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

Fairness in Multi-Agent Sequential Decision-Making

We define a fairness solution criterion for multi-agent decision-making problems, where agents have local interests. This new criterion aims to maximize the worst performance of agents with consideration on the overall performance. We develop a simple linear programming approach and a more scalable game-theoretic approach for computing an optimal fairness policy. This game-theoretic approach formulates this fairness optimization as a two-player, zero-sum game and employs an iterative algorithm for finding a Nash equilibrium, corresponding to an optimal fairness policy. We scale up this approach by exploiting problem structure and value function approximation. Our experiments on resource allocation problems show that this fairness criterion provides a more favorable solution than the utilitarian criterion, and that our game-theoretic approach is significantly faster than linear programming

On the expected number of equilibria in a multi-player multi-strategy evolutionary game

In this paper, we analyze the mean number $E(n,d)$ of internal equilibria in a general $d$-player $n$-strategy evolutionary game where the agents' payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next we characterize the asymptotic behavior of $E(2,d)$, estimating its lower and upper bounds as $d$ increases. Two important consequences are obtained from this analysis. On the one hand, we show that in both cases the probability of seeing the maximal possible number of equilibria tends to zero when $d$ or $n$ respectively goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger $n$ and $d$, numerical results are provided and discussed.Comment: 26 pages, 1 figure, 1 table. revised versio

Identification and Estimation of Discrete Games of Complete Information

We discuss the identification and estimation of discrete games of complete information. Following Bresnahan and Reiss (1990, 1991), a discrete game is a generalization of a standard discrete choice model where utility depends on the actions of other players. Using recent algorithms to compute all of the Nash equilibria to a game, we propose simulation-based estimators for static, discrete games. With appropriate exclusion restrictions about how covariates enter into payoffs and influence equilibrium selection, the model is identified with only weak parametric assumptions. Monte Carlo evidence demonstrates that the estimator can perform well in moderately-sized samples. As an application, we study the strategic decision of firms in spatially-separated markets to establish a presence on the Internet.

Identification and Estimation of Discrete Games of Complete Information

We discuss the identification and estimation of discrete games with complete information. Following Bresnahan and Reiss, a discrete game is defined to be a generalization of a standard discrete choice model in which utility depends on the actions of other players. Using recent algorithms that compute the complete set of the Nash equilibria, we propose simulation-based estimators for static, discrete games. With appropriate exclusion restrictions about how covariates enter into payoffs and influence equilibrium selection, the model is identified with only weak parametric assumptions. Monte Carlo evidence demonstrates that the estimator can perform well in moderately-sized samples. As an illustration, we study the strategic decisions of firms in spatially-separated markets in establishing a presence on the InternetEmpirical Industrial Organization, Simulation Based Estimation, Homotopies

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49-73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N → ∞, the expected number of Nash equilibria is approximately (√(π log N) / 2)(M-1)/√ M. Letting M = N → ∞, the expected number of Nash equilibria is exp(NM + O(log N)), where M ≈ 0.281644 is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies. © 2004 Elsevier Inc. All rights reserved