1. In turn, V i (t i ?a i ) is supermodular if and only if it is concave (the bidder is risk averse), as desired. In the positive value interdependence, independent values setting, we established above that E[v i (a i ,t)|win with a i ]Pr(a i wins) is supermodular in (a i ,t i ); adding the function V i (?a i )(1?Pr(a i wins)) does not affect this conclusion;since V i is increasing) and supermodular in (a i ,t i )
2. Asymmetric All-Pay Auctions with Incomplete Information: The Two-Player Case;Erwin Amann;Games and Economic Behavior,1996
3. Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information;Susan Athey;MIT Working Paper Number,1997
4. Characterizing Properties of Stochastic Objective Functions;?;MIT Working Paper Number,1998
5. Comparative Statics Under Uncertainty: Single Crossing Properties and LogSupermodularity;?;MIT Working Paper Number,1998