Sharp identification regions in games

Abstract

We study identification in static, simultaneous move finite games of complete information, where the presence of multiple Nash equilibria may lead to partial identification of the model parameters. The identification regions for these parameters proposed in the related literature are known not to be sharp. Using the theory of random sets, we show that the sharp identification region can be obtained as the set of minimizers of the distance from the conditional distribution of game's outcomes given covariates, to the conditional Aumann expectation given covariates of a properly defined random set. This is the random set of probability distributions over action profiles given profit shifters implied by mixed strategy Nash equilibria. The sharp identification region can be approximated arbitrarily accurately through a finite number of moment inequalities based on the support function of the conditional Aumann expectation. When only pure strategy Nash equilibria are played, the sharp identification region is exactly determined by a finite number of moment inequalities. We discuss how our results can be extended to other solution concepts, such as for example correlated equilibrium or rationality and rationalizability.We show that calculating the sharp identification region using our characterization is computationally feasible. We also provide a simple algorithm which finds the set of inequalities that need to be checked in order to insure sharpness. We use examples analyzed in the literature to illustrate the gains in identification afforded by our method.

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“…An econometrically implementable choice structure is implemented by assuming the difference between the payoffs for the two choices is additive in 43 See Berry and Tamer (2006) for an overview, and Beresteanu, Molchanov, and Molinari (2008) and Galichon and Henry (2008) for recent advances in partial identification. the different factors that have been defined for the linear model, i.e.…”

Set Identification in Models with Multiple Equilibria

“…This results allows the computation of the identified feature of models with multiple equilibria and a finite number of observable outcomes, as it reduces the problem to that of checking a finite number of moment inequalities. A related representation was developed independently by Beresteanu, Molchanov, and Molinari (2008) who emphasize the characterization of the identified set as an Aumann integral.…”

Identification of Social Interactions

“…Identification for binary choice models has been studied in detail by 43 See Berry and Tamer (2006) for an overview, and Beresteanu, Molchanov, and Molinari (2008) and Galichon and Henry (2008) for recent advances in partial identification. Brock andDurlauf (2001a,b, 2007); other contributions include Soetevent and Kooreman (2007).…”