Quantitative Economics: Jul, 2011, Volume 2, Issue 2

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

Andres Aradillas-Lopez

We study a simultaneous, complete-information game played by p = 1
P
agents. Each p has an ordinal decision variable Yp ∈ Ap = {0 1
Mp }, where
Mp can be unbounded, Ap is p’s action space, and each element in Ap is an ac-
tion, that is, a potential value for Yp . The collective action space is the Cartesian
product A = P Ap . A profile of actions y ∈ A is a Nash equilibrium (NE) pro-
p=1
file 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
ME 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, higher values of Yq 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.
Keywords. Ordered response game, nonparametric identification, bounds, entry
models.
JEL classification. C14, C35, C71.

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