Aggregating the single crossing property: theory and applications to comparative statics and Bayesian games

The single crossing property plays a crucial role in monotone comparative statics (Milgrom and Shannon (1994)), yet in some important applications the property cannot be directly assumed or easily derived.  Difficulties often arise because the property cannot be aggregated: the sum of two functions with the single crossing property need not have the same property.  We obtain the precise conditions under which functions with the single crossing property add up to functions with this property.  We apply our results to certain Bayesian games when establishing the monotonicity of strategies is an important step in proving equilibrium existence.  In particular, we find conditions under which first-price auctions have monotone equilibria, generalizing the result of Reny and Zamir (2004).