Nash Equilibrium and Dynamics
Sergiu Hart
June 2008
Conference in Honor of John Nash’s 80th Birthday
Opening Panel
SERGIU HART c© 2008 – p. 1
NASH EQUILIBRIUMAND DYNAMICS
Sergiu HartCenter for the Study of Rationality
Dept of Economics Dept of MathematicsThe Hebrew University of Jerusalem
[email protected]://www.ma.huji.ac.il/hart
SERGIU HART c© 2008 – p. 2
Nash equilibrium
SERGIU HART c© 2008 – p. 3
nash equilibrium
SERGIU HART c© 2008 – p. 4
Nash equilibrium
SERGIU HART c© 2008 – p. 5
Nash equilibrium
John Nash, Ph.D. Dissertation, Princeton 1950
SERGIU HART c© 2008 – p. 5
Nash equilibrium
EQUILIBRIUM POINT:
John Nash, Ph.D. Dissertation, Princeton 1950
SERGIU HART c© 2008 – p. 5
Nash equilibrium
EQUILIBRIUM POINT:
"Each player’s strategy is optimalagainst those of the others."
John Nash, Ph.D. Dissertation, Princeton 1950
SERGIU HART c© 2008 – p. 5
Nash equilibrium
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
"absence of coalitions,communication, and side-payments"
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
"absence of coalitions,communication, and side-payments"
→ "Nash Program":non-cooperative foundation andimplementation of cooperative approaches
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
"MASS-ACTION" INTERPRETATION
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
"MASS-ACTION" INTERPRETATION
"RATIONAL" INTERPRETATION
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
"MASS-ACTION" INTERPRETATION
"RATIONAL" INTERPRETATION
"prediction of the behavior to beexpected of rational playing thegame"
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
"MASS-ACTION" INTERPRETATION
"RATIONAL" INTERPRETATION
"prediction of the behavior to beexpected of rational playing thegame"
"we need to assume the players knowthe full structure of the game
SERGIU HART c© 2008 – p. 6
Nash equilibrium
NON-COOPERATIVE
"MASS-ACTION" INTERPRETATION
"RATIONAL" INTERPRETATION
"prediction of the behavior to beexpected of rational playing thegame"
"we need to assume the players knowthe full structure of the game ... quitestrongly a rationalistic and idealisinginterpretation"
SERGIU HART c© 2008 – p. 6
Nash equilibrium
SERGIU HART c© 2008 – p. 7
Nash equilibrium
If each player
SERGIU HART c© 2008 – p. 7
Nash equilibrium
If each player
knows his own payoff function
SERGIU HART c© 2008 – p. 7
Nash equilibrium
If each player
knows his own payoff function
is rational
SERGIU HART c© 2008 – p. 7
Nash equilibrium
If each player
knows his own payoff function
is rational
knows the (pure) strategy choices of theothers
SERGIU HART c© 2008 – p. 7
Nash equilibrium
If each player
knows his own payoff function
is rational
knows the (pure) strategy choices of theothers
Then these choices constitutea (pure) NASH EQUILIBRIUM
SERGIU HART c© 2008 – p. 7
Nash equilibrium
If each player
knows his own payoff function
is rational
knows the (pure) strategy choices of theothers
Then these choices constitutea (pure) NASH EQUILIBRIUM
Aumann and Brandenburger 1995SERGIU HART c© 2008 – p. 7
Dynamics
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general"
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games(rather than: for specific classes of games)
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games
"leading to Nash equilibrium"
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games
"leading to Nash equilibrium" : at a Nashequilibrium (or close to it) from some time on
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games
"leading to Nash equilibrium" : at a Nashequilibrium (or close to it) from some time on
"natural"
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games
"leading to Nash equilibrium" : at a Nashequilibrium (or close to it) from some time on
"natural" :not of the "exhaustive search" variety
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games
"leading to Nash equilibrium" : at a Nashequilibrium (or close to it) from some time on
"natural" :not of the "exhaustive search" varietysimple, efficient (time, computation, ...)
SERGIU HART c© 2008 – p. 8
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
"general" : in all games
"leading to Nash equilibrium" : at a Nashequilibrium (or close to it) from some time on
"natural" :not of the "exhaustive search" varietysimple, efficient (time, computation, ...)"uncoupled"
SERGIU HART c© 2008 – p. 8
Uncoupled dynamics
UNCOUPLED DYNAMICS
SERGIU HART c© 2008 – p. 9
Uncoupled dynamics
UNCOUPLED DYNAMICS :
Each player knows only his own payoff function
Hart and Mas-Colell 2003SERGIU HART c© 2008 – p. 9
Uncoupled dynamics
UNCOUPLED DYNAMICS :
Each player knows only his own payoff function(does not know the others’ payoff functions)
Hart and Mas-Colell 2003SERGIU HART c© 2008 – p. 9
Uncoupled dynamics
UNCOUPLED DYNAMICS :
Each player knows only his own payoff function(does not know the others’ payoff functions)
impossibility results
Hart and Mas-Colell 2003, 2006SERGIU HART c© 2008 – p. 9
Uncoupled dynamics
UNCOUPLED DYNAMICS :
Each player knows only his own payoff function(does not know the others’ payoff functions)
impossibility results
how long? exponential time
Hart and Mansour 2008SERGIU HART c© 2008 – p. 9
Dynamics
SERGIU HART c© 2008 – p. 10
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
SERGIU HART c© 2008 – p. 10
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
RESULT:
SERGIU HART c© 2008 – p. 10
Dynamics
FACT:
There are no general, natural dynamicsleading to Nash equilibrium
RESULT:
There cannot be general, natural dynamicsleading to Nash equilibrium
SERGIU HART c© 2008 – p. 10
Correlated equilibrium
SERGIU HART c© 2008 – p. 11
Correlated equilibrium
CORRELATED EQUILIBRIUM
Aumann 1974SERGIU HART c© 2008 – p. 11
Correlated equilibrium
CORRELATED EQUILIBRIUM :
Nash equilibrium when players receivepayoff-irrelevant information before the game
Aumann 1974SERGIU HART c© 2008 – p. 11
Correlated equilibrium
CORRELATED EQUILIBRIUM :
Nash equilibrium when players receivepayoff-irrelevant information before the game
There are general, natural dynamicsleading to correlated equilibria
SERGIU HART c© 2008 – p. 11
Correlated equilibrium
CORRELATED EQUILIBRIUM :
Nash equilibrium when players receivepayoff-irrelevant information before the game
There are general, natural dynamicsleading to correlated equilibria
"Law of Conservation of Coordination"
SERGIU HART c© 2008 – p. 11
Correlated equilibrium
CORRELATED EQUILIBRIUM :
Nash equilibrium when players receivepayoff-irrelevant information before the game
There are general, natural dynamicsleading to correlated equilibria
"Law of Conservation of Coordination":
There must be some "coordination" –either in the solution concept,or in the dynamic
SERGIU HART c© 2008 – p. 11
HAPPY BIRTHDAY, JOHN!
SERGIU HART c© 2008 – p. 12
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