Graphical GamesGraphical Games

Kjartan A. JónssonKjartan A. Jónsson

Nash equilibriumNash equilibrium

Nash equilibriumNash equilibrium N players playing a dominant strategy is N players playing a dominant strategy is

a Nash equilibriuma Nash equilibrium When one has a dominant strategy and When one has a dominant strategy and

the other chooses accordingly is also the other chooses accordingly is also Nash equilibriumNash equilibrium

Computationally expensive for n Computationally expensive for n playersplayers

Computing Nash equilibriumComputing Nash equilibrium

Ex: 2 action gameEx: 2 action game Tabular Tabular

representationrepresentation Consider all Consider all

possible actions possible actions from all playersfrom all players

n playersn players

ExpensiveExpensive

n

n

1

2 2

Nash equilibrium: ProposalNash equilibrium: Proposal

Ex: 2 action gameEx: 2 action game Tree graphTree graph

Consider only Consider only actions from actions from neighborsneighbors

n playersn players kk neighbors neighbors

Then propagate Then propagate result upwardsresult upwards

Less expensiveLess expensive

n

k

1

2 2

CEORoot

Manager AK=1

Manager BK=2

Employee Ak=1

Employee Bk=2

Employee Ck=1

Employee Dk=2

Employee Ek=3

Abstract Tree AlgorithmAbstract Tree Algorithm

Downstream Pass:Downstream Pass: Each node V receives Each node V receives

T(v,ui) from each UiT(v,ui) from each Ui V computes T(w,v) and V computes T(w,v) and

witness lists for each witness lists for each T(w,v) = 1T(w,v) = 1

Upstream Pass:Upstream Pass: V receives values (w,v) V receives values (w,v)

from W, T(w,v) = 1from W, T(w,v) = 1 V picks witness V picks witness u u for for

T(w,v), passes (v,ui) to T(w,v), passes (v,ui) to UiUi

U1 U2 U3

W

V

T(w,v) = 1 <--> an “upstream” Nash where V = v given W = w <--> u: T(v,ui) = 1 for all i, and v is a best response to u,w

Borrowed from Michael Kearns

ProblemProblem

““Since v and ui are continues Since v and ui are continues variables, it is not obvious that the variables, it is not obvious that the table T(v,ui) can be represented table T(v,ui) can be represented compactly, or finitely, for arbitrary compactly, or finitely, for arbitrary vertices in a tree”vertices in a tree”

SolutionsSolutions ““Approximate”Approximate” ““Exact”Exact”

ApproximationApproximation

Approximation algorithmApproximation algorithm Run time: polynomial in 2^kRun time: polynomial in 2^k Represent an approx. to every NashRepresent an approx. to every Nash Generates random Nash or specific NashGenerates random Nash or specific Nash

ExactExact

Extension to exact Extension to exact algorithmalgorithm

Run time: exponentialRun time: exponential Each table is a finite Each table is a finite

union of rectanglesunion of rectangles Exponential in depthExponential in depth

BenefitsBenefits

We can represent a multiplayer We can represent a multiplayer game using a graphgame using a graph Natural relationship between graphical Natural relationship between graphical

games and modern probabilistic games and modern probabilistic modeling more toolsmodeling more tools

Local Markov Networks language to Local Markov Networks language to express correlated equilibriaexpress correlated equilibria

Future researchFuture research

Efficient algorithm for Exact Nash Efficient algorithm for Exact Nash ComputationComputation

Strategy-proofStrategy-proof Loose now to win laterLoose now to win later

Cooperative and behavioral actionsCooperative and behavioral actions Cooperation between a set of playersCooperation between a set of players

ConclusionConclusion

Theoretically: works fineTheoretically: works fine Practically?Practically?

An employee in division A can influence An employee in division A can influence division B (email correspondence)division B (email correspondence)

Circled graphCircled graph Considered in both divisionsConsidered in both divisions IgnoredIgnored

ReferencesReferences

Book: Algorithmic Game Theory, Book: Algorithmic Game Theory, chapter on Graphical Gameschapter on Graphical Games

Paper: Graphical Models for Game Paper: Graphical Models for Game Theory – Michael Kearns, Michael L. Theory – Michael Kearns, Michael L. Littman, Satinder SinghLittman, Satinder Singh

Presentation: by Michael Kearns Presentation: by Michael Kearns (NIPS-gg.ppt)(NIPS-gg.ppt)