Algorithms for Strong Nash Equilibrium with More than Two Agents

Strong Nash equilibrium (SNE) is an appealing solution concept when rational agents can form coalitions. A strategy profile is an SNE if no coalition of agents can benefit by deviating. We present the first general– purpose algorithms for SNE finding in games with more than two agents. An SNE must simultaneously be a Nash equilibrium (NE) and the optimal solution of multiple non–convex optimization problems. This makes even the derivation of necessary and sufficient mathematical equilibrium constraints difficult. We show that forcing an SNE to be resilient only to pure–strategy deviations by coalitions, unlike for NEs, is only a necessary condition here. Second, we show that the application of Karush–Kuhn–Tucker conditions leads to another set of necessary conditions that are not sufficient. Third, we show that forcing the Pareto efficiency of an SNE for each coalition with respect to coalition correlated strategies is sufficient but not necessary. We then develop a tree search algorithm for SNE finding. At each node, it calls an oracle to suggest a candidate SNE and then verifies the candidate. We show that our new necessary conditions can be leveraged to make the oracle more powerful. Experiments validate the overall approach and show that the new conditions significantly reduce search tree size compared to using NE conditions alone. Introduction Equilibrium computation in non–cooperative games has recently received significant attention in artificial intelligence and computer science at large. Many papers have focused on the computational study of Nash equilibrium (NE) (Shoham and Leyton-Brown 2008), showing that searching for it is PPAD–complete (Daskalakis, Goldberg, and Papadimitriou 2006) even with two agents (Chen, Deng, and Teng 2009) and designing various algorithms. Two– agent games can be solved by linear complementarity programming (Lemke and Howson 1964), support enumeration (Porter, Nudelman, and Shoham 2009), mixed–integer linear programming (Sandholm, Gilpin, and Conitzer 2005), or local search (Gatti et al. 2012). With more agents, common methods are nonlinear complementarity programming, Copyright c © 2013, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. simplicial subdivision, homotopy (Shoham and LeytonBrown 2008), and support enumeration (Thompson, Leung, and Leyton-Brown 2011). The strong Nash equilibrium (SNE) concept strengthens NE by requiring the strategy profile to be resilient also to multilateral deviations, including deviations by the grand coalition that contains all the agents (Aumann 1960). It captures the situation in which agents can form coalitions and change their strategies multilaterally in a coordinated way. For a given game, an SNE may or may not exist. Searching for it is NP–complete when the number of agents is a constant (Conitzer and Sandholm 2008; Gatti, Rocco, and Sandholm 2013). An SNE must be simultaneously an NE and the optimal solution of multiple non–convex optimization problems (Hoefer and Skopalik 2010). This makes even the derivation of necessary and sufficient mathematical equilibrium constraints a difficult (and currently open) task. Some results have been proven about the computation of pure– strategy SNEs in specific classes of games, e.g., congestion games (Holzman and Law-Yone 1997; Hayrapetyan, Tardos, and Wexler 2006; Rozenfeld and Tennenholtz 2006; Hoefer and Skopalik 2010), connection games (Epstein, Feldman, and Mansour 2007), maxcut games (Gourvès and Monnot 2009), and continuous games (Nessah and Tian 2012). The only prior algorithm that works also for mixed strategies is very recent (Gatti, Rocco, and Sandholm 2013). It is only for 2–agent games. It is a special kind of tree search algorithm, and at each node it calls anNP–complete oracle—a variation of MIP Nash (Sandholm, Gilpin, and Conitzer 2005)—that returns an NE (if one exists) in a given subspace of the agents’ utilities and then verifies whether the returned NE is an SNE. With more than two agents, MIP Nash cannot be used because the problem of finding an NE is itself already nonlinear. In this paper, we provide two necessary, but non– sufficient, conditions for the existence of an SNE, conditions that can be used to test whether a game admits an SNE. • We provide a mixed–integer nonlinear program to find NEs that are resilient to pure–strategy multilateral deviations. • We provide a nonlinear program to find an NE that satisfies Karush–Kuhn–Tucker conditions (Miettinen 1999). We also provide a sufficient, but non–necessary, condition Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence