Paul W. Goldberg

The Complexity of Computing a Nash Equilibrium

In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwers fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwers problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489-532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwers problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof.

Four Strikes Against Physical Mapping of DNA

Physical mapping is a central problem in molecular biology and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NP-complete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the k-consecutive ones problem for k > or = 2. These models have been chosen to reflect various features typical in biological data, including false-negative and positive errors, small width of the map, and chimericism.

Reducibility among equilibrium problems

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.

Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers

The Vapnik-Chervonenkis (V-C) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the V-C dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are automatic for discrete concept classes, but hitherto little has been known about what general conditions guarantee polynomial bounds on V-C dimension for classes in which concepts and examples are represented by tuples of real numbers. In this paper, we show that for two general kinds of concept class the V-C dimension is polynomially bounded in the number of real numbers used to define a problem instance. One is classes where the criterion for membership of an instance in a concept can be expressed as a formula (in the first-order theory of the reals) with fixed quantification depth and exponentially-bounded length, whose atomic predicates are polynomial inequalities of exponentially-bounded degree, The other is classes where containment of an instance in a concept is testable in polynomial time, assuming we may compute standard arithmetic operations on reals exactly in constant time. Our results show that in the continuous case, as in the discrete, the real barrier to efficient learning in the Occam sense is complexity-theoretic and not information-theoretic. We present examples to show how these results apply to concept classes defined by geometrical figures and neural nets, and derive polynomial bounds on the V-C dimension for these classes.

Bounds for the convergence rate of randomized local search in a multiplayer load-balancing game

This paper studies a load balancing game introduced by Koutsoupias and Papadimitriou, that is intended to model a set of users who share several internet-based resources. Some of the recent work on this topic has considered the problem of constructing Nash equilibria, which are choices of actions where each user has optimal utility given the actions of the other users. A related (harder) problem is to find sequences of utility-improving moves that lead to a Nash equilibrium, starting from some given assignment of resources to users.We consider the special case where all resources are the same as each other. It is known already that there exist efficient algorithms for finding Nash equilibria; our contribution here is to show furthermore that Nash equilibria for this type of game are reached rapidly by Randomized Local Search, a simple generic method for local optimization. Our motivation for studying Randomized Local Search is that (as we show) it can be realised by a simple distributed network of users that act selfishly, have no central control and only interact via the effect they have on the cost functions of resources.

A Tractable and Expressive Class of Marginal Contribution Nets and Its Applications

Coalitional games raise a number of important questions from the point of view of computer science, key among them being how to represent such games compactly, and how to efficiently compute solution concepts assuming such representations. Marginal contribution nets (MC-nets), introduced by Ieong and Shoham, are one of the simplest and most influential representation schemes for coalitional games. MC-nets are a rulebased formalism, in which rules take the form pattern value, where “pattern ” is a Boolean condition over agents, and “value ” is a numeric value. Ieong and Shoham showed that, for a class of what we will call “basic” MC-nets, where patterns are constrained to be a conjunction of literals, marginal contribution nets permit the easy computation of solution concepts such as the Shapley value. However, there are very natural classes of coalitional games that require an exponential number of such basic MC-net rules. We present read-once MC-nets, a new class of MC-nets that is provably more compact than basic MC-nets, while retaining the attractive computational properties of basic MC-nets. We show how the techniques we develop for read-once MC-nets can be applied to other domains, in particular, computing solution concepts in network flow games on series-parallel networks (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the computational complexity of weighted voting games

Coalitional games provide a useful tool for modeling cooperation in multiagent systems. An important special class of coalitional games is weighted voting games, in which each player has a weight (intuitively corresponding to its contribution), and a coalition is successful if the sum of its members’ weights meets or exceeds a given threshold. A key question in coalitional games is finding coalitions and payoff division schemes that are stable, i.e., no group of players has any rational incentive to leave. In this paper, we investigate the computational complexity of stability-related questions for weighted voting games. We study problems involving the core, the least core, and the nucleolus, distinguishing those that are polynomial-time computable from those that are NP-hard or coNP-hard, and providing pseudopolynomial and approximation algorithms for some of the computationally hard problems.

Uncoordinated Two-Sided Matching Markets

Various economic interactions can be modeled as two-sided markets. A central solution concept for these markets is stable matchings, introduced by Gale and Shapley. It is well known that stable matchings can be computed in polynomial time, but many real-life markets lack a central authority to match agents. In those markets, matchings are formed by actions of self-interested agents. Knuth introduced uncoordinated two-sided markets and showed that the uncoordinated better response dynamics may cycle. However, Roth and Vande Vate showed that the random better response dynamics converges to a stable matching with probability one, but they did not address the question of convergence time. In this paper, we give an exponential lower bound for the convergence time of the random better response dynamics in two-sided markets. We also extend the results for the better response dynamics to the best response dynamics; i.e., we present a cycle of best responses and prove that the random best response dynamics converges to a stable matching with probability one, but its convergence time is exponential. Additionally, we identify the special class of correlated matroid two-sided markets with real-life applications for which we prove that the random best response dynamics converges in expected polynomial time.

Bounds for the query complexity of approximate equilibria

We analyze the number of payoff queries needed to compute approximate equilibria of multi-player games. We find that query complexity is an effective tool for distinguishing the computational difficulty of alternative solution concepts, and we develop new techniques for upper- and lower bounding the query complexity. For binary-choice games, we show logarithmic upper and lower bounds on the query complexity of approximate correlated equilibrium. For well-supported approximate correlated equilibrium (a restriction where a players behavior must always be approximately optimal, in the worst case over draws from the distribution) we show a linear lower bound, thus separating the query complexity of well supported approximate correlated equilibrium from the standard notion of approximate correlated equilibrium. Finally, we give a query-efficient reduction from the problem of computing an approximate well-supported Nash equilibrium to the problem of verifying a well supported Nash equilibrium, where the additional query overhead is proportional to the description length of the game. This gives a polynomial-query algorithm for computing well supported approximate Nash equilibria (and hence correlated equilibria) in concisely represented games. We identify a class of games (which includes congestion games) in which the reduction can be made not only query efficient, but also computationally efficient.

Exact Learning of Discretized Geometric Concepts

We first present an algorithm that uses membership and equivalence queries to exactly identify a discretized geometric concept defined by the union of m axis-parallel boxes in d-dimensional discretized Euclidean space where each coordinate can have n discrete values. This algorithm receives at most md counterexamples and uses time and membership queries polynomial in m and log n for any constant d. Furthermore, all equivalence queries can be formulated as the union of O(md log m) axis-parallel boxes. Next, we show how to extend our algorithm to efficiently learn, from only equivalence queries, any discretized geometric concept generated from any number of halfspaces with any number of known (to the learner) slopes in a constant dimensional space. In particular, our algorithm exactly learns (from equivalence queries only) unions of discretized axis-parallel boxes in constant dimensional space in polynomial time. Furthermore, this equivalence query only algorithm can be modified to handle a polynomial number of lies in the counterexamples provided by the environment. Finally, we introduce a new complexity measure that better captures the complexity of the union of