Methods for Empirical Game-Theoretic Analysis

An emerging empirical methodology bridges the gap between game theory and simulation for practical strategic reasoning. Game-Theoretic Analysis Game-theoretic analysis typically takes at its starting point, most naturally, a description of its subject—the game, a formal model of a multiagent interaction. The recent surge in interest among AI researchers in game theory has led to numerous advances in game modeling (Gal & Pfeffer 2004; Kearns, Littman, & Singh 2001; Koller & Milch 2003; Leyton-Brown & Tennenholtz 2003) and solution techniques (Gilpin & Sandholm 2006; Porter, Nudelman, & Shoham 2004), substantially expanding the class of games amenable to computational analysis. Nevertheless, a great many games of interest lie well beyond the boundary of tractable modeling and reasoning. Complexity may be manifest in the number of agents or the size of their strategy sets (policy spaces), or the degree of incomplete and imperfect information. The issue here is not merely computational complexity of the analysis task (e.g., finding equilibrium), but actually the apparent impracticality of producing an explicit game model amenable to automated reasoning. For example, consider the Trading Agent Competition Supply Chain Management (TAC/SCM) game (Eriksson, Finne, & Janson 2006). This is a well-defined six-player symmetric game of imperfect information, with interaction rules and exogenous stochastic processes described in a brief specification document. There is nothing particularly unusual about this game, nevertheless it presents a difficult challenge for game-theoretic analysis. The policy spaces and payoff functions are clearly induced by the specified rules, but the description is quite indirect. Even given complete policies for all six agents, there is no apparent means to derive the expected payoffs, short of sampling from the stochastic environment using an available game simulator. In this case, each sample point takes an hour to compute. Copyright c © 2006, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. Empirical Games The approach we have been pursuing in my research group for the past few years1 is to take the game simulator as the fundamental input, and perform strategic reasoning through interleaved simulation and game-theoretic analysis. The basic object of analysis is an empirical game, a description of the interaction scenario where payoff information comes in the form of data from observations or simulations. Constructing and reasoning about such games presents many interesting subproblems, which can be addressed by existing as well as new methods from simulation, statistics, search, and of course, standard game-theoretic analysis. I find it useful to decompose empirical game-theoretic analysis into three basic steps. Many of the research contributions in this area manifest as techniques applicable to one of these subproblems, or results from approaches taken to them in a given domain. Parametrize Strategy Space Often the complexity of a game resides in vast policy spaces available to agents. Large spaces can arise, for example, from continuous or multi-dimensional action sets, as well as from imperfect information (when actions are conditioned on observation histories). It is often useful in such cases to approximate the game by restricting the strategy space, and structuring the space to admit a sensible search procedure. Results from analysis of restricted subgames often provide insight into the original game. Arguably, all applications of game theory in the social sciences employ stylized abstractions, which are manually designed restricted versions of actual games. From our perspective the interesting question is how to automate the abstraction process starting from a rich but intractable game specification. One generic approach to generating candidate strategies is to start from some baseline or skeletal structure, and systematically introduce parametric variations. Some examples of natural baselines include: 1. Truthful revelation. For example, in an auction game, the baseline would be to bid one’s true value. In the firstSimilar or identical techniques have also been employed by other researchers, especially those working experimentally with multiagent systems. Our main claim is in systematizing the methodology, in explicit game-theoretic terms.