### Abstract

We study a simultaneous, complete-information game played by p=1,...,P agents. Each p has an ordinal decision variable Y_{p}∈A{script}_{p}={0,1,...,M_{p}}, where M_{p} 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 Y_{p}. The collective action space is the Cartesian product A{script}=∏_{p=1}^{P}A{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 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, higher values of Y_{q} 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 ℳ_{ℰ} rarely selects such NE. Our empirical illustration is a multiple entry game played by Home Depot and Lowe's.

Original language | English (US) |
---|---|

Pages (from-to) | 135-171 |

Number of pages | 37 |

Journal | Quantitative Economics |

Volume | 2 |

Issue number | 2 |

DOIs | |

State | Published - Jul 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Economics and Econometrics

### Cite this

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*Quantitative Economics*, vol. 2, no. 2, pp. 135-171. https://doi.org/10.3982/QE74

**Nonparametric probability bounds for nash equilibrium actions in a simultaneous discrete game.** / Aradillas-Lopez, Andres.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Nonparametric probability bounds for nash equilibrium actions in a simultaneous discrete game

AU - Aradillas-Lopez, Andres

PY - 2011/7/1

Y1 - 2011/7/1

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

VL - 2

SP - 135

EP - 171

JO - Quantitative Economics

JF - Quantitative Economics

SN - 1759-7323

IS - 2

ER -