### Quantitative Economics, Volume 2, Issue 2 (July 2011)

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

*Andres Aradillas-Lopez*

#### Abstract

We study a simultaneous, complete-information game played by p = 1

P

agents. Each p has an ordinal decision variable Yp âˆˆ Ap = {0 1

Mp }, where

Mp can be unbounded, Ap is pâ€™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 âˆˆ 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 pâ€™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 Loweâ€™s.

Keywords. Ordered response game, nonparametric identification, bounds, entry

models.

JEL classification. C14, C35, C71.

P

agents. Each p has an ordinal decision variable Yp âˆˆ Ap = {0 1

Mp }, where

Mp can be unbounded, Ap is pâ€™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 âˆˆ 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 pâ€™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 Loweâ€™s.

Keywords. Ordered response game, nonparametric identification, bounds, entry

models.

JEL classification. C14, C35, C71.

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