Graphical Models for Game Theory

• Undirected graph G capturing local interactions• Each player represented by a vertex• N_i(G) = neighbors of i in G (includes i)• Assume: M_i(a) expressible as M’_i(a’) over only N_i(G)• Graphical game: (G,{M’_i})• Compact representation of game• Exponential in max degree (<< # of players)• Ex’s: geography, organizational structure, networks• Analogy to Bayes nets: special structure

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An Abstract Tree Algorithm

• Downstream Pass:– Each node V receives

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

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

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

from W s.t. T(w,v) = 1– V picks witness u for

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

U1 U2 U3

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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

How to represent?How to compute?

An Approximation Algorithm

• Discretize u and v in T(v,u), 1 represents approximate Nash

• Main technical lemma: If k is max degree, grid resolution ~ /k preserves global -Nash equilibria

• An efficient algorithm:– Polynomial in n (fixed k)– Represent an approx. to every Nash– Can generate random Nash, or specific

Nash

U1 U2 U3

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V

• Table dimensions are probability of playing 0• Black shows T(v,u) = 1• Ms want to match, Os to unmatch• Relative value modulated by parent values• =0.01, = 0.05

Extension to exact algorithm:each table is a finite union ofrectangles, exponential in depth

NashProp for Arbitrary Graphs

• Two-phase algorithm:– Table-passing phase– Assignment-passing phase

• Table-passing phase:– Initialization: T[0](w,v) = 1 for all (w,v)– Induction: T[r+1](w,v) = 1 iff u:

• T[r](v,ui) = 1 for all i• V=v a best response to W=w, U=u

• Table consistency stronger than best response

U1 U2 U3

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V

Convergence of Table-Passing

• Table-passing obeys contraction:– {(w,v):T[r+1](w,v) = 1} contained in {(w,v):T[r](w,v) = 1}

• Tables converge and are balanced• Discretization scheme: tables converge quickly• Never eliminate an equilibrium• Tables give a reduced search space• Assignment-passing phase:

– Use graph to propagate a solution consistent with tables– Backtracking local search

• Allow and to be parameters• Alternative approach [Vickrey&Koller]:

– Constraint propagation on junction tree

Graphical Games: Related Work

• Koller and Milch: graphical influence diagrams• La Mura: game networks• Vickrey & Koller: other methods on graphical games• Leyton-Brown: action-graph games