TY - JOUR
T1 - Nonparametric probability bounds for nash equilibrium actions in a simultaneous discrete game
AU - Aradillas-Lopez, Andres
PY - 2011/7
Y1 - 2011/7
N2 - We study a simultaneous, complete-information game played by p=1,...,P agents. Each p has an ordinal decision variable Yp∈A{script}p={0,1,...,Mp}, where Mp can be unbounded, A{script}p is p's action space, and each element in A{script}p is an action, that is, a potential value for Yp. The collective action space is the Cartesian product A{script}=∏p=1PA{script}p. A profile of actions y∈A{script} 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{script} 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{script} 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.
AB - We study a simultaneous, complete-information game played by p=1,...,P agents. Each p has an ordinal decision variable Yp∈A{script}p={0,1,...,Mp}, where Mp can be unbounded, A{script}p is p's action space, and each element in A{script}p is an action, that is, a potential value for Yp. The collective action space is the Cartesian product A{script}=∏p=1PA{script}p. A profile of actions y∈A{script} 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{script} 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{script} 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.
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U2 - 10.3982/QE74
DO - 10.3982/QE74
M3 - Article
AN - SCOPUS:84883494585
SN - 1759-7323
VL - 2
SP - 135
EP - 171
JO - Quantitative Economics
JF - Quantitative Economics
IS - 2
ER -