Heterogeneous Risk/Loss Aversion in Complete Information All-Pay Auctions

Abstract

We extend previous theoretical work on n-players complete information all-pay auction to incorporate heterogeneous risk and loss averse utility functions. We provide sufficient and necessary conditions for the existence of equilibria with a given set of active players with any strictly increasing utility functions and characterize the players' equilibrium mixed strategies. Assuming that players can be ordered by their risk aversion (player a is more risk averse than player b if whenever player b prefers a certain payment over a given lottery so will player a), we find that, in equilibrium, the more risk averse players either bid higher (in terms of first order stochastic dominance of their mixed strategy cumulative distribution) than the less risk averse players and win with higher ex-ante probability — or they drop out. Furthermore, while each player's expected bid decreases with the other players' risk aversion, her expected bid increases with her own risk aversion. Thus, increasing a player's risk aversion creates two opposing effects on total expected bid. A sufficient condition for the total expected bid to decrease with a player's risk aversion is that this player is relatively more risk averse compared to the rest of the players. Our findings have important implications for the literature on gender differences in competitiveness and for gender diversity in firms that use personnel contests for promotions.