Mathematical Programming Formulations to Compute Steady States in Two-Player Extensive-Form Games

The most common solution concept for a strategic interaction situation is the Nash equilibrium, in which no agent can do better by deviating unilaterally. However, the Nash equilibrium underlays on the assumption of common information that is hardly verified in many practical situations. When information is not common, rational agents are assumed to learn from their observations to derive beliefs over their opponents' play and payoffs. In these situations, there are steady states composed of beliefs and strategies in which the strategies do not constitute a Nash equilibrium. These stable states are called in the game theory literature self-confirming equilibria. They are such that every agent plays the best response to her beliefs and these are correct on the equilibrium path, while off the equilibrium path they may be incorrect. We present some mathematical programming formulations for computing self-confirming equilibria and its refinements in two-player extensive-form games and we study their properties.

[1]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[2]  Vincent Conitzer,et al.  Mixed-Integer Programming Methods for Finding Nash Equilibria , 2005, AAAI.

[3]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

[4]  C. E. Lemke.  Some pivot schemes for the linear complementarity problem , 1978 .

[5]  David M. Kreps,et al.  Sequential Equilibria Author ( s ) : , 1982 .

[6]  Nicola Basilico,et al.  Computing Bayes-Nash Equilibria through Support Enumeration Methods in Bayesian Two-Player Strategic-Form Games , 2009, 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology.

[7]  Yoav Shoham,et al.  Multiagent Systems - Algorithmic, Game-Theoretic, and Logical Foundations , 2009 .

[8]  Michael P. Wellman,et al.  Self-Confirming Price Prediction for Bidding in Simultaneous Ascending Auctions , 2005, UAI.

[9]  Yoav Shoham,et al.  Simple search methods for finding a Nash equilibrium , 2004, Games Econ. Behav..

[10]  Moshe Tennenholtz,et al.  Learning equilibrium as a generalization of learning to optimize , 2007, Artif. Intell..

[11]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[12]  D. Fudenberg,et al.  Payoff Information and Self-Confirming Equilibrium , 1999 .

[13]  D. Koller,et al.  Efficient Computation of Equilibria for Extensive Two-Person Games , 1996 .

[14]  Michael P. Wellman,et al.  Conjectural Equilibrium in Multiagent Learning , 1998, Machine Learning.

[15]  Yoav Shoham,et al.  Run the GAMUT: a comprehensive approach to evaluating game-theoretic algorithms , 2004, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, 2004. AAMAS 2004..

[16]  Peter Bro Miltersen,et al.  Computing sequential equilibria for two-player games , 2006, SODA '06.

[17]  D. Fudenberg,et al.  Self-confirming equilibrium , 1993 .

[18]  H. Kuk.  On equilibrium points in bimatrix games , 1996 .