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

We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finite-state Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well studied stochastic models, including Stochastic Context-Free Grammars (SCFG) and Multi-Type Branching Processes (MT-BP). We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminates? These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or in-between, and

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

We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems

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

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.

IIS-0905390. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. We also acknowledge Intel Corporation and IBM for their machine gifts. Keywords: Game theory, continuous games, games of imperfect information, equilibrium We present a new approach for solving large (even infinite) multiplayer games of imperfect information. The key idea behind our approach is that we include additional inputs in the form of qualitative models of equilibrium strategies (how the signal space should be qualitatively partitioned into action regions). In addition, we show that our approach can lead to strong strategies in large finite games that we approximate with infinite games. We prove that our main algorithm is correct even if given a set of qualitative models (satisfying a technical property) of which only some are accurate. We also show how to check the output in settings where all of the models might be wrong (under a weak assumption). Our algorithms can compute equilibria in several classes of games for which no prior algorithms have been developed, and we demonstrate that they run efficiently in practice. In the course of our analysis, we also develop the first mixed-integer programming formulations for computing an epsilon-equilibrium in general multiplayer normal and extensive-form

Dedicated to Christos Papadimitriou, the eternal adolescent We survey recent joint work with Christos Papadimitriou and Paul Goldberg on the computational complexity of Nash equilibria. We show that finding a Nash equilibrium in normal form games is computationally intractable, but in an unusual way. It does belong to the class NP; but Nash’s theorem, showing that a Nash equilibrium always exists, makes the possibility that it is also NP-complete rather unlikely. We show instead that the problem is as hard computationally as finding Brouwer fixed points, in a precise technical sense, giving rise to a new complexity class called PPAD. The existence of the Nash equilibrium was established via Brouwer’s fixed-point theorem; hence, we provide a computational converse to Nash’s theorem. To alleviate the negative implications of this result for the predictive power of the Nash equilibrium, it seems natural to study the complexity of approximate equilibria: an efficient approximation scheme would imply that players could in principle come arbitrarily close to a Nash equilibrium given enough time. We review recent work on computing approximate equilibria and conclude by studying how symmetries may affect the structure and approximation of Nash equilibria. Nash showed that every symmetric game has a symmetric equilibrium. We complement this theorem with a rich set of structural results for a broader, and more interesting class of games with symmetries, called anonymous games. 1

We consider Fisher and Arrow-Debreu markets under additively-separable, piecewise-linear, concave utility functions, and obtain the following results: • For both market models, if an equilibrium exists, there is one that is rational and can be written using polynomially many bits. • There is no simple necessary and sufficient condition for the existence of an equilibrium: The problem of checking for existence of an equilibrium is NP-complete for both market models; the same holds for existence of an ɛ-approximate equilibrium, for ɛ = O(n −5). • Under standard (mild) sufficient conditions, the problem of finding an exact equilibrium is in PPAD for both market models. • Finally, building on the techniques of [CDDT09] we prove that under these sufficient conditions, finding an equilibrium for Fisher markets is PPAD-hard.