2,181 research outputs found
Structure of Extreme Correlated Equilibria: a Zero-Sum Example and its Implications
We exhibit the rich structure of the set of correlated equilibria by
analyzing the simplest of polynomial games: the mixed extension of matching
pennies. We show that while the correlated equilibrium set is convex and
compact, the structure of its extreme points can be quite complicated. In
finite games the ratio of extreme correlated to extreme Nash equilibria can be
greater than exponential in the size of the strategy spaces. In polynomial
games there can exist extreme correlated equilibria which are not finitely
supported; we construct a large family of examples using techniques from
ergodic theory. We show that in general the set of correlated equilibrium
distributions of a polynomial game cannot be described by conditions on
finitely many moments (means, covariances, etc.), in marked contrast to the set
of Nash equilibria which is always expressible in terms of finitely many
moments
Characterization and computation of equilibria in infinite games
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 79-82).Broadly, we study continuous games (those with continuous strategy spaces and utility functions) with a view towards computation of equilibria. We cover all of the game-theoretic background needed to understand these results in detail. Then we present new work, which can be divided into three parts. First, it is known that arbitrary continuous games may have arbitrarily complicated equilibria, so we investigate some properties of games with polynomial utility functions and a class of games with polynomial-like utility functions called separable games. We prove new bounds on the complexity of equilibria of separable games in terms of the complexity of the utility functions. In order to measure this complexity we propose a new definition for the rank of a continuous game; when applied to the case of finite games this improves on the results known in that setting. Furthermore, we prove a characterization theorem showing that several conditions which are necessary for a game to possess a finite-dimensional representation each define the class of separable games precisely, providing evidence that separable games are the natural class of continuous games in which to study computation. The characterization theorem also provides a natural connection between separability and the notion of the rank of a game. Second, we apply this theory to give an algorithm for computing e-Nash equilibria of two-player separable games with continuous strategy spaces. While a direct comparison to corresponding algorithms for finite games is not possible, the asymptotic running time in the complexity of the game grows slower for our algorithm than for any known algorithm for finite games.(cont.) Nonetheless, as in finite games, computing e-Nash equilibria still appears to be difficult for infinite games. Third, we consider computing approximate correlated equilibria in polynomial games. To do so, we first prove several new characterizations of correlated equilibria in continuous games which may be of independent interest. Then we introduce three algorithms for approximating correlated equilibria of polynomial games arbitrarily accurately. These include two discretization algorithms for computing a sample correlated equilibrium: a naive linear programming approach called static discretization which operates without regard to the structure of the game, and a semidefinite programming approach called adaptive discretization which exploits the structure of the game to achieve far better performance in practice. The third algorithm consists of a nested sequence of semidefinite programs converging to a description of the entire set of correlated equilibria.by Noah D. Stein.S.M
Separable and Low-Rank Continuous Games
In this paper, we study nonzero-sum separable games, which are continuous
games whose payoffs take a sum-of-products form. Included in this subclass are
all finite games and polynomial games. We investigate the structure of
equilibria in separable games. We show that these games admit finitely
supported Nash equilibria. Motivated by the bounds on the supports of mixed
equilibria in two-player finite games in terms of the ranks of the payoff
matrices, we define the notion of the rank of an n-player continuous game and
use this to provide bounds on the cardinality of the support of equilibrium
strategies. We present a general characterization theorem that states that a
continuous game has finite rank if and only if it is separable. Using our rank
results, we present an efficient algorithm for computing approximate equilibria
of two-player separable games with fixed strategy spaces in time polynomial in
the rank of the game
Correlated Equilibria in Continuous Games: Characterization and Computation
We present several new characterizations of correlated equilibria in games
with continuous utility functions. These have the advantage of being more
computationally and analytically tractable than the standard definition in
terms of departure functions. We use these characterizations to construct
effective algorithms for approximating a single correlated equilibrium or the
entire set of correlated equilibria of a game with polynomial utility
functions.Comment: Games and Economic Behavior, In Press, Accepted Manuscript, Available
online 16 April 201
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