Stochastic orders and the anatomy of competitive selection

This paper determines the conditions under which stochastic orderings of random variables, e.g., stochastic dominance and the monotone likelihood ordering, are preserved or reversed by conditioning on competitive selection. A new stochastic order over unconditional distributions is introduced—geometric dominance—which is sufficient for a random variable, conditioned on competitive selection, to be stochastically dominant and both necessary and sufficient for it to be dominant under the monotone likelihood ratio ordering. Using geometric dominance, we provide an “anatomy of selection bias” by identifying the conditions under which competitive selection conditioning preserves and reverses unconditional stochastic order relations. We show that, for all standard error distributions (e.g., Normal, Logistic, Laplace), competitive selection preserves stochastic order relations. One implication of this result is that, even in the presence of self-selection bias, the sign of treatment affects can be identified by standard uncorrected OLS and Logit/Probit models. Another is that, for almost all “textbook” distribution families, MLRP ordering is preserved by competitive selection

Effciency and surplus bounds in Cournot competition

We derive bounds on the ratios of deadweight loss and consumer surplus to producer surplus under Cournot competition. To do so, we introduce a parameterization of the degree of curvature of market demand using the parallel concepts of ½-concavity and ½-convexity. The ”more concave” is demand, the larger the share of producer surplus in overall surplus, the smaller is consumer surplus relative to producer surplus, and the lower the ratio of deadweight loss to producer surplus. Deadweight loss over total potential surplus is at …rst increasing with demand concavity, then eventually decreasing. The analysis is extended to asymmetric …rm costs.Cournot equilibrium, social surplus analysis, deadweight loss, market performance.

Strategic information transmission and stochastic orders

I develop new results on uniqueness and comparative statics of equilibria in the Crawford and Sobel (1982) strategic information transmission game. For a class of utility functions, I demonstrate that logconcavity of the density implies uniqueness of equilibria inducing a given number of Receiver actions. I provide comparative statics results with respect to the distribution of types for distributions that are comparable in the likelihood ratio order, implying, e.g., that advice from a better informed Sender induces the Receiver to choose actions that are more spread out

Reputation and Turnover

We consider a repeated duopoly game where each firm privately chooses its investment in quality, and realized quality is a noisy indicator of the firm’s investment. We focus on dynamic reputation equilibria, whereby consumers ‘discipline’ a firm by switching to its rival in the case that the realized quality of its product is too low. This type of equilibrium is characterized by consumers’ tolerance level - the level of product quality below which consumers switch to the rival firm - and firms’ investment in quality. Given consumers’ tolerance level, we determine when a dynamic equilibrium that gives higher welfare than the static equilibrium exists. We also derive comparative statics properties, and characterize a set of investment levels and, hence, layoffs that our equilibria sustain.Reputation, consumer switching, moral hazard, repeated games

Efficiency and surplus bounds in Cournot competition

We derive bounds on the ratios of deadweight loss and consumer surplus to producer surplus under Cournot competition. To do so, we introduce a parameterization of the degree of curvature of market demand using the parallel concepts of ?-concavity and ?-convexity. The ?more concave? is demand, the larger the share of producer surplus in overall surplus, the smaller is consumer surplus relative to producer surplus, and the lower the ratio of deadweight loss to producer surplus. Deadweight loss over total potential surplus is at Þrst increasing with demand concavity, then eventually decreasing. The analysis is extended to asymmetric Þrm costs.Cournot equilibrium, social surplus analysis, deadweight loss, market performance.

Ratio Orderings and Comparative Statics

Monotone ratio orderings are refinements of first order stochastic dominance that allow monotone comparative statics results in games of incomplete information. We develop analogous refinements for second order stochastic dominance based on the monotonicity of the cumulative probability ratio and the unimodality of the likelihood and probability ratios. We go on to investigate comparative statics in first price auctions, both private and common value, of the effects of more precise information in the sense of the new orderings. We find that almost all types bid more aggressively under the new distribution than they did under the old, but the highest types may bid less. This leads to higher expected revenue in a simple common value auction, but to an ambiguous result in the private value case.monotone likelihood ratio, monotone probability ratio, conditional stochastic dominance, generalized Lorenz order, comparative statics, first price auctions, common value auctions

To Bundle or Not to Bundle

Comparing monopoly bundling with separate sales is relatively straightforward in an environment with a large number of goods. In this paper we show that results that are similar to the asymptotic results can be obtained in the more realistic case with a given finite number of goods provided that the distributions of valuations are symmetric and log-concave.Monopoly Pricing, Bundling, Peakedness

Stochastic comparisons of nonhomogeneous processes

The purpose of this paper is to describe various conditions on the parameters of pairs of nonhomogeneous Poisson or birth processes under which the corresponding epoch or inter-epoch times are stochastically ordered in various senses. We derive results involving the usual stochastic order, the multivariate hazard rate order, the multivariate likelihood ratio order, and the multivariate mean residual life order. A sample of applications involving generalized Yule processes, load-sharing models, and minimal repairs in reliability theory, illustrate the usefulness of the new results