Asymmetric endogenous prize contests

We consider a two-player contest in which one contestant has a headstart advantage, but both can exert further effort. We allow the prize to depend on total performance in the contest and consider the respective cases in which efforts are productive and destructive of prize value. When the contest success function takes a logit form, and marginal cost is increasing in effort, we show that a Nash equilibrium exists and is unique both in productive and destructive endogenous prize contests. In equilibrium, the underdog expends more resources to win the prize, but still his probability of winning remains below that of the favorite. In a productive contest, the underdog behaves more aggressively and wins the prize more often in comparison to a fixed-value contest. Thus, the degree of competitive balance-defined as the level of uncertainty of the outcome-depends upon the (fixed or endogenous) prize nature of the contest.