Graphical Models for Game Theory

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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 (ltlt of players)
• Exs geography, organizational structure,
  networks
• Analogy to Bayes nets special structure

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An Abstract Tree Algorithm
U1
U2
U3
T(w,v) 1 lt--gt an upstream Nash
where V v given W w
lt--gt u T(v,ui) 1 for all i, and
v is a best response to u,w
V
W

• 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

How to represent? How to compute?
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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 t e/k preserves global e-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
V
W
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• 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
• t 0.01, e 0.05

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Extension to exact algorithm each table is a
finite union of rectangles, exponential in depth
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NashProp for Arbitrary Graphs

• Two-phase algorithm
• Table-passing phase
• Assignment-passing phase
• Table-passing phase
• Initialization T0(w,v) 1 for all (w,v)
• Induction Tr1(w,v) 1 iff u
• Tr(v,ui) 1 for all i
• Vv a best response to Ww, Uu
• Table consistency stronger than best response

U1
U2
U3
V
W
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Convergence of Table-Passing

• Table-passing obeys contraction
• (w,v)Tr1(w,v) 1 contained in
  (w,v)Tr(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 e and t to be parameters
• Alternative approach VickreyKoller
• Constraint propagation on junction tree

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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