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

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 U1U2U3 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 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 U1U2U3 W 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 of rectangles, 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 U1U2U3 W 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