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The Empirical Implications of Rank in Bimatrix Games
, 2013
"... We study the structural complexity of bimatrix games, formalized via rank, from an empirical perspective. We consider a setting where we have data on player behavior in diverse strategic situations, but where we do not observe the relevant payoff functions. We prove that high complexity (high rank) ..."
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We study the structural complexity of bimatrix games, formalized via rank, from an empirical perspective. We consider a setting where we have data on player behavior in diverse strategic situations, but where we do not observe the relevant payoff functions. We prove that high complexity (high rank) has empirical consequences when arbitrary data is considered. Additionally, we prove that, in more restrictive classes of data (termed laminar), any observation is rationalizable using a lowrank game: specifically a zerosum game. Hence complexity as a structural property of a game is not always testable. Finally, we prove a general result connecting the structure of the feasible data sets with the highest rank that may be needed to rationalize a set of observations.
General revealed Preference Theory
, 2010
"... We provide general conditions under which an economic theory has a universal axiomatization: one that leads to testable implications. Roughly speaking, if we obtain a universal axiomatization when we assume that unobservable parameters (such as preferences) are observable, then we can obtain a unive ..."
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Cited by 1 (1 self)
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We provide general conditions under which an economic theory has a universal axiomatization: one that leads to testable implications. Roughly speaking, if we obtain a universal axiomatization when we assume that unobservable parameters (such as preferences) are observable, then we can obtain a universal axiomatization purely on observables. The result "explains" classical revealed preference theory, as applied to individual rational choice. We obtain new applications to Nash equilibrium theory and Pareto optimal choice.
The Complexity of Nash Equilibria as Revealed by Data
"... In this paper we initiate the study of the computational complexity of Nash equilibria in bimatrix games that are specified via data. This direction is motivated by an attempt to connect the emerging work on the computational complexity of Nash equilibria with the perspective of revealed preference ..."
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In this paper we initiate the study of the computational complexity of Nash equilibria in bimatrix games that are specified via data. This direction is motivated by an attempt to connect the emerging work on the computational complexity of Nash equilibria with the perspective of revealed preference theory, where inputs are data about observed behavior, rather than explicit payoffs. Our results draw such connections for large classes of data sets, and provide a formal basis for studying these connections more generally. In particular, we derive three structural conditions that are sufficient to ensure that a data set is both consistent with Nash equilibria and that the observed equilibria could have been computed efficiently: (i) small dimensionality of the observed strategies, (ii) small support size of the observed strategies, and (iii) small chromatic number of the data set. Key to these results is a connection between data sets and the player rank of a game, defined to be the minimum rank of the payoff matrices of the players. We complement our results by constructing data sets that require rationalizing games to have high player rank, which suggests that computational constraints may be important empirically as well. 1