We prove that the problem of finding a Nash equilibrium in a two-player game is PPAD-complete. 1

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Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume

Abstract. In this work we study the tractability of well supported approximate Nash Equilibria (SuppNE in short) in bimatrix games. In view of the apparent intractability of constructing Nash Equilibria (NE in short) in polynomial time, even for bimatrix games, understanding the limitations of the approximability of the problem is of great importance. We initially prove that SuppNE are immune to the addition of arbitrary real vectors to the rows (columns) of the row (column) player’s payoff matrix. Consequently we propose a polynomial time algorithm (based on linear programming) that constructs a 0.5−SuppNE for arbitrary win lose games. We then parameterize our technique for win lose games, in order to apply it to arbitrary (normalized) bimatrix games. Indeed, this new technique leads to a weaker φ−SuppNE for win lose games, where φ = √ 5−1 2 the golden ratio. Nevertheless, this parameterized technique extends nicely to a technique for arbitrary [0, 1]−bimatrix games, which assures a 0.658−SuppNE in polynomial time. To our knowledge, these are the first polynomial time algorithms providing ε−SuppNE of normalized or win lose bimatrix games, for some nontrivial constant ε ∈ [0, 1), bounded away from 1.

We give a brief and biased survey of the past, present, and future of research on the interface of theoretical computer science and game theory. 1

Abstract. We study the existence and tractability of a notion of approximate equilibria in bimatrix games, called well supported approximate Nash Equilibria (SuppNE in short). We prove existence of ε−SuppNE for any constant ε ∈ (0,1), with only logarithmic support sizes for both players. Also we propose a polynomial–time construction of SuppNE, both for win lose and for arbitrary (normalized) bimatrix games. The quality of these SuppNE depends on the girth of the Nash Dynamics graph in the win lose game, or a (rounded–off) win lose image of the original normalized game. Our constructions are very successful in sparse win lose games (ie, having a constant number of (0,1)−elements in the bimatrix) with large girth in the Nash Dynamics graph. The same holds also for normalized games whose win lose image is sparse with large girth. Finally we prove the simplicity of constructing SuppNE both in random normalized games and in random win lose games. In the former case we prove that the uniform full mix is an o(1) −SuppNE, while in the case of win lose games, we show that (with high probability) there is either a PNE or a 0.5-SuppNE with support sizes only 2. 1

Abstract—Fictitious play is a simple, well-known, and oftenused algorithm for playing (and, especially, learning to play) games. However, in general it does not converge to equilibrium; even when it does, we may not be able to run it to convergence. Still, we may obtain an approximate equilibrium. In this paper, we study the approximation properties that fictitious play obtains when it is run for a limited number of rounds. We show that if both players randomize uniformly over their actions in the first r rounds of fictitious play, then the result is an ǫ-equilibrium, where ǫ = (r + 1)/(2r). (Since we are examining only a constant number of pure strategies, we know that ǫ < 1/2 is impossible, due to a result of Feder et al.) We show that this bound is tight in the worst case; however, with an experiment on random games, we illustrate that fictitious play usually obtains a much better approximation. We then consider the possibility that the players fail to choose the same r. We show how to obtain the optimal approximation guarantee when both the opponent’s r and the game are adversarially chosen (but there is an upper bound R on the opponent’s r), using a linear program formulation. We show that if the action played in the ith round of fictitious play is chosen with probability proportional to: 1 for i = 1 and 1/(i −1) for all 2 ≤ i ≤ R+1, this gives an approximation guarantee of 1 − 1/(2 + ln R). We also obtain a lower bound of 1 − 4/ln R. This provides an actionable prescription for how long to run fictitious play. I.

We give a full proof of the Kakutani (1941) fixed point theorem that is brief, elementary, and based on game theoretic concepts. This proof points to a new family of algorithms for computing approximate fixed points that have advantages over simplicial subdivision methods. An imitation game is a finite two person normal form game in which the strategy spaces for the two agents are the same and the goal of the second player is to choose the same strategy as the first player. These appear in our proof, but are also interesting from other points of view. They give new insights into the “long” paths of the Lemke-Howson algorithm. They provide a rich class of games with “short” Lemke-Howson paths. They are useful for studying the complexity of a number of problems in computational economics. K

Abstract: An imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths.

Abstract—Equilibria computation is of great importance to many areas such as economics, control theory, and recently computer science. We focus on the computation of Nash equilibria in two-player general-sum normal form games, also called bimatrix games. One efficient method to compute these equilibria is based on enumerating the vertices of the best response polyhedrons of the two players and checking the equilibrium conditions for every pair of vertices. We design and implement a parallel algorithm for computing Nash equilibria in bimatrix games based on vertex enumeration. We analyze the performance of the proposed algorithm by performing extensive experiments on a grid computing system. Keywords-game theory; Nash equilibrium; parallel algorithm; bimatrix game