Quantile analysis of “hazard-rate” game models

Abstract

This paper consists of an econometric analysis of a broad class of games of incomplete information. In these games, a player’s action depends both on her unobservable characteristic (the private information), as well as on the ratio of the distribution of the unobservable characteristic and its density function (which we call the ”hazard-rate”). The goal is to use data on players’ actions to recover the distribution of private information. We show that the structural parameter (the quantile of the unobservable characteristic, denoted by L) can be related to the reduced form parameter (the quantile of the data, denoted by Q) through a differential equation Q(α)=a(L(α),αL′(α)). We analyse the local and global identification of this equation in L. At the same time, under suitable assumptions, we establish the local and global well-posedness of the inverse problem Q=T(L). We solve the differential equation and, as a consequence of the well-posedness, we estimate nonparametrically the quantile and the c.d.f. function of the unobserved variables at a root-n speed of convergence (conditional on the quantile of the data being estimated at root-n). Moreover as the transformation of Q in L is continuous with respect to a topology that implies also the derivatives, we estimate nonparametrically the quantile density and the density the unobserved variables (that are continuous functions with respect to Q) at root-n speed of convergence. We provide also several generalisations of the model accounting for exogenuous variables, implicit expression of the strategy function and integral expression of the strategy function. Our results have several policy applications, including better design of auctions, regulation models, estimation of war of attrition in patent races, and public good contracts.