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).
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- Van Zandt, Timothy & Vives, Xavier, 2007.
"Monotone equilibria in Bayesian games of strategic complementarities,"
Journal of Economic Theory,
Elsevier, vol. 134(1), pages 339-360, May.
- Van Zandt, Timothy & Vives, Xavier, 2003. "Monotone Equilibria in Bayesian Games of Strategic Complementarities," CEPR Discussion Papers 4103, C.E.P.R. Discussion Papers.
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