game dynamics and equilibrium
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Dynamics and EquilibriaSergiu Hart
Presidential Address, GAMES 2008 (July 2008)
Revised and Expanded (November 2009)
Revised (July 2010)
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DYNAMICS AND EQUILIBRIA
Sergiu HartCenter for the Study of Rationality
Dept of Economics Dept of MathematicsThe Hebrew University of Jerusalem
hart@huji.ac.il
http://www.ma.huji.ac.il/hart
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Papers
SERGIUHART c 2008 p. 3
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Papers
Hart and Mas-Colell,Econometrica2000
Hart and Mas-Colell,J Econ Theory2001Hart and Mas-Colell,Amer Econ Rev2003
Hart,Econometrica2005Hart and Mas-Colell,Games Econ Behav2006
Hart and Mansour,Games Econ Behav2010Hart,Games Econ Behav2011
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Papers
Hart and Mas-Colell,Econometrica2000
Hart and Mas-Colell,J Econ Theory2001Hart and Mas-Colell,Amer Econ Rev2003
Hart,Econometrica2005Hart and Mas-Colell,Games Econ Behav2006
Hart and Mansour,Games Econ Behav2010Hart,Games Econ Behav2011
http://www.ma.huji.ac.il/hartSERGIUHART c 2008 p. 3
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Nash Equilibrium
SERGIUHART c 2008 p. 4
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Nash Equilibrium
John Nash, Ph.D. Dissertation, Princeton 1950SERGIUHART c 2008 p. 4
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Nash Equilibrium
EQUILIBRIUM POINT:
John Nash, Ph.D. Dissertation, Princeton 1950SERGIUHART c 2008 p. 4
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Nash Equilibrium
EQUILIBRIUM POINT:
"Each players strategy is optimal
against those of the others."
John Nash, Ph.D. Dissertation, Princeton 1950SERGIUHART c 2008 p. 4
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Dynamics
SERGIUHART c 2008 p. 5
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Dynamics
FACT
SERGIUHART c 2008 p. 5
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Dynamics
FACT
There are no general, natural dynamicsleading to Nash equilibrium
SERGIUHART c 2008 p. 5
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Dynamics
FACT
There are nogeneral, natural dynamicsleading to Nash equilibrium
"general"
SERGIUHART c 2008 p. 5
i
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Dynamics
FACT
There are nogeneral, natural dynamicsleading to Nash equilibrium
"general": in all games
SERGIUHART c 2008 p. 5
D i
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Dynamics
FACT
There are nogeneral, natural dynamicsleading to Nash equilibrium
"general": in all gamesrather than: in specific classes of games
SERGIUHART c 2008 p. 5
D i
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Dynamics
FACT
There are nogeneral, natural dynamicsleading to Nash equilibrium
"general": in all gamesrather than: in specific classes of games:
two-person zero-sum gamestwo-person potential gamessupermodular games. . .
SERGIUHART c 2008 p. 5
D i
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Dynamics
FACT
There are no general, natural dynamicsleading to Nash equilibrium
SERGIUHART c 2008 p. 6
D i
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Dynamics
FACT
There are no general, natural dynamicsleading to Nash equilibrium
"leading to Nash equilibrium"
SERGIUHART c 2008 p. 6
D i
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Dynamics
FACT
There are no general, natural dynamicsleading to Nash equilibrium
"leading to Nash equilibrium":at a Nash equilibrium (or close to it)from some time on
SERGIUHART c 2008 p. 6
D i
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Dynamics
FACT
There are no general, natural dynamicsleading to Nash equilibrium
SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural"
SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural":
SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural":
adaptive(reacting, improving, ...)
SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural":
adaptive(reacting, improving, ...)
simple and efficient
SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural":adaptive(reacting, improving, ...)
simple and efficient:computation(performed at each step)
SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural":adaptive(reacting, improving, ...)
simple and efficient:computation(performed at each step)time(how long to reach equilibrium)
SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural":adaptive(reacting, improving, ...)
simple and efficient:computation(performed at each step)time(how long to reach equilibrium)
information(of each player)SERGIUHART c 2008 p. 7
Dynamics
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Dynamics
FACT
There are no general,naturaldynamicsleading to Nash equilibrium
"natural":adaptive(reacting, improving, ...)
simple and efficient:computation(performed at each step)time(how long to reach equilibrium)
information(of each player)bounded rationalitySERGIUHART c 2008 p. 7
Dynamics
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Dynamics
Dynamics that are NOT "natural":
SERGIUHART c 2008 p. 8
Dynamics
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Dynamics
Dynamics that are NOT "natural":
exhaustive search
(deterministic or stochastic)
SERGIUHART c 2008 p. 8
Exhaustive Search
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Exhaustive Search
SERGIUHART c 2008 p. 9
Exhaustive Search
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Exhaustive Search
SERGIUHART c 2008 p. 9
Exhaustive Search
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Exhaustive Search
SERGIUHART c 2008 p. 9
Exhaustive Search
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Exhaustive Search
SERGIUHART c 2008 p. 9
Exhaustive Search
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Exhaustive Search
SERGIUHART c 2008 p. 9
Exhaustive Search
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Exhaustive Search
SERGIUHART c 2008 p. 9
Exhaustive Search
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Exhaustive Search
SERGIUHART c 2008 p. 9
Exhaustive Search
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aust ve Sea c
SERGIUHART c 2008 p. 9
Exhaustive Search
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SERGIUHART c 2008 p. 9
Einsteins Manuscript
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p
Albert Einstein, 1912
On the Special Theory of Relativity (manuscript)SERGIU HART c 2008 p. 10
Dynamics
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y
Dynamics that are NOT "natural":
exhaustive search
(deterministic or stochastic)
SERGIU HART c 2008 p. 11
Dynamics
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Dynamics that are NOT "natural":
exhaustive search
(deterministic or stochastic)
using amediator
SERGIU HART c 2008 p. 11
Dynamics
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Dynamics that are NOT "natural":
exhaustive search
(deterministic or stochastic)
using amediator
broadcastingthe private informationand then performingjointcomputation
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Dynamics
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Dynamics that are NOT "natural":
exhaustive search
(deterministic or stochastic)
using amediator
broadcastingthe private informationand then performingjointcomputation
fully rational learning
(prior beliefs on the strategies of theopponents, Bayesian updating, optimization)
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Dynamics
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FACT
There are no general,naturaldynamics
leading to Nash equilibrium
"natural":adaptive
simple and efficient:computation(performed at each step)time(how long to reach equilibrium)information(of each player)
SERGIU HART c 2008 p. 12
Dynamics
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FACT
There are no general,naturaldynamics
leading to Nash equilibrium
"natural":adaptive
simple and efficient:computation(performed at each step)time(how long to reach equilibrium)information(of each player)
SERGIU HART c 2008 p. 12
Dynamics
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FACT
There are no general,naturaldynamics
leading to Nash equilibrium
"natural":adaptive
simple and efficient:computation(performed at each step)time(how long to reach equilibrium)information(of each player)
SERGIU HART c 2008 p. 12
Natural Dynamics: Information
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SERGIU HART c 2008 p. 13
Natural Dynamics: Information
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Each player knowsonlyhis own payoff(utility) function
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Natural Dynamics: Information
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Each player knowsonlyhis own payoff(utility) function
(doesnotknow the payoff functionsof the other players)
SERGIU HART c 2008 p. 13
Natural Dynamics: Information
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UNCOUPLED DYNAMICS:
Each player knowsonlyhis own payoff(utility) function
(doesnotknow the payoff functionsof the other players)
Hart and Mas-Colell, AER 2003SERGIU HART c 2008 p. 13
Natural Dynamics: Information
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UNCOUPLED DYNAMICS:
Each player knowsonlyhis own payoff(utility) function
(doesnotknow the payoff functionsof the other players)
(privacy-preserving, decentralized, distributed ...)
Hart and Mas-Colell, AER 2003SERGIU HART c 2008 p. 13
Games
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N-person gamein strategic (normal) form:
Playersi= 1, 2,...,N
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Games
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N-person gamein strategic (normal) form:
Playersi= 1, 2,...,N
For each playeri: Actions
ai in Ai
SERGIU HART c 2008 p. 14
Games
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N-person gamein strategic (normal) form:
Playersi= 1, 2,...,N
For each playeri: Actions
ai in Ai
For each playeri: Payoffs(utilities)
ui(a) ui(a1, a2,...,aN)
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Dynamics
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Time
t= 1, 2,...
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Dynamics
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Time
t= 1, 2,...
At periodteach playerichooses anaction
ait in A
i
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Dynamics
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Time
t= 1, 2,...
At periodteach playerichooses anaction
ait in A
i
according to a probability distribution
it in (Ai)
SERGIU HART c 2008 p. 15
Dynamics
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Fix the set of players1, 2,...,N andtheir action spacesA1, A2,...,AN
SERGIU HART c 2008 p. 16
Dynamics
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Fix the set of players1, 2,...,N andtheir action spacesA1, A2,...,AN
A general dynamic:
SERGIU HART c 2008 p. 16
Dynamics
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Fix the set of players1, 2,...,N andtheir action spacesA1, A2,...,AN
A general dynamic:
it i
t( HISTORY ; GAME )
SERGIU HART c 2008 p. 16
Dynamics
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Fix the set of players1, 2,...,N andtheir action spacesA1, A2,...,AN
A general dynamic:
it i
t( HISTORY ; GAME )
it( HISTORY ; u1,...,ui,...,uN )
SERGIU HART c 2008 p. 16
Uncoupled Dynamics
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Fix the set of players1, 2,...,N andtheir action spacesA1, A2,...,AN
A general dynamic:
it i
t( HISTORY ; GAME )
it( HISTORY ; u1,...,ui,...,uN )
An UNCOUPLEDdynamic:
SERGIU HART c 2008 p. 16
Uncoupled Dynamics
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Fix the set of players1, 2,...,N andtheir action spacesA1, A2,...,AN
Ageneraldynamic:
it i
t( HISTORY ; GAME )
it( HISTORY ; u1,...,ui,...,uN )
An UNCOUPLEDdynamic:
it it( HISTORY ; u
i )
SERGIU HART c 2008 p. 16
Uncoupled Dynamics
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Simplestuncoupled dynamics
SERGIU HART c 2008 p. 17
Uncoupled Dynamics
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Simplestuncoupled dynamics:
i
t fi
(at1; ui
)whereat1 = (a
1
t1, a2
t1,...,aNt1) A
are the actions of all the playersin the previous period
SERGIU HART c 2008 p. 17
Uncoupled Dynamics
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Simplestuncoupled dynamics:
i
t fi
(at1; ui
)whereat1 = (a
1
t1, a2
t1,...,aNt1) A
are the actions of all the playersin the previous period
Only last period matters (1-recall)
SERGIU HART c 2008 p. 17
Uncoupled Dynamics
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Simplestuncoupled dynamics:
i
t fi
(at1; ui
)whereat1 = (a
1
t1, a2
t1,...,aNt1) A
are the actions of all the playersin the previous period
Only last period matters (1-recall)
Timetdoes not matter (stationary)
SERGIU HART c 2008 p. 17
Impossibility
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SERGIU HART c 2008 p. 18
Impossibility
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Theorem. There are NOuncoupleddynamicswith1-recall
it fi(at1; u
i)
that yield almost sure convergence of play to
pure Nash equilibria of the stage game in allgames where such equilibria exist.
SERGIU HART c 2008 p. 18
Impossibility
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Theorem. There are NOuncoupleddynamicswith1-recall
it fi(at1; u
i)
that yield almost sure convergence of play to
pure Nash equilibria of the stage game in allgames where such equilibria exist.
Hart and Mas-Colell, GEB 2006SERGIU HART c 2008 p. 18
Proof
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Consider the following two-person game, which
has a unique pure Nash equilibrium
C1 C2 C3
R1 1,0 0,1 1,0
R2 0,1 1,0 1,0R3 0,1 0,1 1,1
SERGIU HART c 2008 p. 19
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Proof
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Consider the following two-person game, which
has a unique pure Nash equilibrium (R3,C3)
C1 C2 C3
R1 1,0 0,1 1,0
R2 0,1 1,0 1,0R3 0,1 0,1 1,1
Assumeby way of contradictionthat we aregiven an uncoupled,1-recall, stationary dynamicthat yields almost sure convergence to pure
Nash equilibria when these existSERGIU HART c 2008 p. 19
Proof
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Suppose the play at timet1is (R1,C1)
C1 C2 C3
R1 1,0 0,1 1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1SERGIU HART c 2008 p. 20
Proof
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Suppose the play at timet1is (R1,C1)
ROWENAis best replying at (R1,C1)
C1 C2 C3
R1 1,0 0,1 1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1SERGIU HART c 2008 p. 20
Proof
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Suppose the play at timet1is (R1,C1)
ROWENAis best replying at (R1,C1)
ROWENAwill play R1also att
C1 C2 C3
R1 1,0 0,1 1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1SERGIU HART c 2008 p. 20
Proof
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Suppose the play at timet1is (R1,C1)
ROWENAis best replying at (R1,C1)
ROWENAwill play R1also attProof:
Change the payoff function of COLIN sothat (R1,C1)is the unique pure Nash eq.
C1 C2 C3
R1 1,0 0,1 1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1SERGIU HART c 2008 p. 20
Proof
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Suppose the play at timet1is (R1,C1)
ROWENAis best replying at (R1,C1)
ROWENAwill play R1also attProof:
Change the payoff function of COLIN sothat (R1,C1)is the unique pure Nash eq.
C1 C2 C3
R1 1,1 0,1 1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,0SERGIU HART c 2008 p. 20
Proof
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Suppose the play at timet1is (R1,C1)
ROWENAis best replying at (R1,C1)
ROWENAwill play R1also attProof:
Change the payoff function of COLIN sothat (R1,C1)is the unique pure Nash eq.
In the new game, ROWENAmustplay R1
after (R1,C1)(by1-recall,stationarity, anda.s. convergenceto the pure Nash eq.)
SERGIU HART c 2008 p. 20
Proof
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Suppose the play at timet1is (R1,C1)
ROWENAis best replying at (R1,C1)
ROWENAwill play R1also attProof:
Change the payoff function of COLIN sothat (R1,C1)is the unique pure Nash eq.
In the new game, ROWENAmustplay R1
after (R1,C1)(by1-recall,stationarity, anda.s. convergenceto the pure Nash eq.)
Byuncoupledness, the same holds in the
original gameSERGIU HART c 2008 p. 20
Proof
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Suppose the play at timet1is (R1,C1)
ROWENAis best replying at (R1,C1)
ROWENAwill play R1also att
SERGIU HART c 2008 p. 20
Proof
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ROWENAis best replying att1
ROWENAwill play the same action att
SERGIU HART c 2008 p. 20
Proof
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Similarly for COLIN:
A player who is best replying cannot switch
SERGIU HART c 2008 p. 21
Proof
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Similarly for COLIN:
A player who is best replying cannot switch
C1 C2 C3
R1 1,0
0,1
1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1
SERGIU HART c 2008 p. 21
Proof
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Similarly for COLIN:
A player who is best replying cannot switch
C1 C2 C3
R1 1,0
0,1
1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1
SERGIU HART c 2008 p. 21
Proof
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Similarly for COLIN:
A player who is best replying cannot switch
C1 C2 C3
R1 1,0
0,1
1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1
(R3,C3) cannot be reachedSERGIU HART c 2008 p. 21
Proof
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Similarly for COLIN:
A player who is best replying cannot switch
C1 C2 C3
R1 1,0
0,1
1,0
R2 0,1 1,0 1,0
R3 0,1 0,1 1,1
(R3,C3) cannot be reached(unless we start there)
SERGIU HART c 2008 p. 21
Possibility
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SERGIU HART c 2008 p. 22
Possibility
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Theorem. THERE EXISTuncoupleddynamicswith2-RECALL
it fi(at2, at1;ui)
that yield almost sure convergence of play to
pure Nash equilibria of the stage game in everygame where such equilibria exist.
SERGIU HART c 2008 p. 22
Possibility
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Define the strategy of each playerias follows:
SERGIU HART c 2008 p. 23
Possibility
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Define the strategy of each playerias follows:
IF:Everyone played the same in the previoustwo periods: at2 =at1 =a; and
Playeribest replied: ai BRi(ai; ui)
THEN: Attplayeriplaysai again: ait =ai
SERGIU HART c 2008 p. 23
Possibility
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Define the strategy of each playerias follows:
IF:Everyone played the same in the previoustwo periods: at2 =at1 =a; and
Playeribest replied: ai BRi(ai; ui)
THEN: Attplayeriplaysai again: ait =ai
ELSE: Attplayerirandomizes uniformly overAi
SERGIU HART c 2008 p. 23
Possibility
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"Good":
SERGIU HART c 2008 p. 24
Possibility
"G d"
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"Good":
simple
SERGIU HART c 2008 p. 24
Possibility
"G d"
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"Good":
simple
"Bad":
SERGIU HART c 2008 p. 24
Possibility
"G d"
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"Good":
simple
"Bad":
exhaustive search
SERGIU HART c 2008 p. 24
Possibility
"G d"
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"Good":
simple
"Bad":
exhaustive searchall players must use it
SERGIU HART c 2008 p. 24
Possibility
"Good"
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"Good":
simple
"Bad":
exhaustive searchall players must use it
takes a long time
SERGIU HART c 2008 p. 24
Possibility
"Good":
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"Good":
simple
"Bad":
exhaustive searchall players must use it
takes a long time
"Ugly": ...
SERGIU HART c 2008 p. 24
Possibility
"Good":
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Good :
simple
"Bad":
exhaustive searchall players must use it
takes a long time
"Ugly": ...
SERGIU HART c 2008 p. 24
Possibility
"Good":
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Good :
simple
"Bad":
exhaustive searchall players must use it
takes a long time
"Ugly": ...
SERGIU HART c 2008 p. 24
Dynamics
FACT
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FACT
There are no general,naturaldynamics
leading to Nash equilibrium
"natural":adaptive
simple and efficient:
computationtime
information
SERGIU HART c 2008 p. 25
Dynamics
FACT
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FACT
There are no general,naturaldynamics
leading to Nash equilibrium
"natural":adaptive
simple and efficient:
computationtime
information: uncoupledness
SERGIU HART c 2008 p. 25
Dynamics
FACT
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FACT
There are no general,naturaldynamics
leading to Nash equilibrium
"natural":adaptive
simple and efficient:
computation: finite recalltime
information: uncoupledness
SERGIU HART c 2008 p. 25
Dynamics
FACT
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FACT
There are no general,naturaldynamics
leading to Nash equilibrium
"natural":adaptive
simple and efficient:
computation: finite recalltimeto reach equilibrium?information: uncoupledness
SERGIU HART c 2008 p. 25
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM?
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HOW LONG TO EQUILIBRIUM?
SERGIU HART c 2008 p. 26
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM?
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HOW LONG TO EQUILIBRIUM?
Estimate the number of time periods it takes until
a Nash equilibrium is reached
SERGIU HART c 2008 p. 26
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM?
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HOW LONG TO EQUILIBRIUM?
Estimate the number of time periods it takes until
a Nash equilibrium is reached
How?
SERGIU HART c 2008 p. 26
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM?
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O O G O QU U
Estimate the number of time periods it takes until
a Nash equilibrium is reached
How?
An uncoupled dynamic
A distributed computational procedure
SERGIU HART c 2008 p. 26
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM?
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Estimate the number of time periods it takes until
a Nash equilibrium is reached
How?
An uncoupled dynamic
A distributed computational procedure
COMMUNICATION COMPLEXITY
SERGIU HART c 2008 p. 26
Communication Complexity
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SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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p p
SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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p p
START: Each participant has some private
information [INPUTS]
SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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START: Each participant has some private
information [INPUTS]COMMUNICATION: Messages aretransmitted between the participants
SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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START: Each participant has some private
information [INPUTS]COMMUNICATION: Messages aretransmitted between the participants
END: All participants reach agreement onthe result
SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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START: Each participant has some private
information [INPUTS]COMMUNICATION: Messages aretransmitted between the participants
END: All participants reach agreement onthe result
SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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START: Each participant has some private
information [INPUTS]COMMUNICATION: Messages aretransmitted between the participants
END: All participants reach agreement onthe result [OUTPUT]
SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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START: Each participant has some private
information [INPUTS]COMMUNICATION: Messages aretransmitted between the participants
END: All participants reach agreement onthe result [OUTPUT]
COMMUNICATION COMPLEXITY=the minimal number of rounds needed
SERGIU HART c 2008 p. 27
Communication Complexity
Distributed computational procedure
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START: Each participant has some private
information [INPUTS]COMMUNICATION: Messages aretransmitted between the participants
END: All participants reach agreement onthe result [OUTPUT]
COMMUNICATION COMPLEXITY=the minimal number of rounds needed
Yao 1979, Kushilevitz and Nisan 1997SERGIU HART c 2008 p. 27
How Long to Equilibrium
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SERGIU HART c 2008 p. 28
How Long to Equilibrium
Uncoupled dynamic leading to Nashequilibria
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equilibria
SERGIU HART c 2008 p. 28
How Long to Equilibrium
Uncoupled dynamic leading to Nashequilibria
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equilibria
START: Each player knows his own payoff
function [INPUTS]
SERGIU HART c 2008 p. 28
How Long to Equilibrium
Uncoupled dynamic leading to Nashequilibria
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equilibria
START: Each player knows his own payoff
function [INPUTS]
COMMUNICATION: The actions played arecommonly observed
SERGIU HART c 2008 p. 28
How Long to Equilibrium
Uncoupled dynamic leading to Nashequilibria
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equ b a
START: Each player knows his own payoff
function [INPUTS]
COMMUNICATION: The actions played arecommonly observed
END: All players play a Nash equilibrium[OUTPUT]
SERGIU HART c 2008 p. 28
How Long to Equilibrium
Uncoupled dynamic leading to Nashequilibria
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q
START: Each player knows his own payoff
function [INPUTS]
COMMUNICATION: The actions played arecommonly observed
END: All players play a Nash equilibrium[OUTPUT]
COMMUNICATION COMPLEXITY=the minimal number of rounds needed
SERGIU HART c 2008 p. 28
How Long to Equilibrium
Uncoupled dynamic leading to Nashequilibria
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q
START: Each player knows his own payoff
function [INPUTS]
COMMUNICATION: The actions played arecommonly observed
END: All players play a Nash equilibrium[OUTPUT]
COMMUNICATION COMPLEXITY=the minimal number of rounds needed
Conitzer and Sandholm 2004 SERGIU HART c 2008 p. 28
How Long to Equilibrium
An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if
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equilibria is TIME-EFFICIENT if
SERGIU HART c 2008 p. 29
How Long to Equilibrium
An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if
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equilibria is TIME EFFICIENT ifthe TIME IT TAKES
SERGIU HART c 2008 p. 29
How Long to Equilibrium
An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if
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equilibria is TIME EFFICIENT ifthe TIME IT TAKES is POLYNOMIALin thenumber of players
SERGIU HART c 2008 p. 29
How Long to Equilibrium
An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if
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equilibria is TIME EFFICIENT ifthe TIME IT TAKES is POLYNOMIALin thenumber of players (rather than: exponential)
SERGIU HART c 2008 p. 29
How Long to Equilibrium
An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if
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equilibria is TIME EFFICIENT ifthe TIME IT TAKES is POLYNOMIALin thenumber of players (rather than: exponential)
Theorem. There are NO TIME-EFFICIENT
uncoupled dynamics that reach a pure Nashequilibrium in all games where such equilibriaexist.
SERGIU HART c 2008 p. 29
How Long to Equilibrium
An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if
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qthe TIME IT TAKES is POLYNOMIALin thenumber of players (rather than: exponential)
Theorem. There are NO TIME-EFFICIENT
uncoupled dynamics that reach a pure Nashequilibrium in all games where such equilibriaexist.
Hart and Mansour, GEB 2010 SERGIU HART c 2008 p. 29
How Long to Equilibrium
An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if
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qthe TIME IT TAKES is POLYNOMIALin thenumber of players (rather than: exponential)
Theorem. There are NO TIME-EFFICIENT
uncoupled dynamics that reach a pure Nashequilibrium in all games where such equilibriaexist.
In fact:exponential, like exhaustive search
Hart and Mansour, GEB 2010 SERGIU HART c 2008 p. 29
How Long to Equilibrium
Intuition:
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SERGIU HART c 2008 p. 30
How Long to Equilibrium
Intuition:
different games have different equilibria
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different games have different equilibria
SERGIU HART c 2008 p. 30
How Long to Equilibrium
Intuition:
different games have different equilibria
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different games have different equilibria
the dynamic procedure must distinguishbetween them
SERGIU HART c 2008 p. 30
How Long to Equilibrium
Intuition:
different games have different equilibria
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different games have different equilibria
the dynamic procedure must distinguishbetween them
no single player can do so by himself
SERGIU HART c 2008 p. 30
Dynamics and Nash Equilibrium
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SERGIU HART c 2008 p. 31
Dynamics and Nash Equilibrium
FACT
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There are NOgeneral, natural dynamics
leading to Nash equilibrium
SERGIU HART c 2008 p. 31
Dynamics and Nash Equilibrium
FACT
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There are NOgeneral, natural dynamics
leading to Nash equilibrium
RESULT
There CANNOT BEgeneral, natural dynamics
leading to Nash equilibrium
SERGIU HART c 2008 p. 31
Dynamics and Nash Equilibrium
RESULT
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There CANNOT BEgeneral, natural dynamicsleading to Nash equilibrium
SERGIU HART c 2008 p. 32
Dynamics and Nash Equilibrium
RESULT
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There CANNOT BE
general, natural dynamicsleading to Nash equilibrium
Perhaps we are asking too much?
SERGIU HART c 2008 p. 32
Dynamics and Nash Equilibrium
RESULT
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There CANNOT BE
general, natural dynamicsleading to Nash equilibrium
Perhaps we are asking too much?
For instance, the size of the data (the payoff
functions) isexponentialrather thanpolynomial in the number of players
SERGIU HART c 2008 p. 32
Correlated Equilibrium
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SERGIU HART c 2008 p. 33
Correlated Equilibrium
CORRELATED EQUILIBRIUM
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CORRELATED EQUILIBRIUM
Aumann, JME 1974 SERGIU HART c 2008 p. 33
Correlated Equilibrium
CORRELATED EQUILIBRIUM :
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CORRELATED EQUILIBRIUM:
Nash equilibrium when players receivepayoff-irrelevant information
before playing the game
Aumann, JME 1974 SERGIU HART c 2008 p. 33
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen players receive payoff-irrelevant signalsbefore playing the game
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before playing the game
SERGIU HART c 2008 p. 34
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen players receive payoff-irrelevant signalsbefore playing the game
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before playing the game
Examples:
SERGIU HART c 2008 p. 34
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen players receive payoff-irrelevant signalsbefore playing the game
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before playing the game
Examples:
Independent signals
SERGIU HART c 2008 p. 34
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen players receive payoff-irrelevant signalsbefore playing the game
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before playing the game
Examples:
Independent signals Nash equilibrium
SERGIU HART c 2008 p. 34
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen players receive payoff-irrelevant signalsbefore playing the game
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before playing the game
Examples:
Independent signals Nash equilibriumPublic signals (sunspots)
SERGIU HART c 2008 p. 34
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen players receive payoff-irrelevant signalsbefore playing the game
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before playing the game
Examples:
Independent signals Nash equilibriumPublic signals (sunspots) convex combinations of Nash equilibria
SERGIU HART c 2008 p. 34
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Correlated Equilibria
"Chicken" game
LEAVE STAY
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S
LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
SERGIU HART c 2008 p. 35
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Correlated Equilibria
"Chicken" game
LEAVE STAY
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LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
L 0 1/2
1/2 0
a(publicly) correlated equilibrium
SERGIU HART c 2008 p. 35
Correlated Equilibria
"Chicken" game
LEAVE STAY L S
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LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
L 1/3 1/3
S 1/3 0
anothercorrelated equilibrium
after signal Lplay LEAVEafter signal Splay STAY
SERGIU HART c 2008 p. 35
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen the players receive payoff-irrelevant signalsbefore playing the game (Aumann 1974)
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p y g g ( )
Examples:
Independent signals Nash equilibrium
Public signals (sunspots)convex combinations of Nash equilibriaButterflies play the Chicken Game
(Speckled WoodPararge aegeria)
SERGIU HART c 2008 p. 36
Correlated Equilibrium
ACorrelated Equilibriumis a Nash equilibriumwhen the players receive payoff-irrelevant signalsbefore playing the game (Aumann 1974)
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Examples:
Independent signals Nash equilibrium
Public signals (sunspots)convex combinations of Nash equilibriaButterflies play the Chicken Game
(Speckled WoodPararge aegeria)Boston Celtics front line
SERGIU HART c 2008 p. 36
Correlated Equilibrium
Signals (public, correlated) areunavoidable
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SERGIU HART c 2008 p. 37
Correlated Equilibrium
Signals (public, correlated) areunavoidable
Common KnowledgeofRationalityCorrelated Equilibrium (Aumann 1987)
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Correlated Equilibrium (Aumann 1987)
SERGIU HART c 2008 p. 37
Correlated Equilibrium
Signals (public, correlated) areunavoidable
Common KnowledgeofRationalityCorrelated Equilibrium (Aumann 1987)
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Correlated Equilibrium (Aumann 1987)
A joint distributionz is acorrelated equilibrium
si
u(j, si)z(j, si) si
u(k, si)z(j, si)
for alli Nand allj, k Si
SERGIU HART c 2008 p. 37
Dynamics & Correlated Equilibria
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SERGIU HART c 2008 p. 38
Dynamics & Correlated Equilibria
RESULT
THERE EXISTgeneral, natural dynamics
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leading to CORRELATED EQUILIBRIA
SERGIU HART c 2008 p. 38
Dynamics & Correlated Equilibria
RESULT
THERE EXISTgeneral, natural dynamics
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leading to CORRELATED EQUILIBRIA
Regret Matching
Hart and Mas-Colell, Ecca 2000SERGIU HART c 2008 p. 38
Dynamics & Correlated Equilibria
RESULT
THERE EXISTgeneral, natural dynamics
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leading to CORRELATED EQUILIBRIA
Regret Matching
General regret-based dynamics
Hart and Mas-Colell, Ecca 2000, JET 2001SERGIU HART c 2008 p. 38
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Regret Matching
"REGRET": the increase in past payoff, if any,
if diff i ld h b d
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if a different action would have been used
"MATCHING": switching to a different actionwith a probability that is proportional to theregret for that action
SERGIU HART c 2008 p. 39
Dynamics & Correlated Equilibria
THERE EXISTgeneral, natural dynamicsleading to CORRELATED EQUILIBRIA
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SERGIU HART c 2008 p. 40
Dynamics & Correlated Equilibria
THERE EXISTgeneral, natural dynamicsleading to CORRELATED EQUILIBRIA
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"general": in all games
SERGIU HART c 2008 p. 40
Dynamics & Correlated Equilibria
THERE EXISTgeneral,naturaldynamicsleading to CORRELATED EQUILIBRIA
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"general": in all games
"natural":
SERGIU HART c 2008 p. 40
Dynamics & Correlated Equilibria
THERE EXISTgeneral,naturaldynamicsleading to CORRELATED EQUILIBRIA
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"general": in all games
"natural":
adaptive(also: close to "behavioral")
SERGIU HART c 2008 p. 40
Dynamics & Correlated Equilibria
THERE EXISTgeneral,naturaldynamicsleading to CORRELATED EQUILIBRIA
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"general": in all games
"natural":
adaptive(also: close to "behavioral")
simple and efficient:computation, time, information
SERGIU HART c 2008 p. 40
Dynamics & Correlated Equilibria
THERE EXISTgeneral, natural dynamicsleading to CORRELATED EQUILIBRIA
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"general": in all games
"natural":
adaptive(also: close to "behavioral")
simple and efficient:computation, time, information
"leading to correlated equilibria":statistics of play become close to
CORRELATED EQUILIBRIA SERGIU HART c 2008 p. 40
Regret Matching and Beyond
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SERGIU HART c 2008 p. 41
Regret Matching and Beyond
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SERGIU HART c 2008 p. 41
Regret Matching and Beyond
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SERGIU HART c 2008 p. 42
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Regret Matching and Beyond
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SERGIU HART c 2008 p. 44
Regret Matching and Beyond
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SERGIU HART c 2008 p. 45
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Regret Matching and Beyond
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SERGIU HART c 2008 p. 47
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Dynamics and Equilibrium
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SERGIU HART c 2008 p. 49
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Dynamics and Equilibrium
NASH EQUILIBRIUM:afixed-pointof a non-linear map
CORRELATED EQUILIBRIUM:
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CORRELATED EQUILIBRIUM:a solution of finitely manylinear inequalities
SERGIU HART c 2008 p. 49
Dynamics and Equilibrium
NASH EQUILIBRIUM:afixed-pointof a non-linear map
CORRELATED EQUILIBRIUM:
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CORRELATED EQUILIBRIUM:a solution of finitely manylinear inequalities
set-valuedfixed-point (curb sets)?
SERGIU HART c 2008 p. 49
Dynamics and Equilibrium
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SERGIU HART c 2008 p. 50
Dynamics and Equilibrium
"LAW OF CONSERVATION OF COORDINATION":
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SERGIU HART c 2008 p. 50
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Dynamics and Equilibrium
"LAW OF CONSERVATION OF COORDINATION":
Th b
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There must be some COORDINATION
either in the EQUILIBRIUMnotion,
SERGIU HART c 2008 p. 50
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The "Program"
A. Demarcatethe BORDERbetween
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SERGIU HART c 2008 p. 51
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Dynamics and Equilibrium
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SERGIU HART c 2008 p. 52
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Dynamics and Equilibrium
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SERGIU HART c 2008 p. 53
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My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMALMODELS
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CONCEPTS
SERGIU HART c 2008 p. 55
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMALMODELS
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CONCEPTS
SERGIU HART c 2008 p. 55
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