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Nonparametric probability bounds for Nash equilibrium actions in a simultaneous discrete game

Authors

  • Andres Aradillas-Lopez

    1. Department of Economics, University of Wisconsin–Madison; aaradill@ssc.wisc.edu
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    • A previous version of this paper was titled “Nonparametric probability bounds for rationalizable choices in strategic-interaction models with approximate equilibrium beliefs”. I thank four anonymous referees and the coeditor for very helpful comments. Part of this research was conducted while the author was visiting El Colegio de México (Centro de Estudios Económicos) and the Centro de Investigación y Docencia Económicas (División de Economía). Financial support from the National Science Foundation Grant SES-0922062 is gratefully acknowledged.


Abstract

We study a simultaneous, complete-information game played by p=1, …, P agents. Each p has an ordinal decision variable Yp∈��p={0, 1, …, Mp}, where Mp can be unbounded, ��p is p's action space, and each element in ��p is an action, that is, a potential value for Yp. The collective action space is the Cartesian product ��=∏p=1P��p. A profile of actions y∈�� is a Nash equilibrium (NE) profile if y is played with positive probability in some existing NE. Assuming that we observe NE behavior in the data, we characterize bounds for the probability that a prespecified y in �� is a NE profile. Comparing the resulting upper bound with Pr[Y=y] (where Y is the observed outcome of the game), we also obtain a lower bound for the probability that the underlying equilibrium selection mechanism ℳ chooses a NE where y is played given that such a NE exists. Our bounds are nonparametric, and they rely on shape restrictions on payoff functions and on the assumption that the researcher has ex ante knowledge about the direction of strategic interaction (e.g., that for qp, higher values of Yq reduce p's payoffs). Our results allow us to investigate whether certain action profiles in �� are scarcely observed as outcomes in the data because they are rarely NE profiles or because ℳ rarely selects such NE. Our empirical illustration is a multiple entry game played by Home Depot and Lowe's.

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