Distributional preferences in probabilistic and share contests

We analyze Nash equilibria of share and probabilistic contests when players have distributional preferences. If players are sufficiently similar, distributional preferences create multiple equilibria. For the case of only mildly heterogeneous players, equilibrium effort can be lower as well as higher than effort with standard, selfish preferences. These findings can explain three anomalies in empirical tests of the probabilistic prize-seeking game, namely the large variance of effort levels (overspreading), individual spending that exceeds the Nash-equilibrium prediction (overspending), and aggregate spending that exceeds the value of the prize (overdissipation), and they are also in line with the findings for share contests. If players are sufficiently heterogeneous, the game has a unique equilibrium that is more egalitarian than the standard, selfish Nash equilibrium. It turns out that the less talented competitor may win the larger share of the prize if his inequality aversion is sufficiently strong. We analyze how the equilibria evolve if the number of players gets larger and how sequential moves influence behavior. Two new insights follow from the analysis of the sequential-move game. First, sequential moves act as a coordination device if there are multiple simultaneous equilibria, and second the inequality aversion of the more egalitarian player can be used as a commitment device for low effort. This effect can reverse the conventional wisdom that the underdog should lead.