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.