robert aumann’s game and economic theorymath.huji.ac.il/~hart/papers/aumann-p.pdf · robert...

Robert Aumann’sGame and Economic Theory

Sergiu Hart

December 9, 2005Stockholm School of Economics

SERGIU HART c©2005 – p. 1

Robert Aumann’sGame and Economic Theory

Sergiu HartCenter of Rationality,

Dept. of Economics, Dept. of MathematicsThe Hebrew University of Jerusalem

[email protected]://www.ma.huji.ac.il/hart


Aumann’s Short CV

1930: Born in Germany


Aumann’s Short CV


1955: Ph.D. at M.I.T.


Aumann’s Short CV


1955: Ph.D. at M.I.T.

From 1956: Professor at the HebrewUniversity of Jerusalem


Aumann’s Short CV


1955: Ph.D. at M.I.T.


1991: EstablishingThe Center for Rationality


Aumann’s Short CV


1955: Ph.D. at M.I.T.



1998-2003: Founding President of theGame Theory Society


Aumann’s Short CV


1955: Ph.D. at M.I.T.



1998-2003: Founding President of theGame Theory Society

From 2001: Retired (?)


2005


Major Contributions

Repeated games


Major Contributions

Repeated games

Perfect competition


Major Contributions

Repeated games

Perfect competition

Correlated equilibrium


Major Contributions

Repeated games

Perfect competition


Interactive epistemology


Major Contributions

Repeated games

Perfect competition



Cooperative games


Major Contributions

Repeated games

Perfect competition



Cooperative games

Foundations


Major Contributions

Repeated games

Perfect competition



Cooperative games

Foundations

. . .


Repeated Games


The President’s Dilemma



1 to us 4 to them

1 to us

4 to them



TAU PRESIDENT

1 to us 4 to them

HUJ 1 to us

PRESIDENT 4 to them



TAU PRESIDENT

1 to us 4 to them

HUJ 1 to us 1 1

PRESIDENT 4 to them



TAU PRESIDENT

1 to us 4 to them

HUJ 1 to us 1 1 5 0

PRESIDENT 4 to them



TAU PRESIDENT

1 to us 4 to them

HUJ 1 to us 1 1 5 0

PRESIDENT 4 to them 0 5



TAU PRESIDENT

1 to us 4 to them

HUJ 1 to us 1 1 5 0

PRESIDENT 4 to them 0 5 4 4



TAU PRESIDENT

1 to us 4 to them

HUJ 1 to us 1 1 5 0


NASH EQUILIBRIUM



TAU PRESIDENT

1 to us 4 to them

HUJ 1 to us 1 1 5 0


NASH EQUILIBRIUM

PARETO OPTIMUM


The Duopolist’s Dilemma

PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15




4 3

4

3




QUANTITY(TAU)

4 3

QUANTITY4

(HUJ) 3




QUANTITY(TAU)

4 3

QUANTITY4 1 1

(HUJ) 3




QUANTITY(TAU)

4 3

QUANTITY4 1 1 5 0

(HUJ) 3




QUANTITY(TAU)

4 3

QUANTITY4 1 1 5 0

(HUJ) 3 0 5




QUANTITY(TAU)

4 3

QUANTITY4 1 1 5 0

(HUJ) 3 0 5 3 3




QUANTITY(TAU)

4 3

QUANTITY4 1 1 5 0

(HUJ) 3 0 5 3 3

NASH EQUILIBRIUM / COURNOT




QUANTITY(TAU)

4 3

QUANTITY4 1 1 5 0

(HUJ) 3 0 5 3 3

NASH EQUILIBRIUM / COURNOT

PARETO OPTIMUM / CARTELSERGIU HART c©2005 – p. 8

The Folk Theorem

The set ofNASH EQUILIBRIUM

outcomesof the repeated game


The Folk Theorem



equals


The Folk Theorem



equals

the set ofFEASIBLE and INDIVIDUALLY RATIONAL

outcomesof the one-shot game .


The Folk Theorem

PAYOFF TOPLAYER 1

PAYOFF TO PLAYER 2


The Folk Theorem

PAYOFF TOPLAYER 1

PAYOFF TO PLAYER 2

(5, 0)

(0, 5)

(4, 4)

(1, 1)


The Folk Theorem

PAYOFF TOPLAYER 1

PAYOFF TO PLAYER 2

(5, 0)

(0, 5)

(4, 4)

(1, 1)(1, 1)

Feasible


The Folk Theorem

PAYOFF TOPLAYER 1

PAYOFF TO PLAYER 2

(5, 0)

(0, 5)

(4, 4)

(1, 1)(1, 1)

Feasible

r2 = 1

r1 = 1SERGIU HART c©2005 – p. 10

The Folk Theorem

PAYOFF TOPLAYER 1

PAYOFF TO PLAYER 2

(5, 0)

(0, 5)

(4, 4)

(1, 1)(1, 1)

Feasible

r2 = 1

r1 = 1

Feasible & IR


The Folk Theorem

Idea of proof:


The Folk Theorem

Idea of proof:

Coordination on a feasible “master plan” ...


The Folk Theorem

Idea of proof:


... supported by the threat of “punishment”in case of deviation


The Folk Theorem

Idea of proof:


... supported by the threat of “punishment”in case of deviation

Proof:

Strategies, payoffs, ...

Aumann 1959


Repeated Games

“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ...

Aumann 1981SERGIU HART c©2005 – p. 12

Repeated Games

“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ... Its aim is to account forphenomena such as cooperation, altruism,revenge, threats (self-destructive or otherwise),etc.


Repeated Games

“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ... Its aim is to account forphenomena such as cooperation, altruism,revenge, threats (self-destructive or otherwise),etc.—phenomena which may at first seemirrational—


Repeated Games

“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ... Its aim is to account forphenomena such as cooperation, altruism,revenge, threats (self-destructive or otherwise),etc.—phenomena which may at first seemirrational—in terms of the usual ‘selfish’utility-maximizing paradigm of game theoryand neoclassical economics.”


The Folk Theorem

Noncooperative strategic behaviorin the repeated game

yields

Cooperative behavior


The Strong Folk Theorem



The set ofSTRONG NASH EQUILIBRIUM outcomes

of the repeated game



Stable relative to deviations by coalitions








equals

the COREof the one-shot game .


The Perfect Folk Theorem



The set ofPERFECT NASH EQUILIBRIUM outcomes




Stable also off equilibrium (“credible threats”)








equals


outcomes of the one-shot game .






equals


outcomes of the one-shot game .

Aumann & Shapley 1976 || Rubinstein 1976SERGIU HART c©2005 – p. 15

The Folk Theorem

Noncooperative strategic behaviorin the repeated game

yields

Cooperative behavior


Asymmetric Information

When players have private information:




How should an “informed” player takeadvantage of his information?





How should an “uninformed” playerbehave?






One-shot game vs. repeated game







When the information is used







When the information is used=⇒ The information is revealed







When the information is used=⇒ The information is revealed=⇒ The advantage disappears?







When the information is used=⇒ The information is revealed=⇒ The advantage disappears?

How can information be revealed credibly(when mutually advantageous)?



The Folk Theorem ⇒




Individual Rationality





Feasibility




Individual Rationality ?

Feasibility


Individual Rationality / Game 1

MATRIX 1 (probability = 1

2):

L RT 4 0

B 0 0


2):

L RT 0 0

B 0 4




2):

L RT 4 0

B 0 0


2):

L RT 0 0

B 0 4

ROW knows which MATRIX was chosen




2):

L RT 4 0

B 0 0


2):

L RT 0 0

B 0 4

ROW knows which MATRIX was chosenCOL does not know which MATRIX was chosen




2):

L RT 4 0

B 0 0


2):

L RT 0 0

B 0 4

ROW knows which MATRIX was chosenCOL does not know which MATRIX was chosen

Actions, but not payoffs, are observedSERGIU HART c©2005 – p. 19


M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4



M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4

How much can ROW guaran-tee?



M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4


If ROW playsT when M1

B when M2



M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4



B when M2Then COL learns whichMATRIX



M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4


If ROW playsT when M1 (→ R )

B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 0



M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4


If ROW plays “revealing”T when M1 (→ R )




M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4




If ROW plays (1

2, 1

2) “non-

revealing” then ROWgets 1



M1

L RT 4 0

B 0 0

M2

L RT 0 0

B 0 4




If ROW plays (1

2, 1

2) “non-


Ignoring his information is optimal for ROW



M1

L RT 4 4

B 4 0

M2

L RT 0 4

B 4 4



M1

L RT 4 4

B 4 0

M2

L RT 0 4

B 4 4



B when M2



M1

L RT 4 4

B 4 0

M2

L RT 0 4

B 4 4


If ROW plays “revealing”T when M1

B when M2



M1

L RT 4 4

B 4 0

M2

L RT 0 4

B 4 4






M1

L RT 4 4

B 4 0

M2

L RT 0 4

B 4 4




If ROW plays (1

2, 1

2) “non-




M1

L RT 4 4

B 4 0

M2

L RT 0 4

B 4 4




If ROW plays (1

2, 1

2) “non-


Full revelation is optimal for ROW



M1

L C RT 6 4 2

B 6 0 2

M2

L C RT 2 0 6

B 2 4 6



M1

L C RT 6 4 2

B 6 0 2

M2

L C RT 2 0 6

B 2 4 6


“Revealing” yields 2

T when M1 (→ R )

B when M2 (→ L )



M1

L C RT 6 4 2

B 6 0 2

M2

L C RT 2 0 6

B 2 4 6



T when M1 (→ R )

B when M2 (→ L )

“Non-revealing” (1

2, 1

2)

(→ C ) yields 2



M1

L C RT 6 4 2

B 6 0 2

M2

L C RT 2 0 6

B 2 4 6



T when M1 (→ R )

B when M2 (→ L )


2, 1

2)

(→ C ) yields 2

“Partially revealing”yields 3



M1

L C RT 6 4 2

B 6 0 2

M2

L C RT 2 0 6

B 2 4 6



T when M1 (→ R )

B when M2 (→ L )


2, 1

2)

(→ C ) yields 2

“Partially revealing”yields 3

Partial revelation is optimal for ROW


Partial Revelation

when M1:

{

play T with probability 0.80

play B with probability 0.20

when M2:

{




Partial Revelation

when M1:

{



when M2:

{



Prob (M1) Prob (M2)

a priori 0.50 0.50


Partial Revelation

when M1:

{



when M2:

{



Prob (M1) Prob (M2)

a priori 0.50 0.50

after T 0.67 0.33


Partial Revelation

when M1:

{



when M2:

{



Prob (M1) Prob (M2)

a priori 0.50 0.50

after T 0.67 0.33

after B 0.25 0.75



Theorem. The minimax value function of therepeated (zero-sum) game equals the

concavification of the minimax value function ofthe one-shot non-revealing game .

Aumann & Maschler 1966





⇒ Optimal strategy of the informed player(precise amount of information to reveal)






⇒ Optimal strategy of the informed player(precise amount of information to reveal)

⇒ Optimal strategy of the uninformed player



The Amount of Revelation

PROBABILITY

VALUE

bbb

b



PROBABILITY

VALUE

One-shot gamenon-revealing

bbb

b



PROBABILITY

VALUE


Repeated game

bbb

b



PROBABILITY

VALUE


Repeated game

b

prior

bb

b



PROBABILITY

VALUE


Repeated game

b

prior

bb

b

posteriorsSERGIU HART c©2005 – p. 25



Individual Rationality V





Feasibility





Feasibility ?


Feasibility / Game 4

M1

L RT 3, 3 0, 1

B 3, 3 0, 1

M2

L RT 4, 0 3, 3

B 4, 0 3, 3



M1

L RT 3, 3 0, 1

B 3, 3 0, 1

M2

L RT 4, 0 3, 3

B 4, 0 3, 3

To get (3, 3):



M1

L RT 3, 3 0, 1

B 3, 3 0, 1

M2

L RT 4, 0 3, 3

B 4, 0 3, 3

To get (3, 3):

ROW must reveal:T when M1 (→ L )

B when M2 (→ R )



M1

L RT 3, 3 0, 1

B 3, 3 0, 1

M2

L RT 4, 0 3, 3

B 4, 0 3, 3

To get (3, 3):


B when M2 (→ R )

⇒ ROW will play

T also when M2



M1

L RT 3, 3 0, 1

B 3, 3 0, 1

M2

L RT 4, 0 3, 3

B 4, 0 3, 3

To get (3, 3):


B when M2 (→ R )

⇒ ROW will play

T also when M2

Revealing is NOT“incentive compatible”



M1

L RT 3, 3 0, 1

B 3, 3 0, 1

M2

L RT 4, 0 3, 3

B 4, 0 3, 3

To get (3, 3):


B when M2 (→ R )

⇒ ROW will play

T also when M2

Revealing is NOT“incentive compatible”

=⇒ (3, 3) is not feasible !


Equilibria

Equilibria of the repeated (non-zero-sum) game:

Aumann, Maschler & Stearns 1968


Equilibria


Partial revelation



Equilibria


Partial revelation

Partial revelation , followed byjoint randomization



Equilibria


Partial revelation

Partial revelation , followed byjoint randomization

Partial revelation , followed byjoint randomization , followed by

additional partial revelation

. . .



Repeated Games – Summary

The Folk Theorem



The Folk Theorem

The Strong Folk Theorem(Aumann 1959)



The Folk Theorem


The Perfect Folk Theorem(Aumann & Shapley 1976, Rubinstein 1976)



The Folk Theorem



Asymmetric Information: Individual Rationality(Aumann & Maschler 1966)



The Folk Theorem



Asymmetric Information: Individual Rationality(Aumann & Maschler 1966)

Asymmetric Information: Equilibria(Aumann, Maschler & Stearns 1968)


Perfect Competition


The Market

Pieter Bruegel the Elder (1559)SERGIU HART c©2005 – p. 31

Perfect Competition

How should perfect competition be modelled?


Perfect Competition


“... the influence of an individual participant onthe economy cannot be mathematicallynegligible, as long as there are only finitely manyparticipants.


Perfect Competition


“... the influence of an individual participant onthe economy cannot be mathematicallynegligible, as long as there are only finitely manyparticipants. Thus a mathematical modelappropriate to the intuitive notion of perfectcompetition must contain infinitely manyparticipants.


Perfect Competition


“... the influence of an individual participant onthe economy cannot be mathematicallynegligible, as long as there are only finitely manyparticipants. Thus a mathematical modelappropriate to the intuitive notion of perfectcompetition must contain infinitely manyparticipants. We submit that the most naturalmodel for this purpose contains a continuum ofparticipants, similar to the continuum of points ona line or the continuum of particles in a fluid.”


The Equivalence Principle

In markets with a continuum of traders :




The set of Walrasian equilibria

coincides with

the solutions of thecorresponding “cooperative” game





coincides with


(core, value, ...)





coincides with


(core, value, ...)

Aumann 1964, Aumann & Shapley 1974,Aumann 1975, ...



The invisible hand (Adam Smith)

The limit contract curve (Edgeworth)

...



The invisible hand (Adam Smith)

The limit contract curve (Edgeworth)

...

“Intuitively, the Equivalence Principle says thatthe institution of market prices arises naturallyfrom the basic forces at work in a market,(almost) no matter what we assume about theway in which these forces work.”


Correlated Equilibrium



A Correlated Equilibrium is a Nash equilibriumwhen the players receive payoff-irrelevant

signals before playing the game

Aumann 1974





Independent signals

Aumann 1974





Independent signals ⇔ Nash equilibria

Aumann 1974






Public signals (“sunspots”)

Aumann 1974






Public signals (“sunspots”)⇔ convex combinations of Nash equilibria

Aumann 1974






Public signals (“sunspots”)⇔ convex combinations of Nash equilibria

Correlated private signals → new equilibria

Aumann 1974



Coordination, communication




Mechanisms, mediator




Mechanisms, mediator

Signals (public, correlated) are unavoidable


Correlated Equilibria

"Chicken" game

LEAVE STAY

LEAVE 5, 5 3, 6

STAY 6, 3 0, 0



"Chicken" game

LEAVE STAY

LEAVE 5, 5 3, 6

STAY 6, 3 0, 0

A Nash equilibrium



"Chicken" game

LEAVE STAY

LEAVE 5, 5 3, 6

STAY 6, 3 0, 0

Another Nash equilibrium i



"Chicken" game

LEAVE STAY

LEAVE 5, 5 3, 6

STAY 6, 3 0, 0

L 0 1/2

1/2 0

A (publicly) correlated equilibrium



"Chicken" game

LEAVE STAY

LEAVE 5, 5 3, 6

STAY 6, 3 0, 0

L S

L 1/3 1/3

S 1/3 0

Another correlated equilibrium :

After signal L play LEAVE

After signal S play STAY


Interactive Epistemology


Common Knowledge

A fact E is common knowledge among a set ofagents if:


Common Knowledge


Everyone knows E


Common Knowledge


Everyone knows E

Everyone knows that everyone knows E


Common Knowledge


Everyone knows E


Everyone knows that everyone knows thateveryone knows E


Common Knowledge


Everyone knows E



Everyone knows that ... ...everyone knows E


Common Knowledge


Everyone knows E



Everyone knows that ... ...everyone knows E

Lewis 1969 || Aumann 1976SERGIU HART c©2005 – p. 41



Partitions (“semantic”)Sentences (“syntactic”)




Partitions (“semantic”)Sentences (“syntactic”)

The Agreement Theorem :If two people have the same prior,

and their posteriors for an event A arecommon knowledge ,

then their posteriors must be equal .


Rationality

Assume a common prior.


Rationality


If all players are Bayesian rational ,


Rationality


If all players are Bayesian rational ,and this is common knowledge ,


Rationality


If all players are Bayesian rational ,and this is common knowledge ,

then

their play constitutesa correlated equilibrium


Other Contributions


Other Major Contributions

Cooperative games (NTU, core, value,bargaining set, ...)




Subjective probability and utility





Power and taxes





Power and taxes

Coalitions





Power and taxes

Coalitions

Foundations





Power and taxes

Coalitions

Foundations

Mathematics





Power and taxes

Coalitions

Foundations

Mathematics

. . .


The Unified Game Theory



“Unlike other approaches to disciplines likeeconomics or political science, GAME THEORYdoes not use different, ad-hoc constructs todeal with various specific issues, such as perfectcompetition, monopoly, oligopoly, internationaltrade, taxation, voting, deterrence, animalbehavior, and so on.

Aumann interview 2004SERGIU HART c©2005 – p. 46


“Unlike other approaches to disciplines likeeconomics or political science, GAME THEORYdoes not use different, ad-hoc constructs todeal with various specific issues, such as perfectcompetition, monopoly, oligopoly, internationaltrade, taxation, voting, deterrence, animalbehavior, and so on.Rather, it develops methodologies that apply inprinciple to all interactive situations , then seeswhere these methodologies lead in each specificapplication.”

Aumann interview 2004SERGIU HART c©2005 – p. 46

Aumann’s Doctoral Students



1. Bezalel Peleg

2. David Schmeidler

3. Shmuel Zamir

4. Elon Kohlberg

5. Benyamin Shitovitz

6. Zvi Artstein

7. Eugene Wesley



1. Bezalel Peleg

2. David Schmeidler

3. Shmuel Zamir

4. Elon Kohlberg

5. Benyamin Shitovitz

6. Zvi Artstein

7. Eugene Wesley

8. Sergiu Hart

9. Abraham Neyman

10. Yair Tauman

11. Dov Samet

12. Ehud Lehrer

13. Yossi Feinberg

. . .


A Scientist at Play


A Scientist at Worky


A Scientist at Play


robert aumann’s game and economic theorymath.huji.ac.il/~hart/papers/aumann-p.pdf · robert...

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