect to which one we will use.

It is not known a functional form of a Banach limit.

End of Parenthesis.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 16 / 64

Page 84: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

Proposition There exists a linear function H : l∞ �! R (called aBanach limit) such that for all fxng 2 l∞,

lim infn!∞

fxng � H(fxng) � lim supn!∞

fxng.

It follows from the Hahn-Banach Theorem.

Remarks:

There are many Banach limits.

Results will be invariant with respect to which one we will use.

It is not known a functional form of a Banach limit.

End of Parenthesis.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 16 / 64

Page 85: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s [Parenthesis]

Proposition There exists a linear function H : l∞ �! R (called aBanach limit) such that for all fxng 2 l∞,

lim infn!∞

fxng � H(fxng) � lim supn!∞

fxng.

It follows from the Hahn-Banach Theorem.

Remarks:

There are many Banach limits.

Results will be invariant with respect to which one we will use.

It is not known a functional form of a Banach limit.

End of Parenthesis.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 16 / 64

Page 86: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Choose a Banach limit H : l∞ �! R.

Given f 2 F , construct fhi (at (f ))g∞t=1 2 l∞ (since G is bounded).

Find

(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

De�ne

H∞i (f ) = H

(1T

T

∑t=1hi (at (f ))

)∞

T=1

!.

Since G∞ is a game in normal form, we can de�ne F �∞ as the set ofNash equilibria of G∞.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 17 / 64

Page 87: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Choose a Banach limit H : l∞ �! R.

Given f 2 F , construct fhi (at (f ))g∞t=1 2 l∞ (since G is bounded).

Find

(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

De�ne

H∞i (f ) = H

(1T

T

∑t=1hi (at (f ))

)∞

T=1

!.

Since G∞ is a game in normal form, we can de�ne F �∞ as the set ofNash equilibria of G∞.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 17 / 64

Page 88: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Choose a Banach limit H : l∞ �! R.

Given f 2 F , construct fhi (at (f ))g∞t=1 2 l∞ (since G is bounded).

Find

(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

De�ne

H∞i (f ) = H

(1T

T

∑t=1hi (at (f ))

)∞

T=1

!.

Since G∞ is a game in normal form, we can de�ne F �∞ as the set ofNash equilibria of G∞.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 17 / 64

Page 89: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Choose a Banach limit H : l∞ �! R.

Given f 2 F , construct fhi (at (f ))g∞t=1 2 l∞ (since G is bounded).

Find

(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

De�ne

H∞i (f ) = H

(1T

T

∑t=1hi (at (f ))

)∞

T=1

!.

Since G∞ is a game in normal form, we can de�ne F �∞ as the set ofNash equilibria of G∞.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 17 / 64

Page 90: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.3.- Payo¤s

Choose a Banach limit H : l∞ �! R.

Given f 2 F , construct fhi (at (f ))g∞t=1 2 l∞ (since G is bounded).

Find

(1T

T

∑t=1hi (at (f ))

)∞

T=1

2 l∞.

De�ne

H∞i (f ) = H

(1T

T

∑t=1hi (at (f ))

)∞

T=1

!.

Since G∞ is a game in normal form, we can de�ne F �∞ as the set ofNash equilibria of G∞.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 17 / 64

Page 91: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 92: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 93: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.

Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 94: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 95: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 96: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 97: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 98: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 and

for all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

Page 99: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.4.- �Folk�Theorems

Family of results characterizing the set of Nash equilibria or SubgamePerfect equilibria of repeated games (GT , Gλ and G∞) and theirrelationships. For example:

Proposition Let G be the Prisoners�Dilemma. Then, for every T � 1and every f 2 F �T , at (f ) = (D,D) for all t � 1.Proof Let f 2 F �T and assume otherwise; namely, there exists1 � t � T , at (f ) 6= (D,D).

Let s = maxf1 � t � T j at (f ) 6= (D,D)g.

Without loss of generality, assume that as1(f ) = C .

De�ne g1 = fg t1g∞t=1 as follows:

for all 1 � t < s (if any), g t1 = f t1 andfor all t � s and all (a1, ..., at�1) 2 At�1, g t1 (a1, ..., at�1) = D.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 18 / 64

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5.4.- �Folk�Theorems

By de�nition of g1, at (g1, f2) = at (f1, f2) for all 1 � t < s (if any).

Hence,

as2(g1, f2) = f s2 (a1(g1, f2), ..., as�1(g1, f2))

= f s2 (a1(f1, f2), ..., as�1(f1, f2))

= as2(f1, f2).

Thus,

for all 1 � t < s (if any), h1(at (g1, f2)) = h1(at (f1, f2)),for all t > s, h1(at (g1, f2)) � 1 = h1(at (f1, f2)),and

h1(as (g1, f2)) = h1(D, a

s2(g1, f2))

> h1(C , as2(g1, f2))

= h1(C , as2(f1, f2))

= h1(as (f1, f2)).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 19 / 64

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5.4.- �Folk�Theorems

By de�nition of g1, at (g1, f2) = at (f1, f2) for all 1 � t < s (if any).Hence,

as2(g1, f2) = f s2 (a1(g1, f2), ..., as�1(g1, f2))

= f s2 (a1(f1, f2), ..., as�1(f1, f2))

= as2(f1, f2).

Thus,

for all 1 � t < s (if any), h1(at (g1, f2)) = h1(at (f1, f2)),for all t > s, h1(at (g1, f2)) � 1 = h1(at (f1, f2)),and

h1(as (g1, f2)) = h1(D, a

s2(g1, f2))

> h1(C , as2(g1, f2))

= h1(C , as2(f1, f2))

= h1(as (f1, f2)).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 19 / 64

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5.4.- �Folk�Theorems

By de�nition of g1, at (g1, f2) = at (f1, f2) for all 1 � t < s (if any).Hence,

as2(g1, f2) = f s2 (a1(g1, f2), ..., as�1(g1, f2))

= f s2 (a1(f1, f2), ..., as�1(f1, f2))

= as2(f1, f2).

Thus,

for all 1 � t < s (if any), h1(at (g1, f2)) = h1(at (f1, f2)),for all t > s, h1(at (g1, f2)) � 1 = h1(at (f1, f2)),and

h1(as (g1, f2)) = h1(D, a

s2(g1, f2))

> h1(C , as2(g1, f2))

= h1(C , as2(f1, f2))

= h1(as (f1, f2)).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 19 / 64

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5.4.- �Folk�Theorems

By de�nition of g1, at (g1, f2) = at (f1, f2) for all 1 � t < s (if any).Hence,

as2(g1, f2) = f s2 (a1(g1, f2), ..., as�1(g1, f2))

= f s2 (a1(f1, f2), ..., as�1(f1, f2))

= as2(f1, f2).

Thus,

for all 1 � t < s (if any), h1(at (g1, f2)) = h1(at (f1, f2)),

for all t > s, h1(at (g1, f2)) � 1 = h1(at (f1, f2)),and

h1(as (g1, f2)) = h1(D, a

s2(g1, f2))

> h1(C , as2(g1, f2))

= h1(C , as2(f1, f2))

= h1(as (f1, f2)).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 19 / 64

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5.4.- �Folk�Theorems

By de�nition of g1, at (g1, f2) = at (f1, f2) for all 1 � t < s (if any).Hence,

as2(g1, f2) = f s2 (a1(g1, f2), ..., as�1(g1, f2))

= f s2 (a1(f1, f2), ..., as�1(f1, f2))

= as2(f1, f2).

Thus,

for all 1 � t < s (if any), h1(at (g1, f2)) = h1(at (f1, f2)),for all t > s, h1(at (g1, f2)) � 1 = h1(at (f1, f2)),

and

h1(as (g1, f2)) = h1(D, a

s2(g1, f2))

> h1(C , as2(g1, f2))

= h1(C , as2(f1, f2))

= h1(as (f1, f2)).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 19 / 64

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5.4.- �Folk�Theorems

By de�nition of g1, at (g1, f2) = at (f1, f2) for all 1 � t < s (if any).Hence,

as2(g1, f2) = f s2 (a1(g1, f2), ..., as�1(g1, f2))

= f s2 (a1(f1, f2), ..., as�1(f1, f2))

= as2(f1, f2).

Thus,

for all 1 � t < s (if any), h1(at (g1, f2)) = h1(at (f1, f2)),for all t > s, h1(at (g1, f2)) � 1 = h1(at (f1, f2)),and

h1(as (g1, f2)) = h1(D, a

s2(g1, f2))

> h1(C , as2(g1, f2))

= h1(C , as2(f1, f2))

= h1(as (f1, f2)).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 19 / 64

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5.4.- �Folk�Theorems

Therefore,

HT1 (g1, f2) =1T

T

∑t=1h1(at (g1, f2))

>1T

T

∑t=1h1(at (f1, f2))

= HT1 (f1, f2),

which contradicts that (f1, f2) 2 F �T . �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 20 / 64

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5.4.- �Folk�Theorems

Proposition Let G be the Prisoners�Dilemma. Then, tit-for-tat is anequilibrium of G∞.

Proof Let gi be the strategy tit-for-tat for player i = 1, 2.

Then, since for all t � 1, at (g1, g2) = (C ,C ), H∞i (g1, g2) = 3.

Let f1 2 F1 be arbitrary (a symmetric argument works for player 2).For every T � 1,

T

∑t=1h1(at (f1, g2)) = 3 �#f1 � t � T j at (f1, g2) = (C ,C )g

+4 �#f1 � t � T j at (f1, g2) = (D,C )g+0 �#f1 � t � T j at (f1, g2) = (C ,D)g+1 �#f1 � t � T j at (f1, g2) = (D,D)g.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 21 / 64

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5.4.- �Folk�Theorems

Proposition Let G be the Prisoners�Dilemma. Then, tit-for-tat is anequilibrium of G∞.

Proof Let gi be the strategy tit-for-tat for player i = 1, 2.

Then, since for all t � 1, at (g1, g2) = (C ,C ), H∞i (g1, g2) = 3.

Let f1 2 F1 be arbitrary (a symmetric argument works for player 2).For every T � 1,

T

∑t=1h1(at (f1, g2)) = 3 �#f1 � t � T j at (f1, g2) = (C ,C )g

+4 �#f1 � t � T j at (f1, g2) = (D,C )g+0 �#f1 � t � T j at (f1, g2) = (C ,D)g+1 �#f1 � t � T j at (f1, g2) = (D,D)g.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 21 / 64

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5.4.- �Folk�Theorems

Proposition Let G be the Prisoners�Dilemma. Then, tit-for-tat is anequilibrium of G∞.

Proof Let gi be the strategy tit-for-tat for player i = 1, 2.

Then, since for all t � 1, at (g1, g2) = (C ,C ), H∞i (g1, g2) = 3.

Let f1 2 F1 be arbitrary (a symmetric argument works for player 2).

For every T � 1,

T

∑t=1h1(at (f1, g2)) = 3 �#f1 � t � T j at (f1, g2) = (C ,C )g

+4 �#f1 � t � T j at (f1, g2) = (D,C )g+0 �#f1 � t � T j at (f1, g2) = (C ,D)g+1 �#f1 � t � T j at (f1, g2) = (D,D)g.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 21 / 64

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5.4.- �Folk�Theorems

Proposition Let G be the Prisoners�Dilemma. Then, tit-for-tat is anequilibrium of G∞.

Proof Let gi be the strategy tit-for-tat for player i = 1, 2.

Then, since for all t � 1, at (g1, g2) = (C ,C ), H∞i (g1, g2) = 3.

Let f1 2 F1 be arbitrary (a symmetric argument works for player 2).

For every T � 1,

T

∑t=1h1(at (f1, g2)) = 3 �#f1 � t � T j at (f1, g2) = (C ,C )g

+4 �#f1 � t � T j at (f1, g2) = (D,C )g+0 �#f1 � t � T j at (f1, g2) = (C ,D)g+1 �#f1 � t � T j at (f1, g2) = (D,D)g.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 21 / 64

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5.4.- �Folk�Theorems

Proposition Let G be the Prisoners�Dilemma. Then, tit-for-tat is anequilibrium of G∞.

Proof Let gi be the strategy tit-for-tat for player i = 1, 2.

Then, since for all t � 1, at (g1, g2) = (C ,C ), H∞i (g1, g2) = 3.

Let f1 2 F1 be arbitrary (a symmetric argument works for player 2).

For every T � 1,

T

∑t=1h1(at (f1, g2)) = 3 �#f1 � t � T j at (f1, g2) = (C ,C )g

+4 �#f1 � t � T j at (f1, g2) = (D,C )g+0 �#f1 � t � T j at (f1, g2) = (C ,D)g+1 �#f1 � t � T j at (f1, g2) = (D,D)g.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 21 / 64

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5.4.- �Folk�Theorems

Proposition Let G be the Prisoners�Dilemma. Then, tit-for-tat is anequilibrium of G∞.

Proof Let gi be the strategy tit-for-tat for player i = 1, 2.

Then, since for all t � 1, at (g1, g2) = (C ,C ), H∞i (g1, g2) = 3.

Let f1 2 F1 be arbitrary (a symmetric argument works for player 2).For every T � 1,

T

∑t=1h1(at (f1, g2)) = 3 �#f1 � t � T j at (f1, g2) = (C ,C )g

+4 �#f1 � t � T j at (f1, g2) = (D,C )g+0 �#f1 � t � T j at (f1, g2) = (C ,D)g+1 �#f1 � t � T j at (f1, g2) = (D,D)g.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 21 / 64

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5.4.- �Folk�Theorems

By the de�nition of g2 (tit-for-tat),

#ft � T j at (f1, g2) = (D,C )g+#ft � T j at (f1, g2) = (D,D)g

= #ft � T j at1(f1, g2) = Dg

� #ft � T j at2(f1, g2) = Dg+ 1

= #ft j at (f1, g2) = (C ,D)g+#ft j at (f1, g2) = (D,D)g+ 1.

Hence,

#ft j at (f1, g2) = (D,C )g � #ft j at (f1, g2) = (C ,D)g+ 1. (1)

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 22 / 64

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5.4.- �Folk�Theorems

By the de�nition of g2 (tit-for-tat),

#ft � T j at (f1, g2) = (D,C )g+#ft � T j at (f1, g2) = (D,D)g

= #ft � T j at1(f1, g2) = Dg

� #ft � T j at2(f1, g2) = Dg+ 1

= #ft j at (f1, g2) = (C ,D)g+#ft j at (f1, g2) = (D,D)g+ 1.

Hence,

#ft j at (f1, g2) = (D,C )g � #ft j at (f1, g2) = (C ,D)g+ 1. (1)

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 22 / 64

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5.4.- �Folk�Theorems

T

∑t=1h1(at (f1, g2)) = 3 �#ft � T j at (f1, g2) = (C ,C )g

+3 �#ft � T j at (f1, g2) = (D,C )g+3 �#ft � T j at (f1, g2) = (C ,D)g+3 �#ft � T j at (f1, g2) = (D,D)g

9>>>>>=>>>>>;= 3T

+1 �#ft � T j at (f1, g2) = (D,C )g�1 �#ft � T j at (f1, g2) = (C ,D)g

�� 1 by (1)

�2 �#ft � T j at (f1, g2) = (C ,D)g�2 �#ft � T j at (f1, g2) = (D,D)g

�� 0.

Hence,T

∑t=1h1(at (f1, g2)) � 3T + 1.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 23 / 64

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5.4.- �Folk�Theorems

T

∑t=1h1(at (f1, g2)) = 3 �#ft � T j at (f1, g2) = (C ,C )g

+3 �#ft � T j at (f1, g2) = (D,C )g+3 �#ft � T j at (f1, g2) = (C ,D)g+3 �#ft � T j at (f1, g2) = (D,D)g

9>>>>>=>>>>>;= 3T

+1 �#ft � T j at (f1, g2) = (D,C )g�1 �#ft � T j at (f1, g2) = (C ,D)g

�� 1 by (1)

�2 �#ft � T j at (f1, g2) = (C ,D)g�2 �#ft � T j at (f1, g2) = (D,D)g

�� 0.

Hence,T

∑t=1h1(at (f1, g2)) � 3T + 1.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 23 / 64

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5.4.- �Folk�Theorems

Thus,

H∞1 (f1, g2) = H

0@( 1T

T

∑t=1h1(at (f1, g2))

)Tt=1

1A� lim sup

n!∞

1T

T

∑t=1h1(at (f1, g2))

� lim supn!∞

1T(3T + 1) = 3

= H∞1 (g1, g2).

Therefore, for all f1 2 F1, H∞1 (f1, g2) � H∞

1 (g1, g2).

Hence, (g1, g2) 2 F �∞. �

Note that this is independent of the particular Banach limit H chosento evaluate sequences of averages.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 24 / 64

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5.4.- �Folk�Theorems

Thus,

H∞1 (f1, g2) = H

0@( 1T

T

∑t=1h1(at (f1, g2))

)Tt=1

1A� lim sup

n!∞

1T

T

∑t=1h1(at (f1, g2))

� lim supn!∞

1T(3T + 1) = 3

= H∞1 (g1, g2).

Therefore, for all f1 2 F1, H∞1 (f1, g2) � H∞

1 (g1, g2).

Hence, (g1, g2) 2 F �∞. �

Note that this is independent of the particular Banach limit H chosento evaluate sequences of averages.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 24 / 64

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5.4.- �Folk�Theorems

Thus,

H∞1 (f1, g2) = H

0@( 1T

T

∑t=1h1(at (f1, g2))

)Tt=1

1A� lim sup

n!∞

1T

T

∑t=1h1(at (f1, g2))

� lim supn!∞

1T(3T + 1) = 3

= H∞1 (g1, g2).

Therefore, for all f1 2 F1, H∞1 (f1, g2) � H∞

1 (g1, g2).

Hence, (g1, g2) 2 F �∞. �

Note that this is independent of the particular Banach limit H chosento evaluate sequences of averages.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 24 / 64

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5.4.- �Folk�Theorems

Thus,

H∞1 (f1, g2) = H

0@( 1T

T

∑t=1h1(at (f1, g2))

)Tt=1

1A� lim sup

n!∞

1T

T

∑t=1h1(at (f1, g2))

� lim supn!∞

1T(3T + 1) = 3

= H∞1 (g1, g2).

Therefore, for all f1 2 F1, H∞1 (f1, g2) � H∞

1 (g1, g2).

Hence, (g1, g2) 2 F �∞. �

Note that this is independent of the particular Banach limit H chosento evaluate sequences of averages.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 24 / 64

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5.4.- �Folk�Theorems

Objective: To describe, for every bounded game in normal form G , the setof equilibrium payo¤s of Gα for α = T ,λ,∞.

In terms of the payo¤s (h(a))a2A of G .

In general, for α = T ,λ,∞, F �α is extremely large.

Relationships:

F �λ �!λ!1F �∞ and F �T �!λ!1

F �∞?

These collection of results are some times calledAumann-Shapley-Rubinstein Theorems.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 25 / 64

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5.4.- �Folk�Theorems

Objective: To describe, for every bounded game in normal form G , the setof equilibrium payo¤s of Gα for α = T ,λ,∞.

In terms of the payo¤s (h(a))a2A of G .

In general, for α = T ,λ,∞, F �α is extremely large.

Relationships:

F �λ �!λ!1F �∞ and F �T �!λ!1

F �∞?

These collection of results are some times calledAumann-Shapley-Rubinstein Theorems.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 25 / 64

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5.4.- �Folk�Theorems

Objective: To describe, for every bounded game in normal form G , the setof equilibrium payo¤s of Gα for α = T ,λ,∞.

In terms of the payo¤s (h(a))a2A of G .

In general, for α = T ,λ,∞, F �α is extremely large.

Relationships:

F �λ �!λ!1F �∞ and F �T �!λ!1

F �∞?

These collection of results are some times calledAumann-Shapley-Rubinstein Theorems.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 25 / 64

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5.4.- �Folk�Theorems

Objective: To describe, for every bounded game in normal form G , the setof equilibrium payo¤s of Gα for α = T ,λ,∞.

In terms of the payo¤s (h(a))a2A of G .

In general, for α = T ,λ,∞, F �α is extremely large.

Relationships:

F �λ �!λ!1F �∞ and F �T �!λ!1

F �∞?

These collection of results are some times calledAumann-Shapley-Rubinstein Theorems.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 25 / 64

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5.4.- �Folk�Theorems

Objective: To describe, for every bounded game in normal form G , the setof equilibrium payo¤s of Gα for α = T ,λ,∞.

In terms of the payo¤s (h(a))a2A of G .

In general, for α = T ,λ,∞, F �α is extremely large.

Relationships:

F �λ �!λ!1F �∞ and F �T �!λ!1

F �∞?

These collection of results are some times calledAumann-Shapley-Rubinstein Theorems.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 25 / 64

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5.4.- �Folk�Theorems

Objective: To describe, for every bounded game in normal form G , the setof equilibrium payo¤s of Gα for α = T ,λ,∞.

In terms of the payo¤s (h(a))a2A of G .

In general, for α = T ,λ,∞, F �α is extremely large.

Relationships:

F �λ �!λ!1F �∞ and F �T �!λ!1

F �∞?

These collection of results are some times calledAumann-Shapley-Rubinstein Theorems.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 25 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form.

De�nition The payo¤ xi 2 R is individually rational for player i 2 Iif

xi � infa�i2A�i

supai2Ai

hi (ai , a�i ) � Ri .

Remark: If G is bounded, then Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ).

Interpretation: Player i can guarantee Ri by himself.

Punishment idea: the other players choose their actions and then ichooses his best action.

Warning: with mixed strategies, this minimax may be smaller; i.e.,there are games for which

infσ�i2Σ�i

supσi2Σi

Hi (σi , σ�i ) < infa�i2A�i

supai2Ai

hi (ai , a�i ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 26 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form.

De�nition The payo¤ xi 2 R is individually rational for player i 2 Iif

xi � infa�i2A�i

supai2Ai

hi (ai , a�i ) � Ri .

Remark: If G is bounded, then Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ).

Interpretation: Player i can guarantee Ri by himself.

Punishment idea: the other players choose their actions and then ichooses his best action.

Warning: with mixed strategies, this minimax may be smaller; i.e.,there are games for which

infσ�i2Σ�i

supσi2Σi

Hi (σi , σ�i ) < infa�i2A�i

supai2Ai

hi (ai , a�i ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 26 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form.

De�nition The payo¤ xi 2 R is individually rational for player i 2 Iif

xi � infa�i2A�i

supai2Ai

hi (ai , a�i ) � Ri .

Remark: If G is bounded, then Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ).

Interpretation: Player i can guarantee Ri by himself.

Punishment idea: the other players choose their actions and then ichooses his best action.

Warning: with mixed strategies, this minimax may be smaller; i.e.,there are games for which

infσ�i2Σ�i

supσi2Σi

Hi (σi , σ�i ) < infa�i2A�i

supai2Ai

hi (ai , a�i ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 26 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form.

De�nition The payo¤ xi 2 R is individually rational for player i 2 Iif

xi � infa�i2A�i

supai2Ai

hi (ai , a�i ) � Ri .

Remark: If G is bounded, then Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ).

Interpretation: Player i can guarantee Ri by himself.

Punishment idea: the other players choose their actions and then ichooses his best action.

Warning: with mixed strategies, this minimax may be smaller; i.e.,there are games for which

infσ�i2Σ�i

supσi2Σi

Hi (σi , σ�i ) < infa�i2A�i

supai2Ai

hi (ai , a�i ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 26 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form.

De�nition The payo¤ xi 2 R is individually rational for player i 2 Iif

xi � infa�i2A�i

supai2Ai

hi (ai , a�i ) � Ri .

Remark: If G is bounded, then Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ).

Interpretation: Player i can guarantee Ri by himself.

Punishment idea: the other players choose their actions and then ichooses his best action.

Warning: with mixed strategies, this minimax may be smaller; i.e.,there are games for which

infσ�i2Σ�i

supσi2Σi

Hi (σi , σ�i ) < infa�i2A�i

supai2Ai

hi (ai , a�i ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 26 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a bounded game in normal form.

De�nition The payo¤ xi 2 R is individually rational for player i 2 Iif

xi � infa�i2A�i

supai2Ai

hi (ai , a�i ) � Ri .

Remark: If G is bounded, then Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ).

Interpretation: Player i can guarantee Ri by himself.

Punishment idea: the other players choose their actions and then ichooses his best action.

Warning: with mixed strategies, this minimax may be smaller; i.e.,there are games for which

infσ�i2Σ�i

supσi2Σi

Hi (σi , σ�i ) < infa�i2A�i

supai2Ai

hi (ai , a�i ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 26 / 64

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5.4.- �Folk�Theorems

Examples:

Prisoners�Dilemma: Ri = minfmaxf3, 4g,maxf0, 1gg = 1.Battle of Sexes: Ri = minfmaxf3, 0g,maxf0, 1gg = 1.Coordination Game: Ri = minfmaxf1, 0g,maxf0, 2gg = 1.Matching Pennies: Ri = minfmaxf1,�1g,maxf1,�1gg = 1.

Notation: C (G ) = cl�co�h(a) 2 R#I j a 2 A

�.

If G is �nite, C (G ) = conh(a) 2 R#I j a 2 A

o.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 27 / 64

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5.4.- �Folk�Theorems

Examples:

Prisoners�Dilemma: Ri = minfmaxf3, 4g,maxf0, 1gg = 1.

Battle of Sexes: Ri = minfmaxf3, 0g,maxf0, 1gg = 1.Coordination Game: Ri = minfmaxf1, 0g,maxf0, 2gg = 1.Matching Pennies: Ri = minfmaxf1,�1g,maxf1,�1gg = 1.

Notation: C (G ) = cl�co�h(a) 2 R#I j a 2 A

�.

If G is �nite, C (G ) = conh(a) 2 R#I j a 2 A

o.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 27 / 64

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5.4.- �Folk�Theorems

Examples:

Prisoners�Dilemma: Ri = minfmaxf3, 4g,maxf0, 1gg = 1.Battle of Sexes: Ri = minfmaxf3, 0g,maxf0, 1gg = 1.

Coordination Game: Ri = minfmaxf1, 0g,maxf0, 2gg = 1.Matching Pennies: Ri = minfmaxf1,�1g,maxf1,�1gg = 1.

Notation: C (G ) = cl�co�h(a) 2 R#I j a 2 A

�.

If G is �nite, C (G ) = conh(a) 2 R#I j a 2 A

o.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 27 / 64

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5.4.- �Folk�Theorems

Examples:

Prisoners�Dilemma: Ri = minfmaxf3, 4g,maxf0, 1gg = 1.Battle of Sexes: Ri = minfmaxf3, 0g,maxf0, 1gg = 1.Coordination Game: Ri = minfmaxf1, 0g,maxf0, 2gg = 1.

Matching Pennies: Ri = minfmaxf1,�1g,maxf1,�1gg = 1.

Notation: C (G ) = cl�co�h(a) 2 R#I j a 2 A

�.

If G is �nite, C (G ) = conh(a) 2 R#I j a 2 A

o.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 27 / 64

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5.4.- �Folk�Theorems

Examples:

Prisoners�Dilemma: Ri = minfmaxf3, 4g,maxf0, 1gg = 1.Battle of Sexes: Ri = minfmaxf3, 0g,maxf0, 1gg = 1.Coordination Game: Ri = minfmaxf1, 0g,maxf0, 2gg = 1.Matching Pennies: Ri = minfmaxf1,�1g,maxf1,�1gg = 1.

Notation: C (G ) = cl�co�h(a) 2 R#I j a 2 A

�.

If G is �nite, C (G ) = conh(a) 2 R#I j a 2 A

o.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 27 / 64

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5.4.- �Folk�Theorems

Examples:

Prisoners�Dilemma: Ri = minfmaxf3, 4g,maxf0, 1gg = 1.Battle of Sexes: Ri = minfmaxf3, 0g,maxf0, 1gg = 1.Coordination Game: Ri = minfmaxf1, 0g,maxf0, 2gg = 1.Matching Pennies: Ri = minfmaxf1,�1g,maxf1,�1gg = 1.

Notation: C (G ) = cl�co�h(a) 2 R#I j a 2 A

�.

If G is �nite, C (G ) = conh(a) 2 R#I j a 2 A

o.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 27 / 64

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5.4.- �Folk�Theorems

Examples:

Prisoners�Dilemma: Ri = minfmaxf3, 4g,maxf0, 1gg = 1.Battle of Sexes: Ri = minfmaxf3, 0g,maxf0, 1gg = 1.Coordination Game: Ri = minfmaxf1, 0g,maxf0, 2gg = 1.Matching Pennies: Ri = minfmaxf1,�1g,maxf1,�1gg = 1.

Notation: C (G ) = cl�co�h(a) 2 R#I j a 2 A

�.

If G is �nite, C (G ) = conh(a) 2 R#I j a 2 A

o.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 27 / 64

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5.4.- �Folk�Theorems

-

6

h2

h10

1

2

3

4

0 1 2 3 4

rr

rr

h(C ,C )

h(D,D)

h(C ,D)

h(D,C )

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 28 / 64

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5.4.- �Folk�Theorems

-

6

h2

h10

1

2

3

4

0 1 2 3 4

rr

rr

h(C ,C )

h(D,D)

h(C ,D)

h(D,C )

C (G )

PPPPPPPPPBBBBBBBBB

BBBBBBBBBPPPPPPPPP

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 29 / 64

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5.4.- �Folk�Theorems

-

6

hM

hW0

1

2

3

0 1 2 3

r

rr

h(F ,F )

h(B,B)

h(F ,B) = h(B,F )

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 30 / 64

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5.4.- �Folk�Theorems

-

6

hM

hW0

1

2

3

0 1 2 3

r

rr

h(F ,F )

h(B,B)

h(F ,B) = h(B,F )

@@@@@@

���������

���������

C (G )

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 31 / 64

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5.4.- �Folk�Theorems

-

6

h2

h10

1

2

0 1 2

rr

r

h(B,R)

h(T , L)

h(T ,R) = h(B, L)

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 32 / 64

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5.4.- �Folk�Theorems

-

6

h2

h10

1

2

0 1 2

rr

r

h(B,R)

h(T , L)

h(T ,R) = h(B, L)

������

C (G )

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 33 / 64

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5.4.- �Folk�Theorems

-

6

h2

h1

-1

1

-1 1

r

rh(H,H) = h(T ,T )

h(T ,H) = h(H,T )

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 34 / 64

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5.4.- �Folk�Theorems

-

6

h2

h1

-1

1

-1 1

r

rh(H,H) = h(T ,T )

h(T ,H) = h(H,T )

@@@@@@

C (G )

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 35 / 64

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5.4.- �Folk�TheoremsIn�nitely Repeated

TheoremLet G be a bounded game in normal form. Then,n

H∞(f ) 2 R#I j f 2 F �∞o= fx 2 C (G ) j xi � Ri for all i 2 Ig .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 36 / 64

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5.4.- �Folk�TheoremsIn�nitely Repeated

TheoremLet G be a bounded game in normal form. Then,n

H∞(f ) 2 R#I j f 2 F �∞o= fx 2 C (G ) j xi � Ri for all i 2 Ig .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 36 / 64

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5.4.- �Folk�Theorems

-

6

h2

h10

1

2

3

4

0 1 2 3 4

rr

rr

h(C ,C )

h(D,D)

h(C ,D)

h(D,C )

C (G )

PPPPPPPPPBBBBBBBBB

BBBBBBBBBPPPPPPPPP

R2 = 1

R1 = 1

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 37 / 64

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5.4.- �Folk�Theorems

-

6

hW

hM0

1

2

3

0 1 2 3

r

rr

h(F ,F )

h(B,B)

h(F ,B) = h(B,F )

@@@@@@

���������

���������

C (G )

R2 = 1

R1 = 1

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 38 / 64

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5.4.- �Folk�Theorems

-

6

h2

h10

1

2

0 1 2

rr

r

h(B,R)

h(T , L)

h(T ,R) = h(B, L)

������

C (G ) R2 = 1

R1 = 1

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 39 / 64

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5.4.- �Folk�Theorems

-

6

h2

h1

-1

1

-1 1

r

rh(H,H) = h(T ,T )

h(T ,H) = h(H,T )

@@@@@@

C (G )

R2 = 1

R1 = 1

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 40 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form.

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .Proposition 2 Let fatg∞

t=1 be such that at 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞

t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 41 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form.

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .

Proposition 2 Let fatg∞t=1 be such that a

t 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞

t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 41 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form.

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .Proposition 2 Let fatg∞

t=1 be such that at 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.

Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 41 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form.

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .Proposition 2 Let fatg∞

t=1 be such that at 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞

t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 41 / 64

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5.4.- �Folk�Theorems

Let G = (I , (Ai )i2I , (hi )i2I ) be a �nite game in normal form.

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .Proposition 2 Let fatg∞

t=1 be such that at 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞

t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 41 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .

Let f 2 F and i 2 I be arbitrary. De�ne recursively gi 2 Fi as follows:Given a1(f ) let b1i 2 Ai be s.t. hi (b1i , a1(f )�i ) � Ri ; it exists since

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) � maxai2Ai

hi (ai , a1(f )�i ) = hi (b1i , a1(f )�i ).

Then, set g1i = b1i .

Assume gi has been de�ned up to t. Let bt+1i 2 Ai be s.t.hi (b1+1i , f t+1(a1(gi , f�i ), ..., at (gi , f�i ))�i ) � Ri ; as before, it alsoexists. Then, for all (a1, ..., at ) 2 At , set

g t+1i (a1, ..., at ) =�bt+1i if 81 � s � t, as = as (gi , f�i )f t+1i (a1, ..., at ) otherwise.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 42 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .

Let f 2 F and i 2 I be arbitrary. De�ne recursively gi 2 Fi as follows:Given a1(f ) let b1i 2 Ai be s.t. hi (b1i , a1(f )�i ) � Ri ; it exists since

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) � maxai2Ai

hi (ai , a1(f )�i ) = hi (b1i , a1(f )�i ).

Then, set g1i = b1i .

Assume gi has been de�ned up to t. Let bt+1i 2 Ai be s.t.hi (b1+1i , f t+1(a1(gi , f�i ), ..., at (gi , f�i ))�i ) � Ri ; as before, it alsoexists. Then, for all (a1, ..., at ) 2 At , set

g t+1i (a1, ..., at ) =�bt+1i if 81 � s � t, as = as (gi , f�i )f t+1i (a1, ..., at ) otherwise.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 42 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .

Let f 2 F and i 2 I be arbitrary. De�ne recursively gi 2 Fi as follows:

Given a1(f ) let b1i 2 Ai be s.t. hi (b1i , a1(f )�i ) � Ri ; it exists since

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) � maxai2Ai

hi (ai , a1(f )�i ) = hi (b1i , a1(f )�i ).

Then, set g1i = b1i .

Assume gi has been de�ned up to t. Let bt+1i 2 Ai be s.t.hi (b1+1i , f t+1(a1(gi , f�i ), ..., at (gi , f�i ))�i ) � Ri ; as before, it alsoexists. Then, for all (a1, ..., at ) 2 At , set

g t+1i (a1, ..., at ) =�bt+1i if 81 � s � t, as = as (gi , f�i )f t+1i (a1, ..., at ) otherwise.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 42 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .

Let f 2 F and i 2 I be arbitrary. De�ne recursively gi 2 Fi as follows:Given a1(f ) let b1i 2 Ai be s.t. hi (b1i , a1(f )�i ) � Ri ; it exists since

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) � maxai2Ai

hi (ai , a1(f )�i ) = hi (b1i , a1(f )�i ).

Then, set g1i = b1i .

Assume gi has been de�ned up to t. Let bt+1i 2 Ai be s.t.hi (b1+1i , f t+1(a1(gi , f�i ), ..., at (gi , f�i ))�i ) � Ri ; as before, it alsoexists. Then, for all (a1, ..., at ) 2 At , set

g t+1i (a1, ..., at ) =�bt+1i if 81 � s � t, as = as (gi , f�i )f t+1i (a1, ..., at ) otherwise.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 42 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

Proposition 1 If f is an equilibrium of Gα for α = T ,λ,∞ then,Hαi (f ) � Ri for all i 2 I .

Let f 2 F and i 2 I be arbitrary. De�ne recursively gi 2 Fi as follows:Given a1(f ) let b1i 2 Ai be s.t. hi (b1i , a1(f )�i ) � Ri ; it exists since

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) � maxai2Ai

hi (ai , a1(f )�i ) = hi (b1i , a1(f )�i ).

Then, set g1i = b1i .

Assume gi has been de�ned up to t. Let bt+1i 2 Ai be s.t.hi (b1+1i , f t+1(a1(gi , f�i ), ..., at (gi , f�i ))�i ) � Ri ; as before, it alsoexists. Then, for all (a1, ..., at ) 2 At , set

g t+1i (a1, ..., at ) =�bt+1i if 81 � s � t, as = as (gi , f�i )f t+1i (a1, ..., at ) otherwise.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 42 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

It is possible to show that, by the de�nition of gi ,

hi (at (gi , f�i )) � Ri .

Hence, for all α = T ,λ,∞, Hαi (gi , f�i ) � Ri .

Thus, if for α = T ,λ,∞, f 2 G �α then, it must be the case that

Hαi (f ) � Ri .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 43 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

It is possible to show that, by the de�nition of gi ,

hi (at (gi , f�i )) � Ri .

Hence, for all α = T ,λ,∞, Hαi (gi , f�i ) � Ri .

Thus, if for α = T ,λ,∞, f 2 G �α then, it must be the case that

Hαi (f ) � Ri .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 43 / 64

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5.4.- �Folk�TheoremsProposition 1: Intuition

It is possible to show that, by the de�nition of gi ,

hi (at (gi , f�i )) � Ri .

Hence, for all α = T ,λ,∞, Hαi (gi , f�i ) � Ri .

Thus, if for α = T ,λ,∞, f 2 G �α then, it must be the case that

Hαi (f ) � Ri .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 43 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Proposition 2 Let fatg∞t=1 be such that a

t 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.

For every i 2 I , there exists a(i) 2 A such that hi (bi , a(i)�i ) � Ri forall bi 2 Ai . Observe that

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) = maxai2Ai

hi (ai , a(i)�i ) � hi (bi , a(i)�i )

for all bi 2 Ai .For every j 2 I , set f 1j = a1j .Take any function γ : 2I nf?g �! I with the property that for allJ 2 2I nf?g, γ(J) 2 J.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 44 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Proposition 2 Let fatg∞t=1 be such that a

t 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.

For every i 2 I , there exists a(i) 2 A such that hi (bi , a(i)�i ) � Ri forall bi 2 Ai . Observe that

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) = maxai2Ai

hi (ai , a(i)�i ) � hi (bi , a(i)�i )

for all bi 2 Ai .

For every j 2 I , set f 1j = a1j .Take any function γ : 2I nf?g �! I with the property that for allJ 2 2I nf?g, γ(J) 2 J.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 44 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Proposition 2 Let fatg∞t=1 be such that a

t 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.

For every i 2 I , there exists a(i) 2 A such that hi (bi , a(i)�i ) � Ri forall bi 2 Ai . Observe that

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) = maxai2Ai

hi (ai , a(i)�i ) � hi (bi , a(i)�i )

for all bi 2 Ai .For every j 2 I , set f 1j = a1j .

Take any function γ : 2I nf?g �! I with the property that for allJ 2 2I nf?g, γ(J) 2 J.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 44 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Proposition 2 Let fatg∞t=1 be such that a

t 2 A for all t � 1 and

lim infn!∞

1T

T

∑t=1hi (at ) � Ri for all i 2 I then, there exists an f 2 F such that

(1) f is an equilibrium of G∞ and (2) at (f ) = at for all t � 1.

For every i 2 I , there exists a(i) 2 A such that hi (bi , a(i)�i ) � Ri forall bi 2 Ai . Observe that

Ri = mina�i2A�i

maxai2Ai

hi (ai , a�i ) = maxai2Ai

hi (ai , a(i)�i ) � hi (bi , a(i)�i )

for all bi 2 Ai .For every j 2 I , set f 1j = a1j .Take any function γ : 2I nf?g �! I with the property that for allJ 2 2I nf?g, γ(J) 2 J.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 44 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Let (b1, ..., bt ) 2 At be arbitrary. Let s = minf1 � r � t j br 6= arg,J = fk 2 I j bsk 6= askg and i = γ(J).

De�ne

f t+1j (b1, ..., bt ) =�at+1j if 81 � r � t, br = ara(i)j otherwise.

It is easy to show that for all t � 1, at (f ) = at (namely, (ii) isproven).

For any gi 2 Fi either at (f ) = at (gi , f�i ) for all t � 1, in which caseH∞i (f ) = Hi (gi , f�i ) or else there existss = minft � 1 j at (gi , f�i ) 6= at (f )g. Then, J = fig andγ(fig) = i . Thus,

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 45 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Let (b1, ..., bt ) 2 At be arbitrary. Let s = minf1 � r � t j br 6= arg,J = fk 2 I j bsk 6= askg and i = γ(J).

De�ne

f t+1j (b1, ..., bt ) =�at+1j if 81 � r � t, br = ara(i)j otherwise.

It is easy to show that for all t � 1, at (f ) = at (namely, (ii) isproven).

For any gi 2 Fi either at (f ) = at (gi , f�i ) for all t � 1, in which caseH∞i (f ) = Hi (gi , f�i ) or else there existss = minft � 1 j at (gi , f�i ) 6= at (f )g. Then, J = fig andγ(fig) = i . Thus,

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 45 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Let (b1, ..., bt ) 2 At be arbitrary. Let s = minf1 � r � t j br 6= arg,J = fk 2 I j bsk 6= askg and i = γ(J).

De�ne

f t+1j (b1, ..., bt ) =�at+1j if 81 � r � t, br = ara(i)j otherwise.

It is easy to show that for all t � 1, at (f ) = at (namely, (ii) isproven).

For any gi 2 Fi either at (f ) = at (gi , f�i ) for all t � 1, in which caseH∞i (f ) = Hi (gi , f�i ) or else there existss = minft � 1 j at (gi , f�i ) 6= at (f )g. Then, J = fig andγ(fig) = i . Thus,

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 45 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

Let (b1, ..., bt ) 2 At be arbitrary. Let s = minf1 � r � t j br 6= arg,J = fk 2 I j bsk 6= askg and i = γ(J).

De�ne

f t+1j (b1, ..., bt ) =�at+1j if 81 � r � t, br = ara(i)j otherwise.

It is easy to show that for all t � 1, at (f ) = at (namely, (ii) isproven).

For any gi 2 Fi either at (f ) = at (gi , f�i ) for all t � 1, in which caseH∞i (f ) = Hi (gi , f�i ) or else there existss = minft � 1 j at (gi , f�i ) 6= at (f )g. Then, J = fig andγ(fig) = i . Thus,

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 45 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

H∞i (gi , f�i ) = H

��1T

T∑t=1hi (at (gi , f�i )

�∞

T=1

�� lim supT!∞

1T ∑T

t=1 hi (at (gi , f�i )

� lim supT!∞1T [s maxfhi (a) j a 2 Ag+ (T � s)Ri ]

� lim sup 1T s maxfhi (a) j a 2 Ag+ lim sup

1T (T � s)Ri

� lim supT!∞1T TRi

= Ri� lim inf

n!∞1T ∑T

t=1 hi (at ) by hypothesis

� lim infn!∞

1T ∑T

t=1 hi (at (f )) by (ii)

� H∞i (f ).

But since gi was arbitrary, f 2 F �∞.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 46 / 64

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5.4.- �Folk�TheoremsProposition 2: Intuition

H∞i (gi , f�i ) = H

��1T

T∑t=1hi (at (gi , f�i )

�∞

T=1

�� lim supT!∞

1T ∑T

t=1 hi (at (gi , f�i )

� lim supT!∞1T [s maxfhi (a) j a 2 Ag+ (T � s)Ri ]

� lim sup 1T s maxfhi (a) j a 2 Ag+ lim sup

1T (T � s)Ri

� lim supT!∞1T TRi

= Ri� lim inf

n!∞1T ∑T

t=1 hi (at ) by hypothesis

� lim infn!∞

1T ∑T

t=1 hi (at (f )) by (ii)

� H∞i (f ).

But since gi was arbitrary, f 2 F �∞.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 46 / 64

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5.4.- �Folk�TheoremsProposition 3: Intuition

Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 3 follows (after some work to deal with convex combinationswith non-rational coe¢ cients) from the following result which in turnfollows from a more general result (Caratheodory Theorem).

Result Let X = cofx1, ..., xK g � Rn. For every x 2 X there exist

y1, ..., yn+1 2 fx1, ..., xK g and p1, ..., pn+1 � 0 such thatn+1

∑j=1

pj = 1 with

the property that x =n+1

∑j=1

pjy j .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 47 / 64

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5.4.- �Folk�TheoremsProposition 3: Intuition

Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 3 follows (after some work to deal with convex combinationswith non-rational coe¢ cients) from the following result which in turnfollows from a more general result (Caratheodory Theorem).

Result Let X = cofx1, ..., xK g � Rn. For every x 2 X there exist

y1, ..., yn+1 2 fx1, ..., xK g and p1, ..., pn+1 � 0 such thatn+1

∑j=1

pj = 1 with

the property that x =n+1

∑j=1

pjy j .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 47 / 64

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5.4.- �Folk�TheoremsProposition 3: Intuition

Proposition 3 For every x 2 C (G ) there exists a sequence fatg∞t=1

such that at 2 A for all t � 1 and for all i 2 I , limT!∞

1T

T

∑t=1hi (at ) exists

and it is equal to xi .

Proposition 3 follows (after some work to deal with convex combinationswith non-rational coe¢ cients) from the following result which in turnfollows from a more general result (Caratheodory Theorem).

Result Let X = cofx1, ..., xK g � Rn. For every x 2 X there exist

y1, ..., yn+1 2 fx1, ..., xK g and p1, ..., pn+1 � 0 such thatn+1

∑j=1

pj = 1 with

the property that x =n+1

∑j=1

pjy j .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 47 / 64

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5.4.- �Folk�TheoremsProposition 4: Intuition

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

For α = T ,∞ the statement obviously holds.

For α = λ, observe that for every t � 1, 0 � (1� λ)λt�1 � 1 and

(1� λ)∞

∑t=1

λt�1 = (1� λ) 11�λ = 1. Thus, each (1� λ)λt�1 can be

seen as the coe¢ cient of an (in�nite) convex combination: Thus,

Hλ(f ) = (1� λ)∞

∑t=1

λt�1h(at (f )) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 48 / 64

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5.4.- �Folk�TheoremsProposition 4: Intuition

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

For α = T ,∞ the statement obviously holds.

For α = λ, observe that for every t � 1, 0 � (1� λ)λt�1 � 1 and

(1� λ)∞

∑t=1

λt�1 = (1� λ) 11�λ = 1. Thus, each (1� λ)λt�1 can be

seen as the coe¢ cient of an (in�nite) convex combination: Thus,

Hλ(f ) = (1� λ)∞

∑t=1

λt�1h(at (f )) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 48 / 64

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5.4.- �Folk�TheoremsProposition 4: Intuition

Proposition 4 For every f 2 F and every α = T ,λ,∞, Hα(f ) 2 C (G ).

For α = T ,∞ the statement obviously holds.

For α = λ, observe that for every t � 1, 0 � (1� λ)λt�1 � 1 and

(1� λ)∞

∑t=1

λt�1 = (1� λ) 11�λ = 1. Thus, each (1� λ)λt�1 can be

seen as the coe¢ cient of an (in�nite) convex combination: Thus,

Hλ(f ) = (1� λ)∞

∑t=1

λt�1h(at (f )) 2 C (G ).

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 48 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .

By Proposition 3, there exists a sequence fatg∞t=1 such that a

t 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .

By Proposition 3, there exists a sequence fatg∞t=1 such that a

t 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .

By Proposition 3, there exists a sequence fatg∞t=1 such that a

t 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).

Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .

By Proposition 3, there exists a sequence fatg∞t=1 such that a

t 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .

By Proposition 3, there exists a sequence fatg∞t=1 such that a

t 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .

By Proposition 3, there exists a sequence fatg∞t=1 such that a

t 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .By Proposition 3, there exists a sequence fatg∞

t=1 such that at 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .By Proposition 3, there exists a sequence fatg∞

t=1 such that at 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.

Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .By Proposition 3, there exists a sequence fatg∞

t=1 such that at 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�Theorems

Proof of the Theorem

�) Let f be an equilibrium of Gα.

By Proposition 1, Hαi (f ) � Ri for all i 2 I .

By Proposition 4, Hα(f ) 2 C (G ).Hence, Hα(f ) 2 fx 2 C (G ) j xi � Ri for all i 2 Ig.

�) Let x 2 C (G ) and assume xi � Ri for all i 2 I .By Proposition 3, there exists a sequence fatg∞

t=1 such that at 2 A for

all t � 1 and for all i 2 I , limT!∞

1T

T∑t=1

hi (at ) = xi .

By Proposition 2, there exists f 2 F such that (1) f is an equilibriumof G∞ and (2) at (f ) = at for all t � 1.Hence, for all i 2 I ,

H∞i (f ) = H

�1T

T∑t=1

hi (at (f ))�∞

T=1

!= limT!∞

1T

T∑t=1

hi (at ) = xi .

Thus, x 2nH∞(f ) 2 R#I j f is an equilibrium of G∞

o. �

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 49 / 64

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5.4.- �Folk�TheoremsDiscounted Repeated

TheoremFor every x 2 C (G ) such that xi > Ri for all i 2 I , there exists λ2 (0, 1)such that for all λ 2 (λ, 1) there exists f 2 F �λ with the property thatHλ(f ) = x .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 50 / 64

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5.4.- �Folk�TheoremsFinitely Repeated

TheoremBenoît and Krishna (1987) Assume that for every i 2 I there existsa�(i) 2 A� such that hi (a�(i)) > Ri . Then, for all x 2 C (G ) such thatxi > Ri for all i 2 I and for every ε > 0 there exists T 2 N such that forall T > T there exists f 2 F �T such that

HT (f )� x < ε.

Benoît, J.P. and V. Krisnha. �Nash Equilibria of Finitely RepeatedGames,� International Journal of Game Theory 16, 1987.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 51 / 64

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5.4.- �Folk�TheoremsFinitely Repeated

TheoremBenoît and Krishna (1987) Assume that for every i 2 I there existsa�(i) 2 A� such that hi (a�(i)) > Ri . Then, for all x 2 C (G ) such thatxi > Ri for all i 2 I and for every ε > 0 there exists T 2 N such that forall T > T there exists f 2 F �T such that

HT (f )� x < ε.

Benoît, J.P. and V. Krisnha. �Nash Equilibria of Finitely RepeatedGames,� International Journal of Game Theory 16, 1987.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 51 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Terminal phase: for Q 2 N,

a�(1), ..., a�(n)| {z }n periods

, ..., a�(1), ..., a�(n)| {z }n periods| {z }

Q -times=Qn periods

.

Observe that for all i 2 N, hi (a�(i)) > Ri and hi (a�(j)) � Ri for allj 2 N.Average payo¤s in the terminal phase: for all i 2 N,

yi =1QnQ

n∑j=1hi (a�(j)) =

1n

n∑j=1hi (a�(j)) > Ri .

Given x 2 C (G ) such that xi > Ri for all i 2 N, choose Q with theproperty that for all i 2 N,

xi +Qyi > supa2A

hi (a) +QRi .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 52 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Terminal phase: for Q 2 N,

a�(1), ..., a�(n)| {z }n periods

, ..., a�(1), ..., a�(n)| {z }n periods| {z }

Q -times=Qn periods

.

Observe that for all i 2 N, hi (a�(i)) > Ri and hi (a�(j)) � Ri for allj 2 N.Average payo¤s in the terminal phase: for all i 2 N,

yi =1QnQ

n∑j=1hi (a�(j)) =

1n

n∑j=1hi (a�(j)) > Ri .

Given x 2 C (G ) such that xi > Ri for all i 2 N, choose Q with theproperty that for all i 2 N,

xi +Qyi > supa2A

hi (a) +QRi .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 52 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Terminal phase: for Q 2 N,

a�(1), ..., a�(n)| {z }n periods

, ..., a�(1), ..., a�(n)| {z }n periods| {z }

Q -times=Qn periods

.

Observe that for all i 2 N, hi (a�(i)) > Ri and hi (a�(j)) � Ri for allj 2 N.

Average payo¤s in the terminal phase: for all i 2 N,

yi =1QnQ

n∑j=1hi (a�(j)) =

1n

n∑j=1hi (a�(j)) > Ri .

Given x 2 C (G ) such that xi > Ri for all i 2 N, choose Q with theproperty that for all i 2 N,

xi +Qyi > supa2A

hi (a) +QRi .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 52 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Terminal phase: for Q 2 N,

a�(1), ..., a�(n)| {z }n periods

, ..., a�(1), ..., a�(n)| {z }n periods| {z }

Q -times=Qn periods

.

Observe that for all i 2 N, hi (a�(i)) > Ri and hi (a�(j)) � Ri for allj 2 N.Average payo¤s in the terminal phase: for all i 2 N,

yi =1QnQ

n∑j=1hi (a�(j)) =

1n

n∑j=1hi (a�(j)) > Ri .

Given x 2 C (G ) such that xi > Ri for all i 2 N, choose Q with theproperty that for all i 2 N,

xi +Qyi > supa2A

hi (a) +QRi .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 52 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Terminal phase: for Q 2 N,

a�(1), ..., a�(n)| {z }n periods

, ..., a�(1), ..., a�(n)| {z }n periods| {z }

Q -times=Qn periods

.

Observe that for all i 2 N, hi (a�(i)) > Ri and hi (a�(j)) � Ri for allj 2 N.Average payo¤s in the terminal phase: for all i 2 N,

yi =1QnQ

n∑j=1hi (a�(j)) =

1n

n∑j=1hi (a�(j)) > Ri .

Given x 2 C (G ) such that xi > Ri for all i 2 N, choose Q with theproperty that for all i 2 N,

xi +Qyi > supa2A

hi (a) +QRi .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 52 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Given ε > 0, choose T 2 N such that there exists a cycle fatg oflength T �Qn such that 1

T �QnT�Qn

∑t=1

h(at )� x < ε.

De�ne f 2 FT : for i 2 N,

for 1 � t � T �Qn.

f ti (�) =�at if all players follow the cycle fatga(j)i if j has deviated,

where a(j) is such that hj (bj , a(j)�j ) � Rj for all bj 2 Aj .for T �Qn+ 1 � t < T .

f ti (�) = terminal phase of Nash equilibria.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 53 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Given ε > 0, choose T 2 N such that there exists a cycle fatg oflength T �Qn such that 1

T �QnT�Qn

∑t=1

h(at )� x < ε.

De�ne f 2 FT : for i 2 N,

for 1 � t � T �Qn.

f ti (�) =�at if all players follow the cycle fatga(j)i if j has deviated,

where a(j) is such that hj (bj , a(j)�j ) � Rj for all bj 2 Aj .for T �Qn+ 1 � t < T .

f ti (�) = terminal phase of Nash equilibria.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 53 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Given ε > 0, choose T 2 N such that there exists a cycle fatg oflength T �Qn such that 1

T �QnT�Qn

∑t=1

h(at )� x < ε.

De�ne f 2 FT : for i 2 N,for 1 � t � T �Qn.

f ti (�) =�at if all players follow the cycle fatga(j)i if j has deviated,

where a(j) is such that hj (bj , a(j)�j ) � Rj for all bj 2 Aj .

for T �Qn+ 1 � t < T .

f ti (�) = terminal phase of Nash equilibria.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 53 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

Given ε > 0, choose T 2 N such that there exists a cycle fatg oflength T �Qn such that 1

T �QnT�Qn

∑t=1

h(at )� x < ε.

De�ne f 2 FT : for i 2 N,for 1 � t � T �Qn.

f ti (�) =�at if all players follow the cycle fatga(j)i if j has deviated,

where a(j) is such that hj (bj , a(j)�j ) � Rj for all bj 2 Aj .for T �Qn+ 1 � t < T .

f ti (�) = terminal phase of Nash equilibria.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 53 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

It is possible to show that for all T su¢ ciently large, all i 2 N, and allgi 2 Fi ,

HTi (f ) � HTi (gi , f�i );namely, f 2 F �T .

Moreover, for su¢ ciently large T , HT (f )� x < ε;

namely, the weight of the terminal phase vanishes.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 54 / 64

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5.4.- �Folk�TheoremsFinitely Repeated: Intuition

It is possible to show that for all T su¢ ciently large, all i 2 N, and allgi 2 Fi ,

HTi (f ) � HTi (gi , f�i );namely, f 2 F �T .

Moreover, for su¢ ciently large T , HT (f )� x < ε;

namely, the weight of the terminal phase vanishes.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 54 / 64

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5.4.- �Folk�Theorems: SPE

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form.

For every t � 1 and i 2 I de�ne the mapping

si : Fi � At �! Fi ,

where, for every (fi , (a1, ..., at )) 2 Fi � At , s(fi , (a1, ..., at ))i 2 Fi isobtained as follows:

s(fi , (a1, ..., at ))1i = ft+1i (a1, ..., at ) and

for all r � 1 and all (b1, ..., br ) 2 Ar ,

s(fi , (a1, ..., at ))r+1i (b1, ..., br ) = f t+r+1i (a1, ..., at , b1, ..., br ).

Notation: for every (f , (a1, ..., at )) 2 F � At , set

s(f , (a1, ..., at )) � (s(fi , (a1, ..., at ))i )i2I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 55 / 64

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5.4.- �Folk�Theorems: SPE

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form.

For every t � 1 and i 2 I de�ne the mapping

si : Fi � At �! Fi ,

where, for every (fi , (a1, ..., at )) 2 Fi � At , s(fi , (a1, ..., at ))i 2 Fi isobtained as follows:

s(fi , (a1, ..., at ))1i = ft+1i (a1, ..., at ) and

for all r � 1 and all (b1, ..., br ) 2 Ar ,

s(fi , (a1, ..., at ))r+1i (b1, ..., br ) = f t+r+1i (a1, ..., at , b1, ..., br ).

Notation: for every (f , (a1, ..., at )) 2 F � At , set

s(f , (a1, ..., at )) � (s(fi , (a1, ..., at ))i )i2I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 55 / 64

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5.4.- �Folk�Theorems: SPE

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form.

For every t � 1 and i 2 I de�ne the mapping

si : Fi � At �! Fi ,

where, for every (fi , (a1, ..., at )) 2 Fi � At , s(fi , (a1, ..., at ))i 2 Fi isobtained as follows:

s(fi , (a1, ..., at ))1i = ft+1i (a1, ..., at ) and

for all r � 1 and all (b1, ..., br ) 2 Ar ,

s(fi , (a1, ..., at ))r+1i (b1, ..., br ) = f t+r+1i (a1, ..., at , b1, ..., br ).

Notation: for every (f , (a1, ..., at )) 2 F � At , set

s(f , (a1, ..., at )) � (s(fi , (a1, ..., at ))i )i2I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 55 / 64

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5.4.- �Folk�Theorems: SPE

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form.

For every t � 1 and i 2 I de�ne the mapping

si : Fi � At �! Fi ,

where, for every (fi , (a1, ..., at )) 2 Fi � At , s(fi , (a1, ..., at ))i 2 Fi isobtained as follows:

s(fi , (a1, ..., at ))1i = ft+1i (a1, ..., at ) and

for all r � 1 and all (b1, ..., br ) 2 Ar ,

s(fi , (a1, ..., at ))r+1i (b1, ..., br ) = f t+r+1i (a1, ..., at , b1, ..., br ).

Notation: for every (f , (a1, ..., at )) 2 F � At , set

s(f , (a1, ..., at )) � (s(fi , (a1, ..., at ))i )i2I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 55 / 64

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5.4.- �Folk�Theorems: SPE

Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form.

For every t � 1 and i 2 I de�ne the mapping

si : Fi � At �! Fi ,

where, for every (fi , (a1, ..., at )) 2 Fi � At , s(fi , (a1, ..., at ))i 2 Fi isobtained as follows:

s(fi , (a1, ..., at ))1i = ft+1i (a1, ..., at ) and

for all r � 1 and all (b1, ..., br ) 2 Ar ,

s(fi , (a1, ..., at ))r+1i (b1, ..., br ) = f t+r+1i (a1, ..., at , b1, ..., br ).

Notation: for every (f , (a1, ..., at )) 2 F � At , set

s(f , (a1, ..., at )) � (s(fi , (a1, ..., at ))i )i2I .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 55 / 64

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5.4.- �Folk�Theorems: SPE

De�nition Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form. Anstrategy f 2 F is a Subgame Perfect Equilibrium (SPE) of Gα, forα = ∞,λ, if for every t � 1 and every (a1, ..., at ) 2 At , s(f , (a1, ..., at )) isa Nash equilibrium of Gα.

TheoremAumann, Shapley, Rubinstein. Let G = (I , (Ai )i2I , (hi )i2I ) be a boundedgame in normal form. Then,nH∞(f ) 2 R#I j f is a SPE of G∞

o= fx 2 C (G ) j xi > Ri for all i 2 Ig .

TheoremFriedman (1971) Let a� 2 A� be such that h(a�) = e. Then, for everyx 2 C (G ) such that xi > ei for all i 2 I , there exists λ2 (0, 1) such thatfor all λ 2 (λ, 1) there exists a SPE f of Gλ with Hλ(f ) = x .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 56 / 64

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5.4.- �Folk�Theorems: SPE

De�nition Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form. Anstrategy f 2 F is a Subgame Perfect Equilibrium (SPE) of Gα, forα = ∞,λ, if for every t � 1 and every (a1, ..., at ) 2 At , s(f , (a1, ..., at )) isa Nash equilibrium of Gα.

TheoremAumann, Shapley, Rubinstein. Let G = (I , (Ai )i2I , (hi )i2I ) be a boundedgame in normal form. Then,nH∞(f ) 2 R#I j f is a SPE of G∞

o= fx 2 C (G ) j xi > Ri for all i 2 Ig .

TheoremFriedman (1971) Let a� 2 A� be such that h(a�) = e. Then, for everyx 2 C (G ) such that xi > ei for all i 2 I , there exists λ2 (0, 1) such thatfor all λ 2 (λ, 1) there exists a SPE f of Gλ with Hλ(f ) = x .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 56 / 64

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5.4.- �Folk�Theorems: SPE

De�nition Let G = (I , (Ai )i2I , (hi )i2I ) be a game in normal form. Anstrategy f 2 F is a Subgame Perfect Equilibrium (SPE) of Gα, forα = ∞,λ, if for every t � 1 and every (a1, ..., at ) 2 At , s(f , (a1, ..., at )) isa Nash equilibrium of Gα.

TheoremAumann, Shapley, Rubinstein. Let G = (I , (Ai )i2I , (hi )i2I ) be a boundedgame in normal form. Then,nH∞(f ) 2 R#I j f is a SPE of G∞

o= fx 2 C (G ) j xi > Ri for all i 2 Ig .

TheoremFriedman (1971) Let a� 2 A� be such that h(a�) = e. Then, for everyx 2 C (G ) such that xi > ei for all i 2 I , there exists λ2 (0, 1) such thatfor all λ 2 (λ, 1) there exists a SPE f of Gλ with Hλ(f ) = x .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 56 / 64

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5.4.- �Folk�Theorems: SPE

TheoremFudenberg and Maskin (1986) Let G = (I , (Ai )i2I , (hi )i2I ) be a boundedgame in normal form and assume dim(C (G )) = n. Then, for all x 2 C (G )such that xi > Ri for all i 2 I , there exists λ2 (0, 1) such that for allλ 2 (λ, 1) there exists a SPE f of Gλ with Hλ(f ) = x .

TheoremBenoît and Krishna (1985) Let G = (I , (Ai )i2I , (hi )i2I ) be a boundedgame in normal form and assume that for each i 2 I there exista�(i), a(i) 2 A� such that hi (a�(i)) > hi (a(i)) and that dim(C (G )) = n.Then, for every x 2 C (G ) such that xi > Ri for all i 2 I and every ε > 0there exists T 2 N such that for all T > T there exists a SPE f 2 F ofGT such that

HT (f )� x < ε.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 57 / 64

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5.4.- �Folk�Theorems

Friedman, J. �A Non-cooperative Equilibrium for Supergames,�TheReview of Economic Studies 38, 1971.

Benoît, J.P. and V. Krisnha. �Finitely Repeated Games,�Econometrica 53, 1985.

Fudenberg, D. and E. Maskin. �The Folk Theorem in RepeatedGames with Discounting or with Incomplete Information,�Econometrica 54, 1986.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 58 / 64

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5.4.- �Folk�Theorems

Friedman, J. �A Non-cooperative Equilibrium for Supergames,�TheReview of Economic Studies 38, 1971.

Benoît, J.P. and V. Krisnha. �Finitely Repeated Games,�Econometrica 53, 1985.

Fudenberg, D. and E. Maskin. �The Folk Theorem in RepeatedGames with Discounting or with Incomplete Information,�Econometrica 54, 1986.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 58 / 64

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5.4.- �Folk�Theorems

Friedman, J. �A Non-cooperative Equilibrium for Supergames,�TheReview of Economic Studies 38, 1971.

Benoît, J.P. and V. Krisnha. �Finitely Repeated Games,�Econometrica 53, 1985.

Fudenberg, D. and E. Maskin. �The Folk Theorem in RepeatedGames with Discounting or with Incomplete Information,�Econometrica 54, 1986.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 58 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.4.- �Folk�TheoremsFinal Remarks

F �λ �!λ!1

F �∞?

NO in general.

YES for the Prisoners�Dilemma.

Often YES.

F �T �!T!∞

F �∞?

NO for the Prisoners�Dilemma.

YES for games with a suboptimal Nash equilibrium.

Bounded rationality (�nite automata).

Large set of players ;-)

Evolution of behavior.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 59 / 64

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5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 230: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 231: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 232: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 233: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 234: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 235: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 236: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 237: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

L. Shapley. �Stochastic Games,�Proceedings of the NationalAcademy of Sciences 39, 1953.

The game that players face over time is not always the same.

Present actions may in�uence future opportunities. For example:

�shing in a lake,

timber industry,

cost-reducing investment decisions,

industry where �rms enter and leave (endogenously),

etc.

Idea: several games may be played, with a transition probability thatmay depend on the pro�le of actions.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 60 / 64

Page 238: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

A stochastic game consists of a �nite set of games in normal formfG1, ...,GK g and a probability distribution p, where for everyk = 1, ...,K :

Gk = (I , (Aki )i2I , (hki )i2I ) is a �nite game in normal form,

without loss of generality, assume Aki \ Ak0i = ? for all i 2 I and

k 6= k 0 (and de�ne Ai =KSk=1

Aki ),

G1 is the initial game;

for every ak 2 Ak , p(ak ) is a probability distribution on fG1, ...,GK g(i.e., p(ak ) 2 ∆K ), where for all k 0 = 1, ...,K ,

p(ak )k 0 is the probability of moving to game Gk 0 if players are at gameGk and choose action ak 2 Ak .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 61 / 64

Page 239: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

A stochastic game consists of a �nite set of games in normal formfG1, ...,GK g and a probability distribution p, where for everyk = 1, ...,K :

Gk = (I , (Aki )i2I , (hki )i2I ) is a �nite game in normal form,

without loss of generality, assume Aki \ Ak0i = ? for all i 2 I and

k 6= k 0 (and de�ne Ai =KSk=1

Aki ),

G1 is the initial game;

for every ak 2 Ak , p(ak ) is a probability distribution on fG1, ...,GK g(i.e., p(ak ) 2 ∆K ), where for all k 0 = 1, ...,K ,

p(ak )k 0 is the probability of moving to game Gk 0 if players are at gameGk and choose action ak 2 Ak .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 61 / 64

Page 240: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

A stochastic game consists of a �nite set of games in normal formfG1, ...,GK g and a probability distribution p, where for everyk = 1, ...,K :

Gk = (I , (Aki )i2I , (hki )i2I ) is a �nite game in normal form,

without loss of generality, assume Aki \ Ak0i = ? for all i 2 I and

k 6= k 0 (and de�ne Ai =KSk=1

Aki ),

G1 is the initial game;

for every ak 2 Ak , p(ak ) is a probability distribution on fG1, ...,GK g(i.e., p(ak ) 2 ∆K ), where for all k 0 = 1, ...,K ,

p(ak )k 0 is the probability of moving to game Gk 0 if players are at gameGk and choose action ak 2 Ak .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 61 / 64

Page 241: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

A stochastic game consists of a �nite set of games in normal formfG1, ...,GK g and a probability distribution p, where for everyk = 1, ...,K :

Gk = (I , (Aki )i2I , (hki )i2I ) is a �nite game in normal form,

without loss of generality, assume Aki \ Ak0i = ? for all i 2 I and

k 6= k 0 (and de�ne Ai =KSk=1

Aki ),

G1 is the initial game;

for every ak 2 Ak , p(ak ) is a probability distribution on fG1, ...,GK g(i.e., p(ak ) 2 ∆K ), where for all k 0 = 1, ...,K ,

p(ak )k 0 is the probability of moving to game Gk 0 if players are at gameGk and choose action ak 2 Ak .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 61 / 64

Page 242: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

A stochastic game consists of a �nite set of games in normal formfG1, ...,GK g and a probability distribution p, where for everyk = 1, ...,K :

Gk = (I , (Aki )i2I , (hki )i2I ) is a �nite game in normal form,

without loss of generality, assume Aki \ Ak0i = ? for all i 2 I and

k 6= k 0 (and de�ne Ai =KSk=1

Aki ),

G1 is the initial game;

for every ak 2 Ak , p(ak ) is a probability distribution on fG1, ...,GK g(i.e., p(ak ) 2 ∆K ), where for all k 0 = 1, ...,K ,

p(ak )k 0 is the probability of moving to game Gk 0 if players are at gameGk and choose action ak 2 Ak .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 61 / 64

Page 243: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

A stochastic game consists of a �nite set of games in normal formfG1, ...,GK g and a probability distribution p, where for everyk = 1, ...,K :

Gk = (I , (Aki )i2I , (hki )i2I ) is a �nite game in normal form,

without loss of generality, assume Aki \ Ak0i = ? for all i 2 I and

k 6= k 0 (and de�ne Ai =KSk=1

Aki ),

G1 is the initial game;

for every ak 2 Ak , p(ak ) is a probability distribution on fG1, ...,GK g(i.e., p(ak ) 2 ∆K ), where for all k 0 = 1, ...,K ,

p(ak )k 0 is the probability of moving to game Gk 0 if players are at gameGk and choose action ak 2 Ak .

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 61 / 64

Page 244: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 245: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 246: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , and

for all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 247: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 248: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 249: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 250: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 251: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 252: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

For every t � 1, denote by Dt the set of histories of length t that areconsistent with the probability distribution p.

A strategy for player i is a sequence f = ff ti g∞t=1 where for all i 2 I ,

f 1i 2 A1i , andfor all t � 1, f t+1i : Dt �! A1i � � � � � AKi with the property that forall (d1, ..., d t ) 2 Dt , f t+1i (d1, ..., d t ) = (a1i , ..., a

Ki ) speci�es an action

of player i for each possible game Gk .

A stationary strategy for player i is a functionsi : fG1, ...,GK g �! Ai such that for all k = 1, ...,K , si (Gk ) 2 Aki .

De�nition of payo¤s accordingly:

�nitely repeated,

in�nitely repeated with discounting,

in�nitely repeated without discounting.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 62 / 64

Page 253: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 254: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 255: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

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5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 257: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 258: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 259: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,

in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 260: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.

based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 261: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,

the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 262: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

TheoremShapley (1953) Let fG1, ...,GK g and p be an stochastic game with theproperty that #I = 2 and for all k = 1, ...,K, Gk is zero sum. Then, thein�nitely repeated game with discounting has a value.

The proof is not constructive (it uses a �x-point argument).

An important part of this literature has tried to show existence ofequilibria with stationary strategies for general settings.

Lokwood (1990)�s characterization with discounting

p(ak ) > 0 for all k = 1, ...,K and all ak 2 Ak .

Massó and Neme (1996)�s characterization with

p(ak )k 0 2 f0, 1g for all k, k 0 = 1, ...,K and all ak 2 Ak ,in�nitely repeated without discounting.based on connected stationary strategies,the set of equilibrium payo¤s is not convex and SPE(NE.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 63 / 64

Page 263: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

Lockwood, B. �The Folk Theorem in Stochastic Games with andwithout Discounting,�Birkbeck College Discussion Paper inEconomics 18, 1990.

Massó, J. and A. Neme. �Equilibrium Payo¤s of Dynamic Games,�International Journal of Game Theory 25, 1996.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 64 / 64

Page 264: Repeated Games Jordi Massó - UABpareto.uab.es/jmasso/pdf/Repeated Games.pdf · Repeated Games Jordi Massó International Doctorate in Economic Analysis (IDEA) ... and Mathematical

5.5.- Stochastic Games

Lockwood, B. �The Folk Theorem in Stochastic Games with andwithout Discounting,�Birkbeck College Discussion Paper inEconomics 18, 1990.

Massó, J. and A. Neme. �Equilibrium Payo¤s of Dynamic Games,�International Journal of Game Theory 25, 1996.

Jordi Massó (International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB))Game Theory: Repeated Games 64 / 64