Comparing the Chosen: Selection Bias When Selection Is Competitive

Consider a decision maker who selects between paired random draws from two unconditional distributions, always selecting the larger draw in the pair. When will the resulting selection-conditioned distributions be ordered by first-order stochastic or monotone likelihood-ratio dominance? In various guises, this question arises in many economic contexts—tournaments, contests, auctions, cheap-talk games, announcement returns, qualitative choice models, and treatment effects under self-selection. This paper develops simple, applicable characterizations of the properties of unconditional distributions which result in dominance conditioned on selection and uses these characterizations to analyze a number of economic selection problems.