We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any non-trivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as a more general arithmetic circuit decision problem that characterizes P-time in a unit-cost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point

One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally hard in the worst-case under reasonable assumptions. In this paper, we study the empirical consequences of strengthening consumer choice theory to enforce that utilities are computationally easy to maximize. We prove the possibly surprising result that computational constraints have no empirical consequences whatsoever for consumer choice theory. That is, a data set is consistent with a utility maximizing consumer if and only if a data set is consistent with a utility maximizing consumer having a utility function that can be maximized in strongly polynomial time. Our result motivates a general approach for posing questions about the empirical content of computational constraints: the revealed preference approach to computational complexity. The approach complements the conventional worst-case view of computational complexity in important ways, and is methodologically close to mainstream economics.

The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2-player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the coefficients in its convex program are “small”, then its solution can be found in strongly polynomial time. We also give a non-succinct linear game whose solution can be found in strongly polynomial time. 1

Abstract — In a landmark paper [32], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [11], [16], [5], [6], [7], [8] has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A major goal of this work is to expand the universe of known PPAD-complete problems. We resolve the computational complexity of a number of outstanding open problems with practical applications. Here is the list of problems we show to be PPAD-complete, along with the domains of practical significance: Fractional Stable

We consider two-player zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a long-standing open question if a polynomial algorithm exists that solves BWR-games. In fact, a pseudo-polynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only two classes are known to have such a pseudo-polynomial algorithm: BW-games (the case with no random nodes) and ergodic BWR-games (in which the game’s value does not depend on the initial position) with constant number of random nodes. We show that the existence of a pseudo-polynomial algorithm for BWR-games with a constant number of random vertices implies smoothed polynomial complexity and the existence of absolute and relative polynomial-time approximation schemes. In particular, we obtain smoothed polynomial complexity and derive absolute and relative approximation schemes for BW-games and ergodic BWR-games (assuming a technical requirement about the probabilities at the random nodes). 1.

We continue the recently initiated study of the computational aspects of weak saddles, an ordinal set-valued solution concept proposed by Shapley. Brandt et al. gave a polynomial-time algorithm for computing weak saddles in a subclass of matrix games, and showed that certain problems associated with weak saddles of bimatrix games are NP-complete. The important question of whether weak saddles can be found efficiently was left open. We answer this question in the negative by showing that finding weak saddles of bimatrix games is NP-hard, under polynomial-time Turing reductions. We moreover prove that recognizing weak saddles is coNP-complete, and that deciding whether a given action is contained in some weak saddle is hard for parallel access to NP and thus not even in NP unless the polynomial hierarchy collapses. Our hardness results are finally shown to carry over to a natural weakening of weak saddles.

This paper studies -- from the perspective of efficient computation -- a type of competition that is widespread throughout the plant and animal kingdoms, higher education, politics and artificial contests. In this setting, an agent gains utility from his relative performance (on some measurable criterion) against other agents, as opposed to his absolute performance. We model this situation using ranking games in which each strategy corresponds to a level of competitiveness, and incurs an upfront cost that is higher for more competitive strategies. We study the Nash equilibria of these games, and polynomial-time algorithms for computing them. For games in which there is no tie between agents’ levels of competitiveness we give a polynomial-time algorithm for computing an exact equilibrium in the 2-player case, and a characterization of Nash equilibria that shows an interesting parallel between these games and unrestricted 2-player games in normal form. When ties are allowed, via a reduction from these games to a subclass of anonymous games, we give polynomial-time approximation schemes for two special cases: constant-sized set of strategies, and constant number of players. The latter result is improved to a fully polynomial-time approximation scheme when the constant number of players only compete to win the game, i.e. to be ranked first.

One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally hard in the worst-case under reasonable assumptions. In this paper, we study the empirical consequences of strengthening consumer choice theory to enforce that utilities are computationally easy to maximize. We prove the possibly surprising result that computational constraints have no empirical consequences whatsoever for consumer choice theory. That is, a data set is consistent with a utility maximizing consumer if and only if a data set is consistent with a utility maximizing consumer having a utility function that can be maximized in strongly polynomial time. Our result motivates a general approach for posing questions about the empirical content of computational constraints: the revealed preference approach to computational complexity. The approach complements the conventional worst-case view of computational complexity in important ways, and is methodologically close to mainstream economics.

The k-means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the k-means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences. 1 k-Means Clustering The problem of clustering data into classes is ubiquitous in computer science, with applications ranging from computational biology over machine learning to image analysis. The k-means method is a very simple and implementation-friendly local improvement heuristic for clustering. It is used to partition a set X of n d-dimensional data points into k clusters. (The number k of clusters is fixed in advance.) In k-means clustering, our goal is not only to get a clustering of the data points, but also to get a center ci for each cluster Xi of the clustering X1,..., Xk. A center can be viewed as a representative of its cluster. We do not require centers to be among the data points, but they can be arbitrary points. The goal is to find a “good ” clustering, where “good ” means that the clustering should minimize the objective function k ∑ ∑ δ(x, ci). i=1 x∈Xi Here, δ denotes a distance measure. In the following, we will mainly use squared Euclidean distances, i.e., δ(x, ci) = ‖x − ci‖2. Of course, given the cluster centers c1,..., ck ∈ Rd, each point x ∈ X should be assigned to the cluster Xi whose center ci is closest to x. On the other hand, given a clustering X1,..., Xk

Nash equilibria of two-player games are much easier to compute in practice than those of n-player games, even though the two problems have the same asymptotic complexity. We used a recent constructive reduction to solve general games using a two-player algorithm. However, the reduction increases the game size too much to be practically usable. An open problem is to find a more compact constructive reduction, which might make this approach feasible. Categories and Subject Descriptors: F.2.0 [Analysis of algorithms and problem complexity]: