Local-Effect Games, and an Algorithm for Computing their Equilibria Navin Bhat Department of Physics University of Toronto Kevin Leyton-Brown Department of Computer Science University of British Columbia 1 Introduction There is much interest in modeling large, real-world problems as games and computing these games’ equilibria. This interest has driven recent advances in algorithms for computing Nash equilibria of general games. One recent, general- purpose algorithm is the continuation method of Govindan and Wilson [2003], a gradient-following algorithm which is based on topological insight into the graph of the Nash equilibrium correspondence by Kohlberg and Mertens [1986]. The worst-case complexity of Govindan and Wilson’s algorithm is open because the worst-case number of gradient-following steps is not known; however, in prac- tice the algorithm’s runtime is dominated by the computation of the gradient (mainly the Jacobian of the payoff function), which is both best- and worst-case exponential in the number of agents. Unfortunately this algorithm, like other state of the art algorithms for computing Nash equilibria, is generally practical only on games having relatively small numbers of players and actions. Researchers have begun to investigate compact game representations that can be leveraged to yield more efficient computation on large games. One in- fluential class of representations exploits (strict) independencies between play- ers’ utility functions; this class includes Graphical Games [Kearns et al., 2001] and Multi-Agent Influence Diagrams [Koller & Milch, 2001]. Recently, Blum et al. [2003] introduced the first general algorithm for computing exact equilib- ria in such games, based on a state-of-the-art algorithm for general games due to Govindan and Wilson (2003). This gradient-following algorithm has as its bottleneck step the exponential-time computation of the Jacobian of the pay- off function. Blum et al.’s algorithm computes this Jacobian in time no worse than exponential in the tree width of the underlying utility function dependency graph—an exponential improvement—and is consequently able to compute equi- libria of considerably larger games. Although real-world games often have very regular structure, strict indepen- dence between agents’ utility functions is not a common assumption in the game theory literature. A second approach to compactly representing games aims to be more realistic by focusing on games in which all agents can potentially affect 1