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We provide efficient algorithms for finding approximate BayesNash equilibria (BNE) in graphical, specifically tree, games of incomplete information. In such games an agent's payoff depends on its private type as well as on the actions of the agents in its local neighborhood in the graph. We consider two classes of such games: (1) arbitrary tree-games with discrete types, and (2) tree-games with continuous types but with constraints on the effect of type on payoffs. For each class we present a message passing on the game-tree algorithm that computes an -BNE in time polynomial in the number of agents and the approximation parameter 1 .

The TAC 2003 supply-chain game presented automated trading agents with a challenging strategic problem. Embedded within a high-dimensional stochastic environment was a pivotal strategic decision about initial procurement of components. Early evidence suggested that the entrant field was headed toward a self-destructive, mutually unprofitable equilibrium. Our agent, Deep Maize, introduced a preemptive strategy designed to neutralize aggressive procurement, perturbing the field to a more profitable equilibrium; it worked. Not only did preemption improve Deep Maize's profitability, it improved profitability for the whole field. Whereas it is perhaps counterintuitive that action designed to prevent others from achieving their goals actually helps them, strategic analysis employing an empirical game-theoretic methodology verifies and provides insight about this outcome.

We formulate the problem of computing equilibria in multiplayer games represented by arbitrary undirected graphs as a constraint satisfaction problem and present two algorithms. The first is PureProp: an algorithm for computing approximate Nash equilibria in complete information one-shot games and approximate Bayes-Nash equilibria in one-shot games of incomplete information. PureProp unifies existing message-passing based algorithms for solving these classes of games. We also address repeated graphical games, and present a second algorithm, PureProp-R, for computing approximate Nash equilibria in these games.

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