Download or read book Complexity Results about Nash Equilibria written by Vincent Conitzer and published by . This book was released on 2002 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "Noncooperative game theory provides a normative framework for analyzing strategic interactions. However, for the toolbox to be operational, the solutions it defines will have to be computed. In this paper, we provide a single reduction that 1) demonstrates NP-hardness of determining whether Nash equilibria with certain natural properties exist, and 2) demonstrates the #P-hardness of counting Nash equilibria (or connected sets of Nash equilibria). We also show that 3) determining whether a pure-strategy Bayes-Nash equilibrium exists is NP-hard, and that 4) determining whether a pure-strategy Nash equilibrium exists in a stochastic (Markov) game is PSP ACE-hard even if the game is invisible (this remains NP-hard if the game is finite). All of our hardness results hold even if there are only two players and the game is symmetric."
Download or read book The Complexity of Nash Equilibria written by Konstantinos Daskalakis and published by . This book was released on 2008 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Automata Languages and Programming written by Peter Widmayer and published by Springer Science & Business Media. This book was released on 2002-06-26 with total page 1100 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 29th International Colloquium on Automata, Languages and Programming, ICALP 2002, held in Malaga, Spain, in July 2002. The 83 revised full papers presented together with 7 invited papers were carefully reviewed and selected from a total of 269 submissions. All current aspects of theoretical computer science are addressed and major new results are presented.
Download or read book Hardness of Approximation Between P and NP written by Aviad Rubinstein and published by . This book was released on 2017 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nash equilibrium is the central solution concept in Game Theory. Since Nash's original paper in 1951, it has found countless applications in modeling strategic behavior of traders in markets, (human) drivers and (electronic) routers in congested networks, nations in nuclear disarmament negotiations, and more. A decade ago, the relevance of this solution concept was called into question by computer scientists~\cite{DGP, CDT}, who proved (under appropriate complexity assumptions) that {\em computing} a Nash equilibrium is an intractable problem. And if centralized, specially designed algorithms cannot find Nash equilibria, why should we expect distributed, selfish agents to converge to one? The remaining hope was that at least approximate Nash equilibria can be efficiently computed. Understanding whether there is an efficient algorithm for {\em approximate Nash equilibrium} has been the central open problem in this field for the past decade. In this thesis, we provide strong evidence that even finding an approximate Nash equilibrium is intractable. We prove several intractability theorems for different settings (two-player games and many-player games) and models (computational complexity, query complexity, and communication complexity). In particular, our main result is that under a plausible and natural complexity assumption (``Exponential Time Hypothesis for \PPAD''), there is no polynomial-time algorithm for finding an approximate Nash equilibrium in two-player games. The problem of approximate Nash equilibrium in a two-player game poses a unique technical challenge: it is a member of the class \PPAD, which captures the complexity of several fundamental total problems, i.e. problems that always have a solution; and it also admits a quasipolynomial ($\approx n^{\log n}$) time algorithm. Either property alone is believed to place this problem far below \NP-hard problems in the complexity hierarchy; having both simultaneously places it just above \P, at what can be called the frontier of intractability. Indeed, the tools we develop in this thesis to advance on this frontier are useful for proving hardness of approximation of several other important problems whose complexity lies between \P~and \NP: \begin{description} \item [Brouwer's fixed point] Given a continuous function $f$ mapping a compact convex set to itself, Brouwer's fixed point theorem guarantees that $f$ has a fixed point, i.e. $x$ such that $f(x) = x$. Our intractability result holds for the relaxed problem of finding an approximate fixed point, i.e. $x$ such that $f(x) \approx x$. \item [Market equilibrium] Market equilibrium is a vector of prices and allocations where the supply meets the demand for each good. %We consider the Arrow-Debreu model where agents are both sellers and buyers of goods. Our intractability result holds for the relaxed problem of finding an approximate market equilibrium, where the supply of each good approximately meets the demand. \item [CourseMatch (A-CEEI)] Approximate Competitive Equilibrium from Equal Income (A-CEEI) is the economic principle underlying CourseMatch, a system for fair allocation of classes to students (currently in use at Wharton, University of Pennsylvania). \item [Densest $k$-subgraph] Our intractability result holds for the following relaxation of the $k$-Clique problem: given a graph containing a $k$-clique, the algorithm has to find a subgraph over $k$ vertices that is ``almost a clique'', i.e. most of the edges are present. \item [Community detection] We consider a well-studied model of communities in social networks, where each member of the community is friends with a large fraction of the community, and each non-member is only friends with a small fraction of the community. \item [VC dimension and Littlestone dimension] The Vapnik-Chervonenkis (VC) dimension is a fundamental measure in learning theory that captures the complexity of a binary concept class. Similarly, the Littlestone dimension is a measure of complexity of online learning. \item [Signaling in zero-sum games] We consider a fundamental problem in signaling, where an informed signaler reveals private information about the payoffs in a two-player zero-sum game, with the goal of helping one of the players.
Download or read book The Computational Complexity of Finding Nash Equilibria in Succinctly Represented Games written by Grant Schoenebeck and published by . This book was released on 2004 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Twenty Lectures on Algorithmic Game Theory written by Tim Roughgarden and published by Cambridge University Press. This book was released on 2016-08-30 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer science and economics have engaged in a lively interaction over the past fifteen years, resulting in the new field of algorithmic game theory. Many problems that are central to modern computer science, ranging from resource allocation in large networks to online advertising, involve interactions between multiple self-interested parties. Economics and game theory offer a host of useful models and definitions to reason about such problems. The flow of ideas also travels in the other direction, and concepts from computer science are increasingly important in economics. This book grew out of the author's Stanford University course on algorithmic game theory, and aims to give students and other newcomers a quick and accessible introduction to many of the most important concepts in the field. The book also includes case studies on online advertising, wireless spectrum auctions, kidney exchange, and network management.
Download or read book Algorithms ESA 2007 written by Lars Arge and published by Springer. This book was released on 2007-09-17 with total page 782 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 15th Annual European Symposium on Algorithms, ESA 2007, held in Eilat, Israel, in October 2007 in the context of the combined conference ALGO 2007. The 63 revised full papers presented together with abstracts of three invited lectures address all current subjects in algorithmics reaching from design and analysis issues of algorithms over to real-world applications and engineering of algorithms in various fields.
Download or read book The Complexity of Nash Equilibria in Multiplayer Zero sum Games and Coordination Games written by Yang Cai (Ph. D.) and published by . This book was released on 2010 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt: We prove a generalization of von Neumann's minmax theorem to the class of separable multiplayer zero-sum games, introduced in [Bregman and Fokin 1998]. These games are polymatrix-that is, graphical games in which every edge is a two-player game between its endpoints-in which every outcome has zero total sum of players' payoffs. Our generalization of the minmax theorem implies convexity of equilibria, polynomial-time tractability, and convergence of no-regret learning algorithms to Nash equilibria. Given that three player zero-sum games are already PPAD-complete, this class of games, i.e. with pairwise separable utility functions, defines essentially the broadest class of multi-player constants sum games to which we can hope to push tractability results. Our result is obtained by establishing a certain game-class collapse, showing that separable constant-sum games are payoff equivalent to pairwise constant-sum polymatrix games-polymatrix games in which all edges are constant-sum games, and invoking a recent result of [Daskalakis, Papadimitriou 2009] for these games. We also explore generalizations to classes of non- constant-sum multi-player games. A natural candidate is polymatrix games with strictly competitive games on their edges. In the two player setting, such games are minmax solvable and recent work has shown that they are merely affine transformations of zero-sum games [Adler, Daskalakis, Papadimitriou 2009]. Surprisingly we show that a polymatrix game comprising of strictly competitive games on its edges is PPAD-complete to solve, proving a striking difference in the complexity of networks of zero-sum and strictly competitive games. Finally, we look at the role of coordination in networked interactions, studying the complexity of polymatrix games with a mixture of coordination and zero-sum games. We show that finding a pure Nash equilibrium in coordination-only polymatrix games is PLS-complete; hence, computing a mixed Nash equilibrium is in PLS n PPAD, but it remains open whether the problem is in P. If, on the other hand, coordination and zero-sum games are combined, we show that the problem becomes PPAD-complete, establishing that coordination and zero-sum games achieve the full generality of PPAD. This work is done in collaboration with Costis Daskalakis.
Download or read book The complexity of Nash Equilibria Local Optima and Pareto Optimal solutions written by Heiko Röglin and published by . This book was released on 2008 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Complexity of Pure Nash Equilibria in Weighted Max congestion Games written by Àlvarez Faura Àlvarez and published by . This book was released on 2008 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Stability and Perfection of Nash Equilibria written by Eric van Damme and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: I have been pleased with the favourable reception of the first edition of this book and I am grateful to have the opportunity to prepare this second edition. In this revised and enlarged edition I corrected some misprints and errors that occurred in the first edition (fortunately I didn't find too many) and I added a large number of notes that give the reader an impression of what kind of results have been obtained since the first edition was printed and that give an indication of the direction the subject is taking. Many of the notes discuss (or refer to papers discussing) applications of the refinements that are considered. Of course, it is the quantity and the quality of the insights and the applications that lend the refinements their validity. Although the guide to the applications is far from complete, the notes certainly allow the reader to form a good judgement of which refinements have really yielded new insights. Hence, as in the first edition, I will refrain from speculating on which refinements of Nash equilibria will survive in the long run. To defend this position let me also cite Binmore [1990] who compares writing about refinements to the Herculean task of defeating the nine-headed Hydra which grew too heads for each that was struck off. It is a pleasure to have the opportunity to thank my secretary, Marjoleine de Wit, who skilfully and, as always, cheerfully typed the manuscript and did the proofreading.
Download or read book Existence and Stability of Nash Equilibrium written by Guilherme Carmona and published by World Scientific. This book was released on 2013 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book aims at describing the recent developments in the existence and stability of Nash equilibrium. The two topics are central to game theory and economics and have been extensively researched. Recent results on existence and stability of Nash equilibrium are scattered and the relationship between them has not been explained clearly. The book will make these results easily accessible and understandable to researchers in the field. Book jacket.
Download or read book Algorithmic Game Theory written by Giuseppe Persiano and published by Springer Science & Business Media. This book was released on 2011-10-07 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the Fourth International Symposium on Algorithmic Game Theory, SAGT 2011, held in Amalfi, Italy, in October 2011. The 26 revised full papers presented together with 2 invited lectures were carefully reviewed and selected from 65 submissions. The papers are organized in topical sections on auctions and advertising, quality of solutions, externalities, mechanism design, complexity, network games, pricing, as well as routing games.
Download or read book The Complexity of Pure Nash Equilibria in Max congestion Games written by Guillem Francès and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We study Network Max-Congestion Games (NMC games, for short), a class of network games where each player tries to minimize the most congested edge along the path he uses as strategy. We focus our study on the complexity of computing a pure Nash equilibria in this kind of games. We show that, for single-commodity games with non-decreasing delay functions, this problem is in P when either all the paths from the source to the target node are disjoint or all the delay functions are equal. For the general case, we prove that the computation of a PNE belongs to the complexity class PLS through a new technique based on generalized ordinal potential functions and a slightly modified definition of the usual local search neighborhood. We further apply this technique to a different class of games (which we call Pareto-efficient) with restricted cost functions. Finally, we also prove some PLS-hardness results, showing that computing a PNE for Pareto-efficient NMC games is indeed a PLS-complete problem.
Download or read book Constraint Processing written by Rina Dechter and published by Morgan Kaufmann. This book was released on 2003-05-05 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: Constraint reasoning has matured over the last three decades with contributions from a diverse community of researchers in artificial intelligence, databases and programming languages, operations research, management science, and applied mathematics. In Constraint Processing, Rina Dechter synthesizes these contributions, as well as her own significant work, to provide the first comprehensive examination of the theory that underlies constraint processing algorithms.
Download or read book Stochastic Multiplayer Games written by Michael Ummels and published by Amsterdam University Press. This book was released on 2010-12 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic games provide a versatile model for reactive systems that are affected by random events. This dissertation advances the algorithmic theory of stochastic games to incorporate multiple players, whose objectives are not necessarily conflicting. The basis of this work is a comprehensive complexity-theoretic analysis of the standard game-theoretic solution concepts in the context of stochastic games over a finite state space. One main result is that the constrained existence of a Nash equilibrium becomes undecidable in this setting. This impossibility result is accompanied by several positive results, including efficient algorithms for natural special cases.
Download or read book Algorithms and Computation written by Xiaotie Deng and published by Springer Science & Business Media. This book was released on 2005-12-09 with total page 1209 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 16th International Symposium on Algorithms and Computation, ISAAC 2005, held in Sanya, Hainan, China in December 2005. The 112 revised full papers presented were carefully reviewed and selected from 549 submissions. The papers are organized in topical sections on computational geometry, computational optimization, graph drawing and graph algorithms, computational complexity, approximation algorithms, internet algorithms, quantum computing and cryptography, data structure, computational biology, experimental algorithm mehodologies and online algorithms, randomized algorithms, parallel and distributed algorithms.
