Download or read book Approximate Pure Nash Equilibria in Congestion Opinion Formation and Facility Location Games written by Matthias Feldotto and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis investigates approximate pure Nash equilibria in different game-theoretic models. In such an outcome, no player can improve her objective by more than a given factor through a deviation to another strategy. In the first part, we investigate two variants of Congestion Games in which the existence of pure Nash equilibria is guaranteed through a potential function argument. However, the computation of such equilibria might be hard. We construct and analyze approximation algorithms that enable the computation of states with low approximation factors in polynomial time. To show their guarantees we use sub games among players, bound the potential function values of arbitrary states and exploit a connection between Shapley and proportional cost shares. Furthermore, we apply and analyze sampling techniques for the computation of approximate Shapley values in different settings. In the second part, we concentrate on the existence of approximate pure Nash equilibria in games in which no pure Nash equilibria exist in general. In the model of Coevolving Opinion Formation Games, we bound the approximation guarantees for natural states nearly independent of the specific definition of the players' neighborhoods by applying a concept of virtual costs. For the special case of only one influential neighbor, we even show lower approximation factors for a natural strategy. Then, we investigate a two-sided Facility Location Game among facilities and clients on a line with an objective function consisting of distance and load. We show tight bounds on the approximation factor for settings with three facilities and infinitely many clients. For the general scenario with an arbitrary number of facilities, we bound the approximation factor for two promising candidates, namely facilities that are uniformly distributed and which are paired. ; eng

Download or read book Network Flow Problems and Congestion Games written by Carol Meyers (Ph. D.) and published by . This book was released on 2006 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: (Cont.) We first address the complexity of finding an optimal minimum cost solution to a congestion game. We consider both network and general congestion games, and we examine several variants of the problem concerning the structure of the game and its associated cost functions. Many of the problem variants are NP-hard, though we do identify several versions of the games that are solvable in polynomial time. We then investigate existence and the price of anarchy of pure Nash equilibria in k-splittable congestion games with linear costs. A k-splittable congestion game is one in which each player may split its flow on at most k different paths. We identify conditions for the existence of equilibria by providing a series of potential functions. For the price of anarchy, we show an asymptotic lower bound of 2.4 for unweighted k-splittable congestion games and 2.401 for weighted k-splittable congestion games, and an upper bound of 2.618 in both cases.

Download or read book Existence And Stability Of Nash Equilibrium written by Guilherme Carmona and published by World Scientific. This book was released on 2012-11-29 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.

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 Approximation Guarantees for Game theoretic Equilibria written by Kshipra Bhawalkar and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Game-theoretic equilibria model natural outcomes of selfish behavior. The concept of the "Price of Anarchy (POA)" captures how well game-theoretic equilibria approximate the socially optimal outcome. In this dissertation, we consider the POA in three different models, as a function of different design parameters and equilibrium concepts. We first consider weighted congestion games where players compete to share resources. The cost of a resource depends on the total weight of the players using that resource and players are allowed to use only certain subsets of the resources. Routing games are a canonical example of such games. We provide a precise characterization of the POA as a function of the resource cost functions. We also explore how the POA is affected by the structure of the permitted strategies for the players. Our bounds apply to pure Nash, mixed Nash, correlated, and coarse-correlated equilibria. We next consider opinion formation games, where players in a social network choose opinions to express to minimize the cost of disagreement with their neighbors' opinions and their own intrinsic beliefs. We obtain bounds on the POA with respect to the pure Nash, mixed Nash, and correlated equilibria. We show that the POA is always at most 2 when the cost functions are convex. We provide a more detailed characterization of the the Price of Anarchy as a function of the costs of disagreement. Finally, we consider combinatorial auctions with item bidding. In this model, several goods are sold simultaneously in independent auctions to buyers who have different values for different bundles of goods. We bound the pure Nash and the Bayes-Nash POA when the individual single-item auctions are second price auctions and buyers' valuations over the bundles are subadditive. We provide the first explicit gap between the worst case pure Nash and Bayes-Nash POA. We also consider how the pure Nash POA varies as a function of the payment rule in the underlying single-item auction.

Download or read book Existence and Properties of Pure Nash Equilibria in Budget Games written by Maximilian Drees and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Approximate Equilibria in Large Games written by Yu Wu and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The complexity of studying Nash equilibrium in large games often scales with the size of the system: as it increases, computing the exact Nash equilibrium can soon become intractable. However, when the number of players in the system approaches infinity and no individual player has a significant impact on the system, we can approximate the system by considering each single player no longer playing against other individual players but a single aggregation of all other players. In this paper, we apply this idea to study and approximate Nash equilibria in two large scale games. In part I, we consider a model of priced resource sharing that combines both queueing behavior and strategic behavior. We study a priority service model where a single server allocates its capacity to agents in proportion to their payment to the system, and users from different classes act to minimize the sum of their cost for processing delay and payment. As the exact processing time of this system is hard to compute and cannot be characterized in closed form, we introduce the concept of aggregate equilibrium to approximate the exact Nash equilibrium, by assuming each individual player plays against a common aggregate priority that characterizes the large system. We then introduce the notion of heavy traffic equilibrium as an alternative approximation of the Nash equilibrium, derived by considering the asymptotic regime where the system load approaches capacity. We show that both aggregate equilibrium and heavy traffic equilibrium are asymptotically exact in heavy traffic. We present some numerical results for both approximate equilibria, and discuss efficiency and revenue, and in particular provide a bound for the price of anarchy of the heavy traffic equilibrium. In part II, we study the reputation system of large scale online marketplace. We develop a large market model to study reputation mechanisms in online marketplaces. We consider two types of sellers: commitment sellers, who are intrinsically honest but may be unable to accurately describe items because of limited expertise; and strategic sellers, who are driven by a profit maximization motive. We focus on stationary equilibria for this dynamic market, in particular, on separating equilibria where strategic sellers are incentivized to describe the items they have for sale truthfully, and characterize the conditions under which such equilibria exist. We then complement our theoretical results with computational analysis and provide insights on the features of markets that may incentivize truthfulness in equilibrium.

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 On Backward Induction Paths and Pure Strategy Nash Equilibria of Congestion Games written by Igal Milchtaich and published by . This book was released on 1996 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 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 Game Theory Alive written by Anna R. Karlin and published by American Mathematical Soc.. This book was released on 2017-04-27 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Algorithmic Game Theory written by Tobias Harks and published by Springer Nature. This book was released on 2020-09-08 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 13th International Symposium on Algorithmic Game Theory, SAGT 2020, held in Augsburg, Germany, in September 2020.* The 21 full papers presented together with 3 abstract papers were carefully reviewed and selected from 53 submissions. The papers are organized in topical sections named: auctions and mechanism design, congestion games and flows over time, markets and matchings, scheduling and games on graphs, and social choice and cooperative games. * The conference was held virtually due to the COVID-19 pandemic.

Download or read book Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing written by Hamed Hatami and published by . This book was released on 2017-06-19 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: STOC '17: Symposium on Theory of Computing Jun 19, 2017-Jun 23, 2017 Montreal, Canada. You can view more information about this proceeding and all of ACM�s other published conference proceedings from the ACM Digital Library: http://www.acm.org/dl.

Download or read book Selfish Routing and the Price of Anarchy written by Tim Roughgarden and published by MIT Press. This book was released on 2023-09-19 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt: An analysis of the loss in performance caused by selfish, uncoordinated behavior in networks. Most of us prefer to commute by the shortest route available, without taking into account the traffic congestion that we cause for others. Many networks, including computer networks, suffer from some type of this "selfish routing." In Selfish Routing and the Price of Anarchy, Tim Roughgarden studies the loss of social welfare caused by selfish, uncoordinated behavior in networks. He quantifies the price of anarchy—the worst-possible loss of social welfare from selfish routing—and also discusses several methods for improving the price of anarchy with centralized control. Roughgarden begins with a relatively nontechnical introduction to selfish routing, describing two important examples that motivate the problems that follow. The first, Pigou's Example, demonstrates that selfish behavior need not generate a socially optimal outcome. The second, the counterintiuitve Braess's Paradox, shows that network improvements can degrade network performance. He then develops techniques for quantifying the price of anarchy (with Pigou's Example playing a central role). Next, he analyzes Braess's Paradox and the computational complexity of detecting it algorithmically, and he describes Stackelberg routing, which improves the price of anarchy using a modest degree of central control. Finally, he defines several open problems that may inspire further research. Roughgarden's work will be of interest not only to researchers and graduate students in theoretical computer science and optimization but also to other computer scientists, as well as to economists, electrical engineers, and mathematicians.

Download or read book Simple Adaptive Strategies From Regret matching To Uncoupled Dynamics written by Sergiu Hart and published by World Scientific. This book was released on 2013-01-22 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume collects almost two decades of joint work of Sergiu Hart and Andreu Mas-Colell on game dynamics and equilibria. The starting point was the introduction of the adaptive strategy called regret-matching, which on the one hand is simple and natural, and on the other is shown to lead to correlated equilibria. This initial finding — boundedly rational behavior that yields fully rational outcomes in the long run — generated a large body of work on the dynamics of simple adaptive strategies. In particular, a natural condition on dynamics was identified: uncoupledness, whereby decision-makers do not know each other's payoffs and utilities (so, while chosen actions may be observable, the motivations are not). This condition turns out to severely limit the equilibria that can be reached. Interestingly, there are connections to the behavioral and neurobiological sciences and also to computer science and engineering (e.g., via notions of “regret”).Simple Adaptive Strategies is self-contained and unified in its presentation. Together with the formal treatment of concepts, theorems, and proofs, significant space is devoted to informal explanations and illuminating examples. It may be used for advanced graduate courses — in game theory, economics, mathematics, computer science, engineering — and for further research.

Download or read book The Design of Competitive Online Algorithms Via a Primal Dual Approach written by Niv Buchbinder and published by Now Publishers Inc. This book was released on 2009 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Extends the primal-dual method to the setting of online algorithms, and shows its applicability to a wide variety of fundamental problems.