Sequential Elimination Contests with All-Pay Auctions

We study a sequential elimination contest where players are filtered prior to the round of competing for prizes. This is motivated by the practice that many crowdsourcing contests have very limited resources of reviewers and want to improve the overall quality of the submissions. We first consider a setting where the designer knows the ranking of the abilities (types) of all $n_1$ registered players, and admit the top $n_2$ players with $2\leq n_2 \leq n_1$ into the contest. The players admitted into the contest update their beliefs about their opponents based on the signal that their abilities are among the top $n_2$. We find that their posterior beliefs, even with IID priors, are correlated and depend on players' private abilities. We explicitly characterize the symmetric and unique Bayesian equilibrium strategy. We find that each admitted player's equilibrium effort is increasing in $n_2$ when $n_2 \in [\lfloor{(n_1+1)/2}\rfloor+1,n_1]$, but not monotone in general when $n_2 \in [2,\lfloor{(n_1+1)/2}\rfloor+1]$. Surprisingly, despite this non-monotonicity, all players exert their highest efforts when $n_2=n_1$. As a sequence, if the designer has sufficient capacity, he should admit all players to maximize their equilibrium efforts. This result holds generally -- it is true under any ranking-based reward structure, ability distribution, and cost function. We also discuss the situation where the designer can only admit $c<n_1$ players. Our numerical results show that, in terms of the expected highest or total efforts, the optimal $n_2$ is either $2$ or $c$. Finally, we extend our model to a two-stage setting, where players with top first-stage efforts can proceed to the second stage competing for prizes. We establish an intriguing negative result in this setting: there does not exist a symmetric and monotone Perfect Bayesian equilibrium.