Chapter 2 Computation of equilibria in finite games

McKelvey, Richard D.; McLennan, Andrew

doi:10.1016/s1574-0021(96)01004-0

Cited by 180 publications

(127 citation statements)

References 55 publications

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“…In the case of two-player polynomial games, such an explicit description is known to be possible using linear matrix inequalities and has been applied to zero-sum polynomial games by Parrilo [19]. While the lack of polyhedral structure in the moment spaces would most likely prohibit the use of a Lemke-Howson type algorithm, a variety of other finite game algorithms may be extendable to this setting; see McKelvey and McLennan for a survey of such algorithms [17].…”

Section: Resultsmentioning

confidence: 99%

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Separable and low-rank continuous games

2008

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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.

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Section: Resultsmentioning

confidence: 99%

“…These methods rely on the polyhedral structure of the mixed strategy spaces of finite games, therefore they seem unlikely to generalize to continuous/separable games. For a survey of algorithms which compute equilibria of finite games, see [17].…”

Section: Introductionmentioning

confidence: 99%

Separable and low-rank continuous games

2008

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“…The …rst step in the procedure requires one to compute the set of all MSNE for given realizations of the payo¤ shifters, S (x; ") : This is a computationally challenging problem, though a well studied one which can be performed using the Gambit software described by McKelvey and McLennan (1996). 31 The complexity of this task grows quickly with the number of players and the number of actions that each player can choose from.…”

Section: A4 Computational Aspects Of the Problemmentioning

confidence: 99%

Sharp identification regions in models with convex moment predictions

Beresteanu¹,

Molchanov²,

Molinari³

2010

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We provide a tractable characterization of the sharp identi…cation region of the parameters in a broad class of incomplete econometric models. Models in this class have set valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous move …nite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for ; we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly de…ned random set. The sharp identi…cation region of ; denoted I ; can then be obtained as the set of minimizers of the distance from a properly speci…ed vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to e¢ ciently verify whether a candidate is in I : We use examples analyzed in the literature to illustrate the gains in identi…cation and computational tractability a¤orded by our method.

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“…3 Equilibrium computation also has a number of well-known applications in economic analysis: predicting the outcome and estimating suitable parameter values of a model, comparing experimental results with model predictions, testing the design of a mechanism, automatically generating conjectures and counterexamples, and so on. These topics are discussed in detail elsewhere [13,36,38,65,84].…”

Section: Who Cares About Computing Equilibria?mentioning

confidence: 99%