Andres Aradillas-Lopez

The Identification Power of Equilibrium in Simple Games

We examine the identification power that (Nash) equilibrium assumptions play in conducting inference about parameters in some simple games. We focus on three static games in which we drop the Nash equilibrium assumption and instead use rationalizability as the basis for strategic play. The first example examines a bivariate discrete game with complete information of the kind studied in entry models. The second example considers the incomplete-information version of the discrete bivariate game. Finally, the third example considers a first-price auction with independent private values. In each example, we study the inferential question of what can be learned about the parameter of interest using a random sample of observations, under level-k rationality, where k is an integer ≥ 1. As k increases, our identified set shrinks, limiting to the identified set under full rationality or rationalizability (as k → ∞). This is related to the concepts of iterated dominance and higher-order beliefs, which are incorporated into the econometric analysis in our framework. We are then able to categorize what can be learned about the parameters in a model under various maintained levels of rationality, highlighting the roles of different assumptions. We provide constructive identification results that lead naturally to consistent estimators.

Identification and Inference in Ascending Auctions With Correlated Private Values

We introduce and apply a new nonparametric approach to identification and inference on data from ascending auctions. We exploit variation in the number of bidders across auctions to nonparametrically identify useful bounds on seller profit and bidder surplus using a general model of correlated private values that nests the standard independent private values (IPV) model. We also translate our identified bounds into closed form and asymptotically valid confidence intervals for several economic measures of interest. Applying our methods to much studied U.S. Forest Service timber auctions, we find evidence of correlation among values after controlling for a rich vector of relevant auction covariates; this correlation causes expected profit, the profit-maximizing reserve price, and bidder surplus to be substantially lower than conventional (IPV) analysis of the data would suggest.

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 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.

A Comment on “Simple Estimators for Invertible Index Models”

The approach described by Ahn et al. (this issue) (henceforth AIPR) constitutes an elegant contribution to the “control function” literature, which focuses on semiparametric models where endogeneity or nonlinearities of unknown form can be captured by a control function (or “control variable”). Like the rest of this literature, the applicability of the method in AIPR presupposes the ability to identify (perhaps nonparametrically) the control function to have the ability to “match” (asymptotically) pairs of observations in a way that identifies the parameters of interest (see Equations (2.15)– (2.19) in AIPR). While the control function approach has been shown to have aided applicability (Ahn and Powell 1993; Honoré and Powell 1994; Blundell and Powell 2004; Honoré and Powell 2005; Imbens and Newey 2009; Hong and Shum 2010; Aradillas-Lopez, Honoré, and Powell 2007), in this note I argue that it is easy to construct examples of microeconometric models where matching is not possible, either because of interval data, missing data, or incomplete models (e.g, structural models with “multiple equilibria”). In doing this, I also want to argue that, while matching is not possible and the methodology in AIPR cannot be applied, the examples I present produce (conditional) moment inequalities which can be used to do inference on the parameters of interest by using recently developed methods involving conditional moment inequalities. This shows that control function models are powerful vehicles for inference even in partially identified settings.