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

Although the study of market equilibria has occupied center stage within Mathematical Economics for over a century, polynomial time algorithms for such questions have so far evaded researchers. We provide the first such algorithm for the linear version of a problem defined by Irving Fisher in 1891. Our algorithm is modeled after Kuhn’s primaldual algorithm for bipartite matching. 1

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

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.

We advance significantly beyond the recent progress on the algorithmic complexity of Nash equilibria by solving two major open problems in the approximation of Nash equilibria and in the smoothed analysis of algorithms. • We show that no algorithm with complexity poly(n, 1 ɛ) can compute an ɛ-approximate Nash equilibrium in a two-player game, in which each player has n pure strategies, unless PPAD ⊆ P. In other words, the problem of computing a Nash equilibrium in a twoplayer game does not have a fully polynomial-time approximation scheme unless PPAD ⊆ P. • We prove that no algorithm for computing a Nash equilibrium in a two-player game can have smoothed complexity poly(n, 1 σ) under input perturbation of magnitude σ, unless PPAD ⊆ RP. In particular, the smoothed complexity of the classic Lemke-Howson algorithm is not polynomial unless PPAD ⊆ RP. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional hypergrid with constant side-length, and show that it can host the embedding of the proof structure of any PPAD problem. We prove a key geometric lemma for finding a discrete fixed-point, a new concept defined on n +1vertices of a unit hypercube. This lemma enables us to overcome the curse of dimensionality in reasoning about fixed-points in high dimensions. 1

We develop a method for establishing the independence of some Zf(a)-formulas from S'2(a). In particular, we show that T'2(a) is not VZ*(a)-conservative over S'2(a). We characterize the Z^-definable functions of T2 as being precisely the functions definable as projections of polynomial local search (PLS) problems. Although it is still an open problem whether bounded arithmetic S2 is finitely axiomatizable, considerable progress on this question has been made: S2 +1 is V2f+1-conservative over T'2 [3], but it is not V2!f+2-conservative unless £f+2 = Ylf+2 [10], and in addition, T2 is not VZf+1-conservative over S'2 unless LogSpace s? = Af+1 [8]. In particular, S2 is not finitely axiomatizable provided that the polynomial-time hierarchy does not collapse [10]. For the theory S2(a) these results imply (with some additional arguments) absolute results: S'2 + (a) is V2f+,(a)-conservative but not VZf+2(a)-conservative over T'2(a), and T'2(a) is not VZf+i(c*)-conservative over S'2(a). Here a represents a new uninterpreted predicate symbol adjoined to the language of arithmetic which may be used in induction formulas; from a computer science perspective, a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)-conservative over S'2(a). This was known for / = 1, 2 by [9,17] (see also [2]), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be used to show the unprovability of other 2f(a)-formulas from S'2(a). Our methods are analogous in spirit to the proof strategy of [8]: prove a witnessing theorem to show that provability of a Zf+1(a)-formula A in S'2(a) implies that it is witnessed by a function of certain complexity and then employ techniques of boolean complexity to construct an oracle a such that the formula A cannot be witnessed by a function of the prescribed complexity. Our formula A shall be 2f(a) and thus we can use the original witnessing theorem of [2]. The boolean complexity used is the same as in [8], namely Hastad's switching lemmas [6].

This paper is a survey and exposition of linear methods for finding Nash equilibria. Above all, these apply to games with two players. In an equilibrium of a twoperson game, the mixed strategy probabilities of one player equalize the expected payoffs for the pure strategies used by the other player. This defines an optimization problem with linear constraints. We do not consider nonlinear methods like simplicial subdivision for approximating fixed points, or systems of inequalities for higher-degree polynomials as they arise for noncooperative games with more than two players. These are surveyed in McKelvey and McLennan (1996)

In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is St-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's move depends on moves of other players. We say that a game has small neighborhood if the " utility function for each player depends only on (the actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph I-I(G). Among other results, we show that if jC has small neighborhood and if I-I(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class _NC ~ of highly parallelizable problems. 1 Introduction and Overview of Results The theory of strategic games and Nash equilibria has important applications in economics and decision making [31, 2]. Determining whether Nash equilibria exist, and effectively computing

We address the fundamental question of whether the Nash equilibria of a game can be computed in polynomial time. We describe certain efficient reductions between this problem for normal form games with a fixed number of players and graphical games with fixed degree. Our main result is that the problem of solving a game for any constant number of players, is reducible to solving a 4-player game.