Game Dynamics and Equilibrium

8/11/2019 Game Dynamics and Equilibrium

1/230

Dynamics and EquilibriaSergiu Hart

Presidential Address, GAMES 2008 (July 2008)

Revised and Expanded (November 2009)

Revised (July 2010)

SERGIUHART c 2008 p. 1


2/230

DYNAMICS AND EQUILIBRIA

Sergiu HartCenter for the Study of Rationality

Dept of Economics Dept of MathematicsThe Hebrew University of Jerusalem

[email protected]

http://www.ma.huji.ac.il/hart



3/230

Papers



4/230

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



5/230

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


6/230

Nash Equilibrium



7/230

Nash Equilibrium

John Nash, Ph.D. Dissertation, Princeton 1950SERGIUHART c 2008 p. 4


8/230

Nash Equilibrium

EQUILIBRIUM POINT:



9/230

Nash Equilibrium

EQUILIBRIUM POINT:

"Each players strategy is optimal

against those of the others."



10/230

Dynamics



11/230

Dynamics

FACT



12/230

Dynamics

FACT

There are no general, natural dynamicsleading to Nash equilibrium



13/230

Dynamics

FACT

There are nogeneral, natural dynamicsleading to Nash equilibrium

"general"


i


14/230

Dynamics

FACT


"general": in all games


D i


15/230

Dynamics

FACT


"general": in all gamesrather than: in specific classes of games


D i


16/230

Dynamics

FACT


"general": in all gamesrather than: in specific classes of games:

two-person zero-sum gamestwo-person potential gamessupermodular games. . .


D i


17/230

Dynamics

FACT



D i


18/230

Dynamics

FACT


"leading to Nash equilibrium"


D i


19/230

Dynamics

FACT


"leading to Nash equilibrium":at a Nash equilibrium (or close to it)from some time on


D i


20/230

Dynamics

FACT



Dynamics


21/230

Dynamics

FACT

There are no general,naturaldynamicsleading to Nash equilibrium

"natural"


Dynamics


22/230

Dynamics

FACT


"natural":


Dynamics


23/230

Dynamics

FACT


"natural":

adaptive(reacting, improving, ...)


Dynamics


24/230

Dynamics

FACT


"natural":

adaptive(reacting, improving, ...)

simple and efficient


Dynamics


25/230

Dynamics

FACT


"natural":adaptive(reacting, improving, ...)

simple and efficient:computation(performed at each step)


Dynamics


26/230

Dynamics

FACT



simple and efficient:computation(performed at each step)time(how long to reach equilibrium)


Dynamics


27/230

Dynamics

FACT




information(of each player)SERGIUHART c 2008 p. 7

Dynamics


28/230

Dynamics

FACT




information(of each player)bounded rationalitySERGIUHART c 2008 p. 7

Dynamics


29/230

Dynamics

Dynamics that are NOT "natural":


Dynamics


30/230

Dynamics


exhaustive search

(deterministic or stochastic)


Exhaustive Search


31/230

Exhaustive Search


Exhaustive Search


32/230

Exhaustive Search


Exhaustive Search


33/230

Exhaustive Search


Exhaustive Search


34/230

Exhaustive Search


Exhaustive Search


35/230

Exhaustive Search


Exhaustive Search


36/230

Exhaustive Search


Exhaustive Search


37/230

Exhaustive Search


Exhaustive Search


38/230

aust ve Sea c


Exhaustive Search


39/230


Einsteins Manuscript


40/230

p

Albert Einstein, 1912

On the Special Theory of Relativity (manuscript)SERGIU HART c 2008 p. 10

Dynamics


41/230

y


exhaustive search


SERGIU HART c 2008 p. 11

Dynamics


42/230


exhaustive search


using amediator


Dynamics


43/230


exhaustive search


using amediator

broadcastingthe private informationand then performingjointcomputation


Dynamics


44/230


exhaustive search


using amediator

broadcastingthe private informationand then performingjointcomputation

fully rational learning

(prior beliefs on the strategies of theopponents, Bayesian updating, optimization)


Dynamics


45/230

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)


Dynamics


46/230

FACT



"natural":adaptive



Dynamics


47/230

FACT



"natural":adaptive



Natural Dynamics: Information


48/230




49/230

Each player knowsonlyhis own payoff(utility) function




50/230


(doesnotknow the payoff functionsof the other players)




51/230

UNCOUPLED DYNAMICS:



Hart and Mas-Colell, AER 2003SERGIU HART c 2008 p. 13



52/230

UNCOUPLED DYNAMICS:



(privacy-preserving, decentralized, distributed ...)

Hart and Mas-Colell, AER 2003SERGIU HART c 2008 p. 13

Games


53/230

N-person gamein strategic (normal) form:

Playersi= 1, 2,...,N


Games


54/230



For each playeri: Actions

ai in Ai


Games


55/230



For each playeri: Actions

ai in Ai

For each playeri: Payoffs(utilities)

ui(a) ui(a1, a2,...,aN)


Dynamics


56/230

Time

t= 1, 2,...


Dynamics


57/230

Time

t= 1, 2,...

At periodteach playerichooses anaction

ait in A

i


Dynamics


58/230

Time

t= 1, 2,...

At periodteach playerichooses anaction

ait in A

i

according to a probability distribution

it in (Ai)


Dynamics


59/230

Fix the set of players1, 2,...,N andtheir action spacesA1, A2,...,AN


Dynamics


60/230


A general dynamic:


Dynamics


61/230


A general dynamic:

it i

t( HISTORY ; GAME )


Dynamics


62/230


A general dynamic:

it i

t( HISTORY ; GAME )

it( HISTORY ; u1,...,ui,...,uN )


Uncoupled Dynamics


63/230


A general dynamic:

it i

t( HISTORY ; GAME )


An UNCOUPLEDdynamic:


Uncoupled Dynamics


64/230


Ageneraldynamic:

it i

t( HISTORY ; GAME )


An UNCOUPLEDdynamic:

it it( HISTORY ; u

i )


Uncoupled Dynamics


65/230

Simplestuncoupled dynamics


Uncoupled Dynamics


66/230

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


Uncoupled Dynamics


67/230


i

t fi

(at1; ui

)whereat1 = (a

1

t1, a2

t1,...,aNt1) A


Only last period matters (1-recall)


Uncoupled Dynamics


68/230


i

t fi

(at1; ui

)whereat1 = (a

1

t1, a2

t1,...,aNt1) A


Only last period matters (1-recall)

Timetdoes not matter (stationary)


Impossibility


69/230


Impossibility


70/230

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.


Impossibility


71/230

Theorem. There are NOuncoupleddynamicswith1-recall

it fi(at1; u

i)


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


72/230

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



73/230

Proof


74/230

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


75/230

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


76/230


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


77/230



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


78/230



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


79/230





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


80/230





In the new game, ROWENAmustplay R1

after (R1,C1)(by1-recall,stationarity, anda.s. convergenceto the pure Nash eq.)


Proof


81/230





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


82/230



ROWENAwill play R1also att


Proof


83/230

ROWENAis best replying att1

ROWENAwill play the same action att


Proof


84/230

Similarly for COLIN:

A player who is best replying cannot switch


Proof


85/230



C1 C2 C3

R1 1,0

0,1

1,0

R2 0,1 1,0 1,0

R3 0,1 0,1 1,1


Proof


86/230



C1 C2 C3

R1 1,0

0,1

1,0

R2 0,1 1,0 1,0

R3 0,1 0,1 1,1


Proof


87/230



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


88/230



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)


Possibility


89/230


Possibility


90/230

Theorem. THERE EXISTuncoupleddynamicswith2-RECALL

it fi(at2, at1;ui)


pure Nash equilibria of the stage game in everygame where such equilibria exist.


Possibility


91/230

Define the strategy of each playerias follows:


Possibility


92/230


IF:Everyone played the same in the previoustwo periods: at2 =at1 =a; and

Playeribest replied: ai BRi(ai; ui)

THEN: Attplayeriplaysai again: ait =ai


Possibility


93/230


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


Possibility


94/230

"Good":


Possibility

"G d"


95/230

"Good":

simple


Possibility

"G d"


96/230

"Good":

simple

"Bad":


Possibility

"G d"


97/230

"Good":

simple

"Bad":

exhaustive search


Possibility

"G d"


98/230

"Good":

simple

"Bad":

exhaustive searchall players must use it


Possibility

"Good"


99/230

"Good":

simple

"Bad":


takes a long time


Possibility

"Good":


100/230

"Good":

simple

"Bad":


takes a long time

"Ugly": ...


Possibility

"Good":


101/230

Good :

simple

"Bad":


takes a long time

"Ugly": ...


Possibility

"Good":


102/230

Good :

simple

"Bad":


takes a long time

"Ugly": ...


Dynamics

FACT


103/230

FACT



"natural":adaptive

simple and efficient:

computationtime

information


Dynamics

FACT


104/230

FACT



"natural":adaptive


computationtime

information: uncoupledness


Dynamics

FACT


105/230

FACT



"natural":adaptive


computation: finite recalltime

information: uncoupledness


Dynamics

FACT


106/230

FACT



"natural":adaptive


computation: finite recalltimeto reach equilibrium?information: uncoupledness


Natural Dynamics: Time

HOW LONG TO EQUILIBRIUM?


107/230






108/230


Estimate the number of time periods it takes until

a Nash equilibrium is reached





109/230




How?





110/230

O O G O QU U



How?

An uncoupled dynamic

A distributed computational procedure





111/230



How?

An uncoupled dynamic

A distributed computational procedure

COMMUNICATION COMPLEXITY


Communication Complexity


112/230



Distributed computational procedure


113/230

p p





114/230

p p

START: Each participant has some private

information [INPUTS]





115/230


information [INPUTS]COMMUNICATION: Messages aretransmitted between the participants





116/230



END: All participants reach agreement onthe result





117/230



END: All participants reach agreement onthe result





118/230



END: All participants reach agreement onthe result [OUTPUT]





119/230




COMMUNICATION COMPLEXITY=the minimal number of rounds needed





120/230





Yao 1979, Kushilevitz and Nisan 1997SERGIU HART c 2008 p. 27

How Long to Equilibrium


121/230



Uncoupled dynamic leading to Nashequilibria


122/230

equilibria





123/230

equilibria

START: Each player knows his own payoff

function [INPUTS]





124/230

equilibria


function [INPUTS]

COMMUNICATION: The actions played arecommonly observed





125/230

equ b a


function [INPUTS]


END: All players play a Nash equilibrium[OUTPUT]





126/230

q


function [INPUTS]








127/230

q


function [INPUTS]




Conitzer and Sandholm 2004 SERGIU HART c 2008 p. 28


An uncoupled dynamic leading to Nashequilibria is TIME-EFFICIENT if


128/230

equilibria is TIME-EFFICIENT if





129/230

equilibria is TIME EFFICIENT ifthe TIME IT TAKES





130/230

equilibria is TIME EFFICIENT ifthe TIME IT TAKES is POLYNOMIALin thenumber of players





131/230

equilibria is TIME EFFICIENT ifthe TIME IT TAKES is POLYNOMIALin thenumber of players (rather than: exponential)





132/230

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.





133/230

qthe TIME IT TAKES is POLYNOMIALin thenumber of players (rather than: exponential)



Hart and Mansour, GEB 2010 SERGIU HART c 2008 p. 29




134/230

qthe TIME IT TAKES is POLYNOMIALin thenumber of players (rather than: exponential)



In fact:exponential, like exhaustive search

Hart and Mansour, GEB 2010 SERGIU HART c 2008 p. 29


Intuition:


135/230



Intuition:

different games have different equilibria


136/230




Intuition:



137/230


the dynamic procedure must distinguishbetween them



Intuition:



138/230


the dynamic procedure must distinguishbetween them

no single player can do so by himself


Dynamics and Nash Equilibrium


139/230



FACT


140/230

There are NOgeneral, natural dynamics




FACT


141/230

There are NOgeneral, natural dynamics


RESULT

There CANNOT BEgeneral, natural dynamics




RESULT


142/230

There CANNOT BEgeneral, natural dynamicsleading to Nash equilibrium



RESULT


143/230

There CANNOT BE

general, natural dynamicsleading to Nash equilibrium

Perhaps we are asking too much?



RESULT


144/230

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


Correlated Equilibrium


145/230



CORRELATED EQUILIBRIUM


146/230

CORRELATED EQUILIBRIUM

Aumann, JME 1974 SERGIU HART c 2008 p. 33


CORRELATED EQUILIBRIUM :


147/230

CORRELATED EQUILIBRIUM:

Nash equilibrium when players receivepayoff-irrelevant information

before playing the game

Aumann, JME 1974 SERGIU HART c 2008 p. 33


ACorrelated Equilibriumis a Nash equilibriumwhen players receive payoff-irrelevant signalsbefore playing the game


148/230






149/230


Examples:





150/230


Examples:

Independent signals





151/230


Examples:

Independent signals Nash equilibrium





152/230


Examples:

Independent signals Nash equilibriumPublic signals (sunspots)





153/230


Examples:

Independent signals Nash equilibriumPublic signals (sunspots) convex combinations of Nash equilibria



154/230

Correlated Equilibria

"Chicken" game

LEAVE STAY


155/230

S

LEAVE 5, 5 3, 6

STAY 6, 3 0, 0



156/230


157/230


"Chicken" game

LEAVE STAY


158/230

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 L S


159/230

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



ACorrelated Equilibriumis a Nash equilibriumwhen the players receive payoff-irrelevant signalsbefore playing the game (Aumann 1974)


160/230

p y g g ( )

Examples:


Public signals (sunspots)convex combinations of Nash equilibriaButterflies play the Chicken Game

(Speckled WoodPararge aegeria)



ACorrelated Equilibriumis a Nash equilibriumwhen the players receive payoff-irrelevant signalsbefore playing the game (Aumann 1974)


161/230

Examples:


Public signals (sunspots)convex combinations of Nash equilibriaButterflies play the Chicken Game

(Speckled WoodPararge aegeria)Boston Celtics front line



Signals (public, correlated) areunavoidable


162/230




Common KnowledgeofRationalityCorrelated Equilibrium (Aumann 1987)


163/230

Correlated Equilibrium (Aumann 1987)




Common KnowledgeofRationalityCorrelated Equilibrium (Aumann 1987)


164/230

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


Dynamics & Correlated Equilibria


165/230



RESULT

THERE EXISTgeneral, natural dynamics


166/230

leading to CORRELATED EQUILIBRIA



RESULT



167/230


Regret Matching

Hart and Mas-Colell, Ecca 2000SERGIU HART c 2008 p. 38


RESULT



168/230


Regret Matching

General regret-based dynamics

Hart and Mas-Colell, Ecca 2000, JET 2001SERGIU HART c 2008 p. 38


169/230


170/230

Regret Matching

"REGRET": the increase in past payoff, if any,

if diff i ld h b d


171/230

if a different action would have been used

"MATCHING": switching to a different actionwith a probability that is proportional to theregret for that action



THERE EXISTgeneral, natural dynamicsleading to CORRELATED EQUILIBRIA


172/230





173/230




THERE EXISTgeneral,naturaldynamicsleading to CORRELATED EQUILIBRIA


174/230


"natural":





175/230


"natural":

adaptive(also: close to "behavioral")





176/230


"natural":


simple and efficient:computation, time, information





177/230


"natural":


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


178/230




179/230




180/230



181/230



182/230




183/230



184/230



185/230



186/230

Dynamics and Equilibrium


187/230



188/230


NASH EQUILIBRIUM:afixed-pointof a non-linear map



189/230

CORRELATED EQUILIBRIUM:a solution of finitely manylinear inequalities



NASH EQUILIBRIUM:afixed-pointof a non-linear map



190/230

CORRELATED EQUILIBRIUM:a solution of finitely manylinear inequalities

set-valuedfixed-point (curb sets)?




191/230



"LAW OF CONSERVATION OF COORDINATION":


192/230



193/230


"LAW OF CONSERVATION OF COORDINATION":

Th b


194/230

There must be some COORDINATION

either in the EQUILIBRIUMnotion,



195/230


196/230

The "Program"

A. Demarcatethe BORDERbetween


197/230



198/230


199/230


200/230



201/230



202/230


203/230


204/230


205/230


206/230



207/230



208/230


209/230


210/230


211/230


212/230


213/230


214/230


215/230


216/230


217/230


218/230


219/230


220/230


221/230


222/230


223/230


224/230


225/230


226/230

My Game Theory

INSIGHTS,

IDEAS,

CONCEPTS

FORMALMODELS


227/230

CONCEPTS


My Game Theory

INSIGHTS,

IDEAS,

CONCEPTS

FORMALMODELS


228/230

CONCEPTS



229/230


230/230

Game Dynamics and Equilibrium

Documents

Transcript of Game Dynamics and Equilibrium