On the computation of stable sets for bimatrix games

In this paper, an alternative definition of stable sets, defined by mertens [mertens, 1989. Stable equilibria – a reformulation. Part i. Definitions and basic properties. Mathematics of operations research 14, 575–625], is given where perturbations are interpreted as restrictions on the strategy space instead of perturbations of the payoffs. This alternative interpretation is then used to compute a special type of stable sets –called standard stable sets – in the context of bimatrix games, exclusively using linear optimization techniques and finite enumerations

Geometry and equilibria in bimatrix games

This thesis studies the application of geometric concepts and methods in the analysis of strategic-form games, in particular bimatrix games. Our focus is on three geometric concepts: the index, geometric algorithms for the computation of Nash equilibria, and polytopes. The contribution of this thesis consists of three parts. First, we present an algorithm for the computation of the index in degenerate bimatrix games. For this, we define a new concept, the “lex-index” of an extreme equilibrium, which is an extension of the standard index. The index of an equilibrium component is easily computable as the sum of the lex-indices of all extreme equilibria of that component. Second, we give several new results on the linear tracing procedure, and its bimatrix game implementation, the van den Elzen-Talman (ET) algorithm. We compare the ET algorithm to two other algorithms: On the one hand, we show that the Lemke-Howson algorithm, the classic method for equilibrium computation in bimatrix games, and the ET algorithm differ substantially. On the other hand, we prove that the ET algorithm, or more generally, the linear tracing procedure, is a special case of the global Newton method, a geometric algorithm for the computation of equilibria in strategic-form games. As the main result of this part of the thesis, we show that there is a generic class of bimatrix games in which an equilibrium of positive index is not traceable by the ET algorithm. This result answers an open question regarding sustainability. The last part of this thesis studies the index in symmetric games. We use a construction of polytopes to prove a new result on the symmetric index: A symmetric equilibrium has symmetric index +1 if and only if it is “potentially unique”, in the sense that there is an extended symmetric game, with additional strategies for the players, where the given symmetric equilibrium is unique

Computing Simply Stable Equilibria

Abstract: For each two-player game, a linear-programming algorithm finds a component of the Nash equilibria and a subset of its perfect equilibria that are simply stable: there are nearby equilibria for each nearby game that perturbs one strategy's probability or payoff more than others. Keywords: Equilibrium, stability, algorithm, linear complementarity. Basically, a set of equilibria is stable if every game nearby has equilibria nearby. They study several specifications of the neighborhood of a game and select the smallest for their definition; In this article we use a different, essentially smaller, neighborhood and therefore a weaker refinement called simple stability. A practical advantage of this refinement is that it enables an elementary procedure for computing a simply-stable set of equilibria of a two-player game. A set of perfect equilibria within a single connected component is simply stable if each game obtained by perturbing some strategy's payoff or minimal probability has equilibria near this set. This criterion is weaker in that it considers only perturbations of pure strategies, rather than mixed strategies as in Kohlberg and Mertens. It is also mildly stronger in that it confines the set to perfect equilibria in a single component, and it allows perturbations of both probabilities and payoffs. 2 These features reflect partially 1 NSF grants SES 8908269 and 9207850 provided financial support and Faruk Gul provided intellectual support. An STSC APL II version of the computer program is available from the author, and a faster C version has been prepared for the game solver Gambit by 2 As in Kohlberg and Mertens, perturbing a strategy's minimal probability perturbs 1 the motivations for Mertens' and Hillas' refinements. We describe a numerical algorithm that constructs a simply-stable set comprising at most 2n extreme points of some component, where n is the number of pure strategies. Jansen, Jurg, and Borm (1990) for two-player games, and After a review of the topic in x1, Part I explains the algorithm in geometric terms, emphasizing the main ideas. Part II presents an algebraic construction and a combinatorial proof that the algorithm works. other players' payoffs from all their strategies, whereas perturbing its payoff gives its player a bonus for using that strategy. Thus, like stability, simple stability weakens hyperstability, which considers all payoff perturbations, by considering a restricted set of perturbations. If a game has only pure strategies (e.g., all mixed strategies are represented explicitly as pure strategies) then simple stability implies stability

Scaling Empirical Game-Theoretic Analysis.

To analyze the incentive structure of strategic multi-agent interactions, such scenarios are often cast as games, where players optimize their payoffs by selecting a strategy in anticipation of the strategic decisions of other players. When our modeling needs are too complex to address analytically, empirical game models, game models in which observations of simulated play are used to estimate payoffs of agents, can be employed to facilitate game-theoretic analysis. This dissertation focuses on extending the capability of the empirical game-theoretic analysis (EGTA) framework for modeling and analyzing large games. My contributions are in three distinct areas: increasing the scale of game simulation through software infrastructure, improving performance of common analytic tasks by bringing them closer to the data, and reducing sampling requirements for statistically confident analysis through sequential sampling algorithms. With the advent of EGTAOnline, an experiment management system for distributed game simulation that I developed, EGTA practitioners no longer limit their studies to what can be conducted on a single computer. Over one billion payoff observations have been added to EGTAOnline's database to date, corresponding to hundreds of distinct experiments. To reduce the cost of analyzing this data, I explored conducting analysis in the database. I found that translating data to an in-memory object representation was a dominant cost for game-theoretic analysis software. By avoiding that cost, conducting analysis in the database improves performance. A further way to improve scalability is to ensure we only gather as much data as is necessary to support analysis. I developed algorithms that interweave sampling and evaluations of statistical confidence, improving on existing ad hoc sampling methods by providing a measure of statistical confidence for analysis and reducing the number of observations taken. In addition to these software and methodological contributions, I present two applications: a strategic analysis of selecting a wireless access point for your traffic, and an investigation of mapping an analytical pricing model to a large simulated stock market.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/110315/1/bcassell_1.pd

Modeling Settlement Bargaining with Algorithmic Game Theory

Past computational models of settlement bargaining have lacked explicit game theoretic foundations. Algorithmic game theory, however, offers techniques that can find perfect Bayesian equilibria even where closed-form mathematical solutions may be intractable. Some recent mathematical models tackle two-sided asymmetric information, including evidentiary signals models, in which the judgment is a sum of both shared and independent private information, and correlated signals models, in which both parties receive noisy signals about the same information. To relax assumptions inherent in these models, this paper employs several progressively more complicated techniques, including iterative elimination of dominated alternatives, no regret learning, and counterfactual regret minimization. Although these algorithms are not guaranteed to produce Nash equilibria in general-sum games like litigation, they nonetheless succeed in producing either exact or close approximate equilibria on discrete versions of the corresponding mathematical models. A single algorithmic game theory model can incorporate a number of features that state-of-the-art mathematical models cannot handle simultaneously, such as two-sided correlated signals of both liability and damages, risk aversion, and options to concede

Approximate Analysis of Large Simulation-Based Games.

Game theory offers powerful tools for reasoning about agent behavior and incentives in multi-agent systems. Traditional approaches to game-theoretic analysis require enumeration of all possible strategies and outcomes. This often constrains game models to small numbers of agents and strategies or simple closed-form payoff descriptions. Simulation-based game theory extends the reach of game-theoretic analysis through the use of agent-based modeling. In the simulation-based approach, the analyst describes an environment procedurally and then computes payoffs by simulation of agent interactions in that environment. I use simulation-based game theory to study a model of credit network formation. Credit networks represent trust relationships in a directed graph and have been proposed as a mechanism for distributed transactions without a central currency. I explore what information is important when agents make initial decisions of whom to trust, and what sorts of networks can result from their decisions. This setting demonstrates both the value of simulation-based game theory—extending game-theoretic analysis beyond analytically tractable models—and its limitations—simulations produce prodigious amounts of data, and the number of simulations grows exponentially in the number of agents and strategies. I propose several techniques for approximate analysis of simulation-based games with large numbers of agents and large amounts of simulation data. First, I show how bootstrap-based statistics can be used to estimate confidence bounds on the results of simulation-based game analysis. I show that bootstrap confidence intervals for regret of approximate equilibria are well-calibrated. Next, I describe deviation-preserving reduction, which approximates an environment with a large number of agents using a game model with a small number of players, and demonstrate that it outperforms previous player reductions on several measures. Finally, I employ machine learning to construct game models from sparse data sets, and provide evidence that learned game models can produce even better approximate equilibria in large games than deviation-preserving reduction.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113587/1/btwied_1.pd

Essays on Economics and Computer Science

146 pagesThis dissertation considers a number of problems in pure and applied game theory. The first chapter considers the problem of how the introduction of fines and monitoring affects welfare in a routing game. I characterize equilibria of the game and discuss network topologies in which the introduction of fines can harm those agents which are not subject to them. The second, and primary, chapter considers the computational aspects of tenable strategy sets. I characterize these set-valued solution concepts using the more familiar framework of perturbed strategies, introduce strong alternatives to the problems of verifying whether a strategy block satisfies the conditions of tenability, and provide some hardness results regarding the verification of fine tenability. Additionally, I show an inclusion relation between the concept of coarse tenability and the notion of stability introduced by Kohlberg and Mertens (1986). Finally, I show how the methods developed for tenability provide an alternative characterization for proper equilibria in bimatrix games. This characterization gives a bound on the perturbations required in the definition of proper equilibria, though such bounds cannot be computed efficiently in general. The third, and final, chapter develops a model of contracting for content creation in an oligopolistic environment of attention intermediaries. I characterize symmetric equilibria in single-homing (exclusive) and multi-homing regimes. The focus is on the trade-off between reductions in incentives offered by intermediaries and the benefits of access to additional content for consumers. I show that when the extent of multi-homing is exogenous in the absence of exclusivity clauses, consumer surplus is always higher with multi-homing than under exclusivity, despite weaker incentives offered by platforms to content creators