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

We study a simultaneous, complete-information game played by p = 1 P agents. Each p has an ordinal decision variable Y p ∈ A p ={ 0 1 M p} , where M p can be unbounded, A p is p’s action space, and each element in A p is an action , that is, a potential value for Y p . The collective action space is the Cartesian product A = ∏ P p = 1 A p . A profile of actions y ∈ A 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 A 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 E 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 q = p ,highervaluesof Y q reduce p’s payoffs). Our results allow us to investigate whether certain action profiles in A are scarcely observed as outcomes in the data because they are rarely NE profiles or because ME rarely selects such NE. Our empirical illustration is a multiple entry game played by Home Depot and Lowe’s.