Optimal contest design for simple agents

We study the optimal design of contests for 'simple' agents, where potential contestants strategically reason about whether or not to participate in the contest, but do not strategize about the quality of their submissions. Consider a population of n agents, where an agent with type (qi, ci chooses between participating and producing a submission of quality qi at cost ci, versus not participating at all, to maximize her utility. How should a principal distribute a total prize V amongst the n ranks to maximize some increasing function of the qualities of elicited submissions in a contest with such simple agents' We first solve the optimal contest design problem in settings where agents have homogenous participation costs ci = c. Here, the contest that maximizes every increasing function of the elicited contributions qi is always a simple contest, awarding equal prizes of V/j* each to the top j* = V/c - Θ (√V/(c ln (V/c))) contestants. This is in contrast with the optimal contest structure in comparable models with strategic effort choices, where the optimal contest is either a winner-take-all contest or awards possibly unequal prizes, depending on the curvature of agents' effort cost functions. We next address the general case with heterogenous costs ci: here, agents' types (qici are inherently two-dimensional, which significantly complicates equilibrium analysis. With heterogenous costs, the optimal contest depends on the objective being maximized; our main result here is that the winner-take-all contest is a 3-approximation of the optimal contest when the principal's objective is to maximize the quality of the best elicited contribution. The proof of this result hinges around a `sub-equilibrium' lemma, which establishes a stochastic dominance relation between the distribution of qualities elicited in an equilibrium and a sub-equilibrium---a strategy profile that is a best response for all agents who choose to participate in that strategy profile; this relation between equilibria and sub-equilibria may be of more general interest.