Abstract not found

We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recently-established equivalence between polynomialtime solvability of normal-form games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPAD-complete class of Brouwer functions. 1

These are the lecture notes for ten lectures to be given at the CBMS

A system with multiple interacting agents (whether artificial or human) is often best analyzed using game-theoretic tools. Unfortunately, while the formal foundations are well-established, standard computational techniques for game-theoretic reasoning are inadequate for dealing with realistic games. This paper describes the Gala system, an implemented system that allows the specification and efficient solution of large imperfect information games. The system contains the first implementation of a recent algorithm, due to Koller, Megiddo, and von Stengel. Experimental results from the system demonstrate that the algorithm is exponentially faster than the standard algorithm in practice, not just in theory. It therefore allows the solution of games that are orders of magnitude larger than were previously possible. The system also provides a new declarative language for compactly and naturally representing games by their rules. As a whole, the Gala system provides the capability for automa...

Abstract. This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

We review the current state of the art of methods for numerical computation of Nash equilibria for finite n-person games. Classical path following methods, such as the Lemke-Howson algorithm for two person games, and Scarf-type fixed point algorithms for n-person games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semi-algebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.

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

Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computationally intractable. The approach is to generate the normal form of the game (the matrix containing the payoff for each strategy combination), and then solve a linear program (LP) or a linear complementarity problem (LCP). The size of the normal form, however, is typically exponential in the size of the game tree, thus making this method impractical in all but the simplest cases. This paper describes a new representation of strategies which results in a practical linear formulation of the problem of two-player games with perfect recall (i.e., games where players never forget anything, which is a standard assumption). Standard LP or LCP solvers can then be applied to find optimal randomized strategies. The resulting algorithms are, in general, exponentially better than the standard ones, both in terms of time and in terms of space.

In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned trac. In a Nash equilibrium, each user sel shly routes its trac on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.

. The Nash equilibria of a two-person, non-zero-sum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, it is first necessary to convert the game to a strategic description such as the normal form. The classical normal form, however, is often exponentially large in the size of the game tree. In this paper we suggest an alternative approach, based on the sequence form of the game. For a game with perfect recall, the sequence form is a linear sized strategic description, which results in an LCP of linear size. For this LCP, we show that an equilibrium is found by Lemke's algorithm, a generalization of the Lemke-Howson method. Keywords. Equilibrium, extensive game, Lemke-Howson algorithm, linear complementarity, sequence form. Computer Science Division, University of California, Berkeley, CA 94720; and IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120 y IBM Almaden Research Center, 650 Harr...