Complexity results for some classes of strategic games

Game theory is a branch of applied mathematics studying the interaction of self-interested entities, so-called agents. Its central objects of study are games, mathematical models of real-world interaction, and solution concepts that single out certain outcomes of a game that are meaningful in some way. The solutions thus produced can then be viewed both from a descriptive and from a normative perspective. The rise of the Internet as a computational platform where a substantial part of today's strategic interaction takes place has spurred additional interest in game theory as an analytical tool, and has brought it to the attention of a wider audience in computer science. An important aspect of real-world decision-making, and one that has received only little attention in the early days of game theory, is that agents may be subject to resource constraints. The young field of algorithmic game theory has set out to address this shortcoming using techniques from computer science, and in particular from computational complexity theory. One of the defining problems of algorithmic game theory concerns the computation of solution concepts. Finding a Nash equilibrium, for example, i.e., an outcome where no single agent can gain by changing his strategy, was considered one of the most important problems on the boundary of P, the complexity class commonly associated with efficient computation, until it was recently shown complete for the class PPAD. This rather negative result for general games has not settled the question, however, but immediately raises several new ones: First, can Nash equilibria be approximated, i.e., is it possible to efficiently find a solution such that the potential gain from a unilateral deviation is small? Second, are there interesting classes of games that do allow for an exact solution to be computed efficiently? Third, are there alternative solution concepts that are computationally tractable, and how does the value of solutions selected by these concepts compare to those selected by established solution concepts? The work reported in this thesis is part of the effort to answer the latter two questions. We study the complexity of well-known solution concepts, like Nash equilibrium and iterated dominance, in various classes of games that are both natural and practically relevant: ranking games, where outcomes are rankings of the players; anonymous games, where players do not distinguish between the other players in the game; and graphical games, where the well-being of any particular player depends only on the actions of a small group other players. In ranking games, we further compare the payoffs obtainable in Nash equilibrium outcomes with those of alternative solution concepts that are easy to compute. We finally study, in general games, solution concepts that try to remedy some of the shortcomings associated with Nash equilibrium, like the need for randomization to achieve a stable outcome

Dynamic strategic interactions : analysis and mechanism design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 225-232).Modern systems, such as engineering systems with autonomous entities, markets, and financial networks, consist of self-interested agents with potentially conflicting objectives. These agents interact in a dynamic manner, modifying their strategies over time to improve their payoffs. The presence of self-interested agents in such systems, necessitates the analysis of the impact of multi-agent decision making on the overall system, and the design of new systems with improved performance guarantees. Motivated by this observation, in the first part of this thesis we focus on fundamental structural properties of games, and exploit them to provide a new framework for analyzing the limiting behavior of strategy update rules in various game-theoretic settings. In the second part, we investigate the design problem of an auctioneer who uses iterative multi-- item auctions for efficient allocation of resources. More specifically, in the first part of the thesis we focus on potential games, a special class of games with desirable equilibrium and dynamic properties, and analyze their preference structure. Exploiting this structure we obtain a decomposition of arbitrary games into three components, which we refer to as the potential, harmonic, and nonstrategic components. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents conflicts between the interests of the players. We make this intuition precise by studying the properties of these two components, and establish that indeed they have quite distinct and remarkable characteristics. The decomposition also allows us to approximate a given game with a potential game. We show that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of a potential game that approximates it. The decomposition provides a valuable tool for the analysis of dynamics in games. Earlier literature established that many natural strategy update rules converge to a Nash equilibrium in potential games. We show that games that are close to a potential game exhibit similar properties. In particular, we focus on three commonly studied discrete-time update rules (better/best response, logit response, and discrete-time fictitious play dynamics), and establish that in near-potential games, the limiting behavior of these update rules can be characterized by an approximate equilibrium set, size of which is proportional to the distance of the original game from a potential game. Since a close potential game to a given game can be systematically found via decomposition, our results suggest a systematic framework for studying the limiting behavior of adaptive dynamics in arbitrary finite strategic form games: the limiting behavior of dynamics in a given game can be characterized by first approximating this game with a potential game, and then analyzing the limiting behavior of dynamics in the potential game. In the second part of the thesis, we change our focus to implementing efficient outcomes in multi-agent settings by using simple mechanisms. In particular, we develop novel efficient iterative auction formats for multi-item environments, where items exhibit value complementarities/substitutabilities. We obtain our results by focusing on a special class of value functions, which we refer to as graphical valuations. These valuations are not fully general, but importantly they capture value complementarity/substitutability in important practical settings, while allowing for a compact representation of the value functions. We start our analysis by first analyzing how the special structure of graphical valuations can be exploited to design simple iterative auction formats. We show that in settings where the underlying value graph is a tree (and satisfies an additional technical condition), a Walrasian equilibrium always exists (even in the presence of value complementarities). Using this result we provide a linear programming formulation of the efficient allocation problem for this class of valuations. Additionally, we demonstrate that a Walrasian equilibrium may not exist, when the underlying value graph is more general. However, we also establish that in this case a more general pricing equilibrium always exists, and provide a stronger linear programming formulation that can be used to identify the efficient allocation for general graphical valuations. We then consider solutions of these linear programming formulations using iterative algorithms. Complementing these iterative algorithms with appropriate payment rules, we obtain iterative auction formats that implement the efficient outcome at an (ex-post perfect) equilibrium. The auction formats we obtain rely on simple pricing rules that, in the most general case, require offering a bidder-specific price for each item, and bidder-specific discounts/markups for pairs of items. Our results in this part of the thesis suggest that when value functions of bidders exhibit some special structure, it is possible to systematically exploit this structure in order to develop simple efficient iterative auction formats.by Utku Ozan Candogan.Ph.D

The Complexity of Games on Highly Regular Graphs (Extended Abstract)

We study from the complexity point of view the problem of finding equilibria in games defined by highly regular graphs with extremely succinct representation, such as the d-dimensional grid; we argue that such games are of interest in the modelling of large systems of interacting agents. We show that the problem of determining whether such a game on the d-dimensional grid has a pure Nash equilibrium depends on d, and the dichotomy is remarkably sharp: It is in P when d = 1, but NEXP-complete for 2. In contrast, we prove that mixed Nash equilibria can be found in deterministic exponential time for any d, by quantifier elimination