The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement. Extending our result for the Harsanyi-Selten process, we show that several other homotopy-based algorithms for finding equilibria of games are also PSPACE-complete to implement. A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in [Savani and von Stengel]. These results show that our techniques can be widely applied and suggest that the PSPACE-completeness of implementing homotopy methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc

Equilibria, Fixed Points, and Complexity Classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes

Computing the Equilibria of Bimatrix Games using Dominance Heuristics

We propose a formulation of a general-sum bimatrix game as a bipartite directed graph with the objective of establishing a correspondence between the set of the relevant structures of the graph (in particular elementary cycles) and the set of the Nash equilibria of the game. We show that finding the set of elementary cycles of the graph permits the computation of the set of equilibria. For games whose graphs have a sparse adjacency matrix, this serves as a good heuristic for computing the set of equilibria. The heuristic also allows the discarding of sections of the support space that do not yield any equilibrium, thus serving as a useful pre-processing step for algorithms that compute the equilibria through support enumeration

An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size $2000 \times 2000$. Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria

Computing all solutions of Nash equilibrium problems with discrete strategy sets

The Nash equilibrium problem is a widely used tool to model non-cooperative games. Many solution methods have been proposed in the literature to compute solutions of Nash equilibrium problems with continuous strategy sets, but, besides some specific methods for some particular applications, there are no general algorithms to compute solutions of Nash equilibrium problems in which the strategy set of each player is assumed to be discrete. We define a branching method to compute the whole solution set of Nash equilibrium problems with discrete strategy sets. This method is equipped with a procedure that, by fixing variables, effectively prunes the branches of the search tree. Furthermore, we propose a preliminary procedure that by shrinking the feasible set improves the performances of the branching method when tackling a particular class of problems. Moreover, we prove existence of equilibria and we propose an extremely fast Jacobi-type method which leads to one equilibrium for a new class of Nash equilibrium problems with discrete strategy sets. Our numerical results show that all proposed algorithms work very well in practice