Post on 21-Dec-2015
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)