Nash Equilibria Gale Strings And Perfect Matchings PDF Download
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Author | : Julian Merschen |
Publisher | : |
Total Pages | : |
Release | : 2011 |
Genre | : |
ISBN | : |
Download Nash Equilibria, Gale Strings, and Perfect Matchings Book in PDF, ePub and Kindle
This thesis concerns the problem 2-NASH of finding a Nash equilibrium of a bimatrix game, for the special class of so-called "hard-to-solve" bimatrix games. The term "hardto-solve" relates to the exponential running time of the famous and often used Lemke- Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the "Gale evenness condition" that characterise the vertices of these polytopes. We define the combinatorial problem "Another completely labeled Gale string" whose solutions define the Nash equilibria of any game defined by cyclic polytopes, including the games where the Lemke-Howson algorithm takes exponential time. We show that "Another completely labeled Gale string" is solvable in polynomial time by a reduction to the "Perfect matching" problem in Euler graphs. We adapt the Lemke-Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemke-Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in finding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difficulties concerning a general polynomial-time algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful.
Author | : Anna R. Karlin |
Publisher | : American Mathematical Soc. |
Total Pages | : 400 |
Release | : 2017-04-27 |
Genre | : Mathematics |
ISBN | : 1470419823 |
Download Game Theory, Alive Book in PDF, ePub and Kindle
We live in a highly connected world with multiple self-interested agents interacting and myriad opportunities for conflict and cooperation. The goal of game theory is to understand these opportunities. This book presents a rigorous introduction to the mathematics of game theory without losing sight of the joy of the subject. This is done by focusing on theoretical highlights (e.g., at least six Nobel Prize winning results are developed from scratch) and by presenting exciting connections of game theory to other fields such as computer science (algorithmic game theory), economics (auctions and matching markets), social choice (voting theory), biology (signaling and evolutionary stability), and learning theory. Both classical topics, such as zero-sum games, and modern topics, such as sponsored search auctions, are covered. Along the way, beautiful mathematical tools used in game theory are introduced, including convexity, fixed-point theorems, and probabilistic arguments. The book is appropriate for a first course in game theory at either the undergraduate or graduate level, whether in mathematics, economics, computer science, or statistics. The importance of game-theoretic thinking transcends the academic setting—for every action we take, we must consider not only its direct effects, but also how it influences the incentives of others.
Author | : Tim Roughgarden |
Publisher | : Cambridge University Press |
Total Pages | : 356 |
Release | : 2016-08-30 |
Genre | : Computers |
ISBN | : 1316781178 |
Download Twenty Lectures on Algorithmic Game Theory Book in PDF, ePub and Kindle
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.
Author | : Bernhard von Stengel |
Publisher | : Cambridge University Press |
Total Pages | : 375 |
Release | : 2021-08-19 |
Genre | : Business & Economics |
ISBN | : 1108843301 |
Download Game Theory Basics Book in PDF, ePub and Kindle
A lively introduction to Game Theory, ideal for students in mathematics, computer science, or economics.
Author | : Nicolo Cesa-Bianchi |
Publisher | : Cambridge University Press |
Total Pages | : 4 |
Release | : 2006-03-13 |
Genre | : Computers |
ISBN | : 113945482X |
Download Prediction, Learning, and Games Book in PDF, ePub and Kindle
This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.
Author | : Shing-Tung Yau |
Publisher | : Il Saggiatore |
Total Pages | : 398 |
Release | : 2010-09-07 |
Genre | : Mathematics |
ISBN | : 0465020232 |
Download The Shape of Inner Space Book in PDF, ePub and Kindle
The leading mind behind the mathematics of string theory discusses how geometry explains the universe we see. Illustrations.
Author | : Peter C. Cramton |
Publisher | : MIT Press (MA) |
Total Pages | : 678 |
Release | : 2006 |
Genre | : Business & Economics |
ISBN | : |
Download Combinatorial Auctions Book in PDF, ePub and Kindle
A synthesis of theoretical and practical research on combinatorial auctions from the perspectives of economics, operations research, and computer science.
Author | : Lars Ljungqvist |
Publisher | : MIT Press |
Total Pages | : 1120 |
Release | : 2004 |
Genre | : Business & Economics |
ISBN | : 9780262122740 |
Download Recursive Macroeconomic Theory Book in PDF, ePub and Kindle
A significant new edition of a text that offers both tools and sample applications; extensive revisions and seven new chapters improve and expand upon the original treatment.
Author | : Ulle Endriss |
Publisher | : Lulu.com |
Total Pages | : 424 |
Release | : 2017 |
Genre | : Computers |
ISBN | : 1326912097 |
Download Trends in Computational Social Choice Book in PDF, ePub and Kindle
Computational social choice is concerned with the design and analysis of methods for collective decision making. It is a research area that is located at the interface of computer science and economics. The central question studied in computational social choice is that of how best to aggregate the individual points of view of several agents, so as to arrive at a reasonable compromise. Examples include tallying the votes cast in an election, aggregating the professional opinions of several experts, and finding a fair manner of dividing a set of resources amongst the members of a group -- Back cover.
Author | : Herbert Gintis |
Publisher | : MIT Press |
Total Pages | : 430 |
Release | : 2005 |
Genre | : Business & Economics |
ISBN | : 9780262072526 |
Download Moral Sentiments and Material Interests Book in PDF, ePub and Kindle
Moral Sentiments and Material Interests presents an innovative synthesis of research in different disciplines to argue that cooperation stems not from the stereotypical selfish agent acting out of disguised self-interest but from the presence of "strong reciprocators" in a social group. Presenting an overview of research in economics, anthropology, evolutionary and human biology, social psychology, and sociology, the book deals with both the theoretical foundations and the policy implications of this explanation for cooperation. Chapter authors in the remaining parts of the book discuss the behavioral ecology of cooperation in humans and nonhuman primates, modeling and testing strong reciprocity in economic scenarios, and reciprocity and social policy. The evidence for strong reciprocity in the book includes experiments using the famous Ultimatum Game (in which two players must agree on how to split a certain amount of money or they both get nothing.)