Dynamics and Equilibria - Einstein Institute of Mathematics @ The

Dynamics and Equilibria
Sergiu Hart
Presidential Address, GAMES 2008 (July 2008)
Revised and Expanded (November 2009)
Revised (2010, 2011, 2012, 2013)
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S ERGIU HART °
DYNAMICS AND E QUILIBRIA
Sergiu Hart
Center for the Study of Rationality
Dept of Economics Dept of Mathematics
The Hebrew University of Jerusalem
[email protected]
http://www.ma.huji.ac.il/hart
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S ERGIU HART °
Papers
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S ERGIU HART °
Papers
Hart and Mas-Colell, Econometrica 2000
Hart and Mas-Colell, J Econ Theory 2001
Hart and Mas-Colell, Amer Econ Rev 2003
Hart, Econometrica 2005
Hart and Mas-Colell, Games Econ Behav 2006
Hart and Mansour, Games Econ Behav 2010
Hart, Games Econ Behav 2011
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S ERGIU HART °
Papers
Hart and Mas-Colell, Econometrica 2000
Hart and Mas-Colell, J Econ Theory 2001
Hart and Mas-Colell, Amer Econ Rev 2003
Hart, Econometrica 2005
Hart and Mas-Colell, Games Econ Behav 2006
Hart and Mansour, Games Econ Behav 2010
Hart, Games Econ Behav 2011
http://www.ma.huji.ac.il/hart
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Book
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Nash Equilibrium
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S ERGIU HART °
Nash Equilibrium
John Nash, Ph.D. Dissertation, Princeton 1950
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Nash Equilibrium
E QUILIBRIUM
POINT :
John Nash, Ph.D. Dissertation, Princeton 1950
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Nash Equilibrium
E QUILIBRIUM
POINT :
"Each player’s strategy is optimal
against those of the others."
John Nash, Ph.D. Dissertation, Princeton 1950
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Dynamics
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S ERGIU HART °
Dynamics
FACT
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"general"
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"general" : in all games
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"general" : in all games
rather than: in specific classes of games
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"general" : in all games
rather than: in specific classes of games:
two-person zero-sum games
two-person potential games
supermodular games
...
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"leading to Nash equilibrium"
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"leading to Nash equilibrium" :
at a Nash equilibrium (or close to it)
from some time on
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural"
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural" :
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural" :
adaptive (reacting, improving, ...)
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural" :
adaptive (reacting, improving, ...)
simple and efficient
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural" :
adaptive (reacting, improving, ...)
simple and efficient:
computation (performed at each step)
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural" :
adaptive (reacting, improving, ...)
simple and efficient:
computation (performed at each step)
time (how long to reach equilibrium)
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading 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)
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading 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 rationality
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S ERGIU HART °
Dynamics
Dynamics that are
NOT
"natural" :
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S ERGIU HART °
Dynamics
Dynamics that are
NOT
"natural" :
exhaustive search
(deterministic or stochastic)
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Exhaustive Search
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S ERGIU HART °
Exhaustive Search
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S ERGIU HART °
Exhaustive Search
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S ERGIU HART °
Exhaustive Search
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S ERGIU HART °
Exhaustive Search
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S ERGIU HART °
Exhaustive Search
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S ERGIU HART °
Exhaustive Search
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S ERGIU HART °
Exhaustive Search
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Exhaustive Search
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Einstein’s Manuscript
Albert Einstein, 1912
On the Special Theory of Relativity (manuscript)
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Dynamics
Dynamics that are
NOT
"natural" :
exhaustive search
(deterministic or stochastic)
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S ERGIU HART °
Dynamics
Dynamics that are
NOT
"natural" :
exhaustive search
(deterministic or stochastic)
using a mediator
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S ERGIU HART °
Dynamics
Dynamics that are
NOT
"natural" :
exhaustive search
(deterministic or stochastic)
using a mediator
broadcasting the private information
and then performing joint computation
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S ERGIU HART °
Dynamics
Dynamics that are
NOT
"natural" :
exhaustive search
(deterministic or stochastic)
using a mediator
broadcasting the private information
and then performing joint computation
fully rational learning
(prior beliefs on the strategies of the
opponents, Bayesian updating, optimization)
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural":
adaptive
simple and efficient:
computation (performed at each step)
time (how long to reach equilibrium)
information (of each player)
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S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural":
adaptive
simple and efficient:
computation (performed at each step)
time (how long to reach equilibrium)
information (of each player)
c 2008 – p. 13
S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural":
adaptive
simple and efficient:
computation (performed at each step)
time (how long to reach equilibrium)
information (of each player)
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S ERGIU HART °
Natural Dynamics: Information
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S ERGIU HART °
Natural Dynamics: Information
Each player knows only his own payoff
(utility) function
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S ERGIU HART °
Natural Dynamics: Information
Each player knows only his own payoff
(utility) function
(does not know the payoff functions
of the other players)
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S ERGIU HART °
Natural Dynamics: Information
UNCOUPLED DYNAMICS :
Each player knows only his own payoff
(utility) function
(does not know the payoff functions
of the other players)
Hart and Mas-Colell, AER 2003
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S ERGIU HART °
Natural Dynamics: Information
UNCOUPLED DYNAMICS :
Each player knows only his own payoff
(utility) function
(does not know the payoff functions
of the other players)
(privacy-preserving, decentralized, distributed ...)
Hart and Mas-Colell, AER 2003
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S ERGIU HART °
Games
N -person game in strategic (normal) form:
Players
i = 1, 2, ..., N
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S ERGIU HART °
Games
N -person game in strategic (normal) form:
Players
i = 1, 2, ..., N
For each player i: Actions
ai in Ai
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S ERGIU HART °
Games
N -person game in strategic (normal) form:
Players
i = 1, 2, ..., N
For each player i: Actions
ai in Ai
For each player i: Payoffs (utilities)
ui (a) ≡ ui (a1 , a2 , ..., aN )
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S ERGIU HART °
Dynamics
Time
t = 1, 2, ...
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S ERGIU HART °
Dynamics
Time
t = 1, 2, ...
At period t each player i chooses an action
ait in Ai
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S ERGIU HART °
Dynamics
Time
t = 1, 2, ...
At period t each player i chooses an action
ait in Ai
according to a probability distribution
σ it in ∆(Ai )
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S ERGIU HART °
Dynamics
Fix the set of players 1, 2, ..., N and
their action spaces A1 , A2 , ..., AN
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S ERGIU HART °
Dynamics
Fix the set of players 1, 2, ..., N and
their action spaces A1 , A2 , ..., AN
A general dynamic:
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S ERGIU HART °
Dynamics
Fix the set of players 1, 2, ..., N and
their action spaces A1 , A2 , ..., AN
A general dynamic:
σ it ≡ σ it ( HISTORY ;
GAME )
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S ERGIU HART °
Dynamics
Fix the set of players 1, 2, ..., N and
their action spaces A1 , A2 , ..., AN
A general dynamic:
σ it ≡ σ it ( HISTORY ;
GAME )
≡ σ it ( HISTORY ; u1 , ..., ui , ..., uN )
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S ERGIU HART °
Uncoupled Dynamics
Fix the set of players 1, 2, ..., N and
their action spaces A1 , A2 , ..., AN
A general dynamic:
σ it ≡ σ it ( HISTORY ;
GAME )
≡ σ it ( HISTORY ; u1 , ..., ui , ..., uN )
An
UNCOUPLED
dynamic:
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S ERGIU HART °
Uncoupled Dynamics
Fix the set of players 1, 2, ..., N and
their action spaces A1 , A2 , ..., AN
A general dynamic:
σ it ≡ σ it ( HISTORY ;
GAME )
≡ σ it ( HISTORY ; u1 , ..., ui , ..., uN )
An
UNCOUPLED
dynamic:
σ it ≡ σ it ( HISTORY ; ui )
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S ERGIU HART °
Uncoupled Dynamics
Simplest uncoupled dynamics
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S ERGIU HART °
Uncoupled Dynamics
Simplest uncoupled dynamics:
σ it ≡ f i (at−1 ; ui )
where at−1 = (a1t−1 , a2t−1 , ..., aN
t−1 ) ∈ A
are the actions of all the players
in the previous period
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S ERGIU HART °
Uncoupled Dynamics
Simplest uncoupled dynamics:
σ it ≡ f i (at−1 ; ui )
where at−1 = (a1t−1 , a2t−1 , ..., aN
t−1 ) ∈ A
are the actions of all the players
in the previous period
Only last period matters (“1-recall” )
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S ERGIU HART °
Uncoupled Dynamics
Simplest uncoupled dynamics:
σ it ≡ f i (at−1 ; ui )
where at−1 = (a1t−1 , a2t−1 , ..., aN
t−1 ) ∈ A
are the actions of all the players
in the previous period
Only last period matters (“1-recall” )
Time t does not matter (“stationary” )
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S ERGIU HART °
Impossibility
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S ERGIU HART °
Impossibility
Theorem. There are
with 1-recall
NO
uncoupled dynamics
σ it ≡ f i (at−1 ; ui )
that yield almost sure convergence of play to
pure Nash equilibria of the stage game in all
games where such equilibria exist.
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S ERGIU HART °
Impossibility
Theorem. There are
with 1-recall
NO
uncoupled dynamics
σ it ≡ f i (at−1 ; ui )
that yield almost sure convergence of play to
pure Nash equilibria of the stage game in all
games where such equilibria exist.
Hart and Mas-Colell, GEB 2006
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S ERGIU HART °
Proof
Consider the following two-person game, which
has a unique pure Nash equilibrium
C1
C2
C3
1,0 0,1
R2 0,1 1,0
R3 0,1 0,1
1,0
1,0
1,1
R1
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S ERGIU HART °
Proof
Consider the following two-person game, which
has a unique pure Nash equilibrium (R3,C3)
C1
C2
C3
1,0 0,1
R2 0,1 1,0
R3 0,1 0,1
1,0
1,0
1,1
R1
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S ERGIU HART °
Proof
Consider the following two-person game, which
has a unique pure Nash equilibrium (R3,C3)
C1
C2
C3
1,0 0,1
R2 0,1 1,0
R3 0,1 0,1
1,0
1,0
1,1
R1
Assume by way of contradiction that we are
given an uncoupled, 1-recall, stationary dynamic
that yields almost sure convergence to pure
Nash equilibria when these exist
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R1
R2
R3
C1
C2
C3
1,0
0,1
0,1
0,1
1,0
0,1
1,0
1,0
1,1
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R OWENA is best replying at (R1,C1)
R1
R2
R3
C1
C2
C3
1,0
0,1
0,1
0,1
1,0
0,1
1,0
1,0
1,1
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R OWENA is best replying at (R1,C1)
⇒ ROWENA will play R1 also at t
R1
R2
R3
C1
C2
C3
1,0
0,1
0,1
0,1
1,0
0,1
1,0
1,0
1,1
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R OWENA is best replying at (R1,C1)
⇒ ROWENA will play R1 also at t
Proof:
Change the payoff function of C OLIN so
that (R1,C1) is the unique pure Nash eq.
R1
R2
R3
C1
C2
C3
1,0
0,1
0,1
0,1
1,0
0,1
1,0
1,0
1,1
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R OWENA is best replying at (R1,C1)
⇒ ROWENA will play R1 also at t
Proof:
Change the payoff function of C OLIN so
that (R1,C1) is the unique pure Nash eq.
R1
R2
R3
C1
C2
C3
1,1
0,1
0,1
0,1
1,0
0,1
1,0
1,0
1,0
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R OWENA is best replying at (R1,C1)
⇒ ROWENA will play R1 also at t
Proof:
Change the payoff function of C OLIN so
that (R1,C1) is the unique pure Nash eq.
In the new game, R OWENA must play R1
after (R1,C1) (by 1-recall, stationarity, and
a.s. convergence to the pure Nash eq.)
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R OWENA is best replying at (R1,C1)
⇒ ROWENA will play R1 also at t
Proof:
Change the payoff function of C OLIN so
that (R1,C1) is the unique pure Nash eq.
In the new game, R OWENA must play R1
after (R1,C1) (by 1-recall, stationarity, and
a.s. convergence to the pure Nash eq.)
By uncoupledness, the same holds in the
original game
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S ERGIU HART °
Proof
Suppose the play at time t−1 is (R1,C1)
R OWENA is best replying at (R1,C1)
⇒ ROWENA will play R1 also at t
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S ERGIU HART °
Proof
R OWENA is best replying at t−1
⇒ ROWENA will play the same action at t
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S ERGIU HART °
Proof
Similarly for C OLIN:
A player who is best replying cannot switch
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S ERGIU HART °
Proof
Similarly for C OLIN:
A player who is best replying cannot switch
C1
C2
l
l 1,0 ↔
l 0,1 l
C3
R1
1,0 ↔ 0,1
1,0 ↔
R2
0,1
1,0 ↔
R3
0,1
1,1 ↔
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S ERGIU HART °
Proof
Similarly for C OLIN:
A player who is best replying cannot switch
C1
C2
l
l 1,0 ↔
l 0,1 l
C3
R1
1,0 ↔ 0,1
1,0 ↔
R2
0,1
1,0 ↔
R3
0,1
1,1 ↔
c 2008 – p. 22
S ERGIU HART °
Proof
Similarly for C OLIN:
A player who is best replying cannot switch
C1
⇒
C2
R1
1,0 ↔ 0,1
1,0 ↔
R2
0,1
1,0 ↔
R3
0,1
(R3,C3)
l
l 1,0 ↔
l 0,1 l
C3
1,1 ↔
cannot be reached
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S ERGIU HART °
Proof
Similarly for C OLIN:
A player who is best replying cannot switch
C1
⇒
C2
l
l 1,0 ↔
l 0,1 l
C3
R1
1,0 ↔ 0,1
1,0 ↔
R2
0,1
1,0 ↔
R3
0,1
1,1 ↔
cannot be reached
(unless we start there)
(R3,C3)
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S ERGIU HART °
2-Recall
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2-Recall: Possibility
Theorem. THERE
with 2- RECALL
EXIST
uncoupled dynamics
σ it ≡ f i (at−2 , at−1 ; ui )
that yield almost sure convergence of play to
pure Nash equilibria of the stage game in every
game where such equilibria exist.
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S ERGIU HART °
Possibility
Define the strategy of each player i as follows:
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S ERGIU HART °
Possibility
Define the strategy of each player i as follows:
IF :
Everyone played the same in the previous
two periods:
at−2 = at−1 = a;
and
Player i best replied:
THEN :
ai ∈ BRi (a−i ; ui )
At t player i plays ai again:
ait = ai
c 2008 – p. 24
S ERGIU HART °
Possibility
Define the strategy of each player i as follows:
IF :
Everyone played the same in the previous
two periods:
at−2 = at−1 = a;
and
Player i best replied:
ai ∈ BRi (a−i ; ui )
ait = ai
THEN :
At t player i plays ai again:
ELSE :
At t player i randomizes uniformly over Ai
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S ERGIU HART °
Possibility
"Good":
c 2008 – p. 25
S ERGIU HART °
Possibility
"Good":
simple
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S ERGIU HART °
Possibility
"Good":
simple
"Bad":
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S ERGIU HART °
Possibility
"Good":
simple
"Bad":
exhaustive search
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S ERGIU HART °
Possibility
"Good":
simple
"Bad":
exhaustive search
all players must use it
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S ERGIU HART °
Possibility
"Good":
simple
"Bad":
exhaustive search
all players must use it
takes a long time
c 2008 – p. 25
S ERGIU HART °
Possibility
"Good":
simple
"Bad":
exhaustive search
all players must use it
takes a long time
"Ugly": ...
c 2008 – p. 25
S ERGIU HART °
Possibility
"Good":
simple
"Bad":
exhaustive search
all players must use it
takes a long time
"Ugly": ...
c 2008 – p. 25
S ERGIU HART °
Possibility
"Good":
simple
"Bad":
exhaustive search
all players must use it
takes a long time
"Ugly": ...
c 2008 – p. 25
S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural":
adaptive
simple and efficient:
computation
time
information
c 2008 – p. 26
S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural":
adaptive
simple and efficient:
computation
time
information: uncoupledness
X
c 2008 – p. 26
S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural":
adaptive
simple and efficient:
computation: finite recall X
time
information: uncoupledness X
c 2008 – p. 26
S ERGIU HART °
Dynamics
FACT
There are no general, natural dynamics
leading to Nash equilibrium
"natural":
adaptive
simple and efficient:
computation: finite recall X
time to reach equilibrium ?
information: uncoupledness X
c 2008 – p. 26
S ERGIU HART °
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM ?
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S ERGIU HART °
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM ?
Estimate the number of time periods it takes until
a Nash equilibrium is reached
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S ERGIU HART °
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM ?
Estimate the number of time periods it takes until
a Nash equilibrium is reached
How?
c 2008 – p. 27
S ERGIU HART °
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM ?
Estimate the number of time periods it takes until
a Nash equilibrium is reached
How?
An uncoupled dynamic
≈
A distributed computational procedure
c 2008 – p. 27
S ERGIU HART °
Natural Dynamics: Time
HOW LONG TO EQUILIBRIUM ?
Estimate the number of time periods it takes until
a Nash equilibrium is reached
How?
An uncoupled dynamic
≈
A distributed computational procedure
⇒ COMMUNICATION COMPLEXITY
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S ERGIU HART °
Communication Complexity
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S ERGIU HART °
Communication Complexity
Distributed computational procedure
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S ERGIU HART °
Communication Complexity
Distributed computational procedure
START : Each participant has some private
[ INPUTS ]
information
c 2008 – p. 28
S ERGIU HART °
Communication Complexity
Distributed computational procedure
START : Each participant has some private
[ INPUTS ]
information
COMMUNICATION : Messages are
transmitted between the participants
c 2008 – p. 28
S ERGIU HART °
Communication Complexity
Distributed computational procedure
START : Each participant has some private
[ INPUTS ]
information
COMMUNICATION : Messages are
transmitted between the participants
END : All participants reach agreement on
the result
c 2008 – p. 28
S ERGIU HART °
Communication Complexity
Distributed computational procedure
START : Each participant has some private
[ INPUTS ]
information
COMMUNICATION : Messages are
transmitted between the participants
END : All participants reach agreement on
the result
c 2008 – p. 28
S ERGIU HART °
Communication Complexity
Distributed computational procedure
START : Each participant has some private
[ INPUTS ]
information
COMMUNICATION : Messages are
transmitted between the participants
END : All participants reach agreement on
[ OUTPUT ]
the result
c 2008 – p. 28
S ERGIU HART °
Communication Complexity
Distributed computational procedure
START : Each participant has some private
[ INPUTS ]
information
COMMUNICATION : Messages are
transmitted between the participants
END : All participants reach agreement on
[ OUTPUT ]
the result
=
the minimal number of rounds needed
COMMUNICATION COMPLEXITY
c 2008 – p. 28
S ERGIU HART °
Communication Complexity
Distributed computational procedure
START : Each participant has some private
[ INPUTS ]
information
COMMUNICATION : Messages are
transmitted between the participants
END : All participants reach agreement on
[ OUTPUT ]
the result
=
the minimal number of rounds needed
COMMUNICATION COMPLEXITY
Yao 1979, Kushilevitz and Nisan 1997
c 2008 – p. 28
S ERGIU HART °
How Long to Equilibrium
c 2008 – p. 29
S ERGIU HART °
How Long to Equilibrium
Uncoupled dynamic leading to Nash
equilibria
c 2008 – p. 29
S ERGIU HART °
How Long to Equilibrium
Uncoupled dynamic leading to Nash
equilibria
START: Each player knows his own payoff
function
[ INPUTS ]
c 2008 – p. 29
S ERGIU HART °
How Long to Equilibrium
Uncoupled dynamic leading to Nash
equilibria
START: Each player knows his own payoff
function
[ INPUTS ]
COMMUNICATION : The actions played are
commonly observed
c 2008 – p. 29
S ERGIU HART °
How Long to Equilibrium
Uncoupled dynamic leading to Nash
equilibria
START: Each player knows his own payoff
function
[ INPUTS ]
COMMUNICATION : The actions played are
commonly observed
END : All players play a Nash equilibrium
[ OUTPUT ]
c 2008 – p. 29
S ERGIU HART °
How Long to Equilibrium
Uncoupled dynamic leading to Nash
equilibria
START: Each player knows his own payoff
function
[ INPUTS ]
COMMUNICATION : The actions played are
commonly observed
END : All players play a Nash equilibrium
[ OUTPUT ]
COMMUNICATION COMPLEXITY =
the minimal number of rounds needed
c 2008 – p. 29
S ERGIU HART °
How Long to Equilibrium
Uncoupled dynamic leading to Nash
equilibria
START: Each player knows his own payoff
function
[ INPUTS ]
COMMUNICATION : The actions played are
commonly observed
END : All players play a Nash equilibrium
[ OUTPUT ]
COMMUNICATION COMPLEXITY =
the minimal number of rounds needed
Conitzer and Sandholm 2004
c 2008 – p. 29
S ERGIU HART °
How Long to Equilibrium
An uncoupled dynamic leading to Nash
equilibria is TIME - EFFICIENT if
c 2008 – p. 30
S ERGIU HART °
How Long to Equilibrium
An uncoupled dynamic leading to Nash
equilibria is TIME - EFFICIENT if
the TIME IT TAKES
c 2008 – p. 30
S ERGIU HART °
How Long to Equilibrium
An uncoupled dynamic leading to Nash
equilibria is TIME - EFFICIENT if
the TIME IT TAKES is POLYNOMIAL in the
number of players
c 2008 – p. 30
S ERGIU HART °
How Long to Equilibrium
An uncoupled dynamic leading to Nash
equilibria is TIME - EFFICIENT if
the TIME IT TAKES is POLYNOMIAL in the
number of players (rather than: exponential)
c 2008 – p. 30
S ERGIU HART °
How Long to Equilibrium
An uncoupled dynamic leading to Nash
equilibria is TIME - EFFICIENT if
the TIME IT TAKES is POLYNOMIAL in the
number of players (rather than: exponential)
Theorem. There are NO TIME - EFFICIENT
uncoupled dynamics that reach a pure Nash
equilibrium in all games where such equilibria
exist.
c 2008 – p. 30
S ERGIU HART °
How Long to Equilibrium
An uncoupled dynamic leading to Nash
equilibria is TIME - EFFICIENT if
the TIME IT TAKES is POLYNOMIAL in the
number of players (rather than: exponential)
Theorem. There are NO TIME - EFFICIENT
uncoupled dynamics that reach a pure Nash
equilibrium in all games where such equilibria
exist.
Hart and Mansour, GEB 2010
c 2008 – p. 30
S ERGIU HART °
How Long to Equilibrium
An uncoupled dynamic leading to Nash
equilibria is TIME - EFFICIENT if
the TIME IT TAKES is POLYNOMIAL in the
number of players (rather than: exponential)
Theorem. There are NO TIME - EFFICIENT
uncoupled dynamics that reach a pure Nash
equilibrium in all games where such equilibria
exist.
In fact: exponential, like exhaustive search
Hart and Mansour, GEB 2010
c 2008 – p. 30
S ERGIU HART °
How Long to Equilibrium
Intuition:
c 2008 – p. 31
S ERGIU HART °
How Long to Equilibrium
Intuition:
different games have different equilibria
c 2008 – p. 31
S ERGIU HART °
How Long to Equilibrium
Intuition:
different games have different equilibria
the dynamic procedure must distinguish
between them
c 2008 – p. 31
S ERGIU HART °
How Long to Equilibrium
Intuition:
different games have different equilibria
the dynamic procedure must distinguish
between them
no single player can do so by himself
c 2008 – p. 31
S ERGIU HART °
Dynamics and Nash Equilibrium
c 2008 – p. 32
S ERGIU HART °
Dynamics and Nash Equilibrium
FACT
There are NO general, natural dynamics
leading to Nash equilibrium
c 2008 – p. 32
S ERGIU HART °
Dynamics and Nash Equilibrium
FACT
There are NO general, natural dynamics
leading to Nash equilibrium
RESULT
There
general, natural dynamics
leading to Nash equilibrium
CANNOT BE
c 2008 – p. 32
S ERGIU HART °
Dynamics and Nash Equilibrium
RESULT
There
general, natural dynamics
leading to Nash equilibrium
CANNOT BE
c 2008 – p. 33
S ERGIU HART °
Dynamics and Nash Equilibrium
RESULT
There
general, natural dynamics
leading to Nash equilibrium
CANNOT BE
Perhaps we are asking too much?
c 2008 – p. 33
S ERGIU HART °
Dynamics and Nash Equilibrium
RESULT
There
general, natural dynamics
leading to Nash equilibrium
CANNOT BE
Perhaps we are asking too much?
For instance, the size of the data (the payoff
functions) is exponential rather than
polynomial in the number of players
c 2008 – p. 33
S ERGIU HART °
Correlated Equilibrium
c 2008 – p. 34
S ERGIU HART °
Correlated Equilibrium
C ORRELATED
EQUILIBRIUM
Aumann, JME 1974
c 2008 – p. 34
S ERGIU HART °
Correlated Equilibrium
C ORRELATED
EQUILIBRIUM
:
Nash equilibrium when players receive
payoff-irrelevant information
before playing the game
Aumann, JME 1974
c 2008 – p. 34
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when players receive payoff-irrelevant signals
before playing the game
c 2008 – p. 35
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when players receive payoff-irrelevant signals
before playing the game
Examples:
c 2008 – p. 35
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when players receive payoff-irrelevant signals
before playing the game
Examples:
Independent signals
c 2008 – p. 35
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when players receive payoff-irrelevant signals
before playing the game
Examples:
Independent signals
⇔
Nash equilibrium
c 2008 – p. 35
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when players receive payoff-irrelevant signals
before playing the game
Examples:
Independent signals ⇔ Nash equilibrium
Public signals (“sunspots”)
c 2008 – p. 35
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when players receive payoff-irrelevant signals
before playing the game
Examples:
Independent signals ⇔ Nash equilibrium
Public signals (“sunspots”) ⇔
convex combinations of Nash equilibria
c 2008 – p. 35
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when players receive payoff-irrelevant signals
before playing the game
Examples:
Independent signals ⇔ Nash equilibrium
Public signals (“sunspots”) ⇔
convex combinations of Nash equilibria
Butterflies play the Chicken Game
(“Speckled Wood” Pararge aegeria)
c 2008 – p. 35
S ERGIU HART °
Correlated Equilibria
"Chicken" game
LEAVE
STAY
LEAVE
STAY
5, 5
6, 3
3, 6
0, 0
c 2008 – p. 36
S ERGIU HART °
Correlated Equilibria
"Chicken" game
LEAVE
STAY
LEAVE
STAY
5, 5
6, 3
3, 6
0, 0
a Nash equilibrium
c 2008 – p. 36
S ERGIU HART °
Correlated Equilibria
"Chicken" game
LEAVE
STAY
LEAVE
STAY
5, 5
6, 3
3, 6
0, 0
another Nash equilibriumi
c 2008 – p. 36
S ERGIU HART °
Correlated Equilibria
"Chicken" game
LEAVE
STAY
LEAVE
STAY
5, 5
6, 3
3, 6
0, 0
L
0 1/2
1/2 0
a (publicly) correlated equilibrium
c 2008 – p. 36
S ERGIU HART °
Correlated Equilibria
"Chicken" game
LEAVE
STAY
LEAVE
STAY
5, 5
6, 3
3, 6
0, 0
L
L
S
S
1/3 1/3
1/3 0
another correlated equilibrium
after signal
L
play
LEAVE
after signal
S
play
STAY
c 2008 – p. 36
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when the players receive payoff-irrelevant signals
before playing the game (Aumann 1974)
Examples:
Independent signals ⇔ Nash equilibrium
Public signals (“sunspots”) ⇔
convex combinations of Nash equilibria
Butterflies play the Chicken Game
(“Speckled Wood” Pararge aegeria)
c 2008 – p. 37
S ERGIU HART °
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibrium
when the players receive payoff-irrelevant signals
before playing the game (Aumann 1974)
Examples:
Independent signals ⇔ Nash equilibrium
Public signals (“sunspots”) ⇔
convex combinations of Nash equilibria
Butterflies play the Chicken Game
(“Speckled Wood” Pararge aegeria)
Boston Celtics’ front line
c 2008 – p. 37
S ERGIU HART °
Correlated Equilibrium
Signals (public, correlated) are unavoidable
c 2008 – p. 38
S ERGIU HART °
Correlated Equilibrium
Signals (public, correlated) are unavoidable
Common Knowledge of Rationality ⇔
Correlated Equilibrium (Aumann 1987)
c 2008 – p. 38
S ERGIU HART °
Correlated Equilibrium
Signals (public, correlated) are unavoidable
Common Knowledge of Rationality ⇔
Correlated Equilibrium (Aumann 1987)
A joint distribution z is a correlated equilibrium
X
s−i
⇔
u(j, s−i )z(j, s−i ) ≥
X
u(k, s−i )z(j, s−i )
s−i
for all i ∈ N and all j, k ∈ S i
c 2008 – p. 38
S ERGIU HART °
Dynamics & Correlated Equilibria
c 2008 – p. 39
S ERGIU HART °
Dynamics & Correlated Equilibria
RESULT
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
c 2008 – p. 39
S ERGIU HART °
Dynamics & Correlated Equilibria
RESULT
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
Regret Matching
Hart and Mas-Colell, Ec’ca 2000
c 2008 – p. 39
S ERGIU HART °
Dynamics & Correlated Equilibria
RESULT
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
Regret Matching
General regret-based dynamics
Hart and Mas-Colell, Ec’ca 2000, JET 2001
c 2008 – p. 39
S ERGIU HART °
Regret Matching
c 2008 – p. 40
S ERGIU HART °
Regret Matching
" REGRET ": the increase in past payoff, if any,
if a different action would have been used
c 2008 – p. 40
S ERGIU HART °
Regret Matching
" REGRET ": the increase in past payoff, if any,
if a different action would have been used
" MATCHING ": switching to a different action
with a probability that is proportional to the
regret for that action
c 2008 – p. 40
S ERGIU HART °
Dynamics & Correlated Equilibria
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
c 2008 – p. 41
S ERGIU HART °
Dynamics & Correlated Equilibria
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
"general": in all games
c 2008 – p. 41
S ERGIU HART °
Dynamics & Correlated Equilibria
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
"general": in all games
"natural":
c 2008 – p. 41
S ERGIU HART °
Dynamics & Correlated Equilibria
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
"general": in all games
"natural":
adaptive (also: close to "behavioral")
c 2008 – p. 41
S ERGIU HART °
Dynamics & Correlated Equilibria
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
"general": in all games
"natural":
adaptive (also: close to "behavioral")
simple and efficient:
computation, time, information
c 2008 – p. 41
S ERGIU HART °
Dynamics & Correlated Equilibria
THERE EXIST
leading to
general, natural dynamics
CORRELATED EQUILIBRIA
"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
c 2008 – p. 41
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 42
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 42
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 43
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 44
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 45
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 46
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 47
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 48
S ERGIU HART °
Regret Matching and Beyond
c 2008 – p. 49
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 50
S ERGIU HART °
Dynamics and Equilibrium
N ASH EQUILIBRIUM:
a fixed-point of a non-linear map
c 2008 – p. 50
S ERGIU HART °
Dynamics and Equilibrium
N ASH EQUILIBRIUM:
a fixed-point of a non-linear map
C ORRELATED EQUILIBRIUM:
a solution of finitely many linear inequalities
c 2008 – p. 50
S ERGIU HART °
Dynamics and Equilibrium
N ASH EQUILIBRIUM:
a fixed-point of a non-linear map
C ORRELATED EQUILIBRIUM:
a solution of finitely many linear inequalities
set-valued fixed-point (curb sets)?
c 2008 – p. 50
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 51
S ERGIU HART °
Dynamics and Equilibrium
"L AW
OF
C ONSERVATION
OF
C OORDINATION":
c 2008 – p. 51
S ERGIU HART °
Dynamics and Equilibrium
"L AW
OF
C ONSERVATION
There must be some
OF
C OORDINATION":
COORDINATION
—
c 2008 – p. 51
S ERGIU HART °
Dynamics and Equilibrium
"L AW
OF
C ONSERVATION
There must be some
either in the
OF
C OORDINATION":
COORDINATION
EQUILIBRIUM
—
notion,
c 2008 – p. 51
S ERGIU HART °
Dynamics and Equilibrium
"L AW
OF
C ONSERVATION
There must be some
either in the
OF
C OORDINATION":
COORDINATION
EQUILIBRIUM
or in the
—
notion,
DYNAMIC
c 2008 – p. 51
S ERGIU HART °
The "Program"
c 2008 – p. 52
S ERGIU HART °
The "Program"
A . Demarcate the BORDER between
c 2008 – p. 52
S ERGIU HART °
The "Program"
A . Demarcate the BORDER between
classes of dynamics where
convergence to equilibria
CAN be obtained
c 2008 – p. 52
S ERGIU HART °
The "Program"
A . Demarcate the BORDER between
classes of dynamics where
convergence to equilibria
CAN be obtained, and
classes of dynamics where
convergence to equilibria
CANNOT be obtained
c 2008 – p. 52
S ERGIU HART °
The "Program"
A . Demarcate the BORDER between
classes of dynamics where
convergence to equilibria
CAN be obtained, and
classes of dynamics where
convergence to equilibria
CANNOT be obtained
B . Find NATURAL dynamics for the various
equilibrium concepts
c 2008 – p. 52
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 53
S ERGIU HART °
Dynamics and Equilibrium
Nash Equilibrium
c 2008 – p. 53
S ERGIU HART °
Dynamics and Equilibrium
Nash Equilibrium
" DYNAMICALLY DIFFICULT "
c 2008 – p. 53
S ERGIU HART °
Dynamics and Equilibrium
Nash Equilibrium
" DYNAMICALLY DIFFICULT "
Correlated Equilibrium
c 2008 – p. 53
S ERGIU HART °
Dynamics and Equilibrium
Nash Equilibrium
" DYNAMICALLY DIFFICULT "
Correlated Equilibrium
" DYNAMICALLY EASY "
c 2008 – p. 53
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
Dynamics and Equilibrium
c 2008 – p. 54
S ERGIU HART °
My Game Theory
c 2008 – p. 55
S ERGIU HART °
My Game Theory
c 2008 – p. 55
S ERGIU HART °
My Game Theory
c 2008 – p. 55
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
c 2008 – p. 55
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 55
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °
My Game Theory
INSIGHTS,
IDEAS,
CONCEPTS
FORMAL
MODELS
c 2008 – p. 56
S ERGIU HART °