Reward Design in Risk-Taking Contests

Abstract

Following the risk-taking model of Seel and Strack, n players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the problem of a principal who aims to induce a desirable equilibrium performance of the players by choosing how much reward is attributed to each rank. Specifically, we determine optimal reward schemes for principals interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape.

Mean Field Contest with Singularity

“…Players are ranked according to their level of stopping and paid a reward which is a decreasing function of the rank. This is an infinite-player version of the n-player game studied in [32] which in turn extends the Seel-Strack model [36] where only the top-ranked player receives a reward. First, we establish existence and uniqueness of a mean field equilibrium for any given reward function.…”

“…Second, we solve the problem of optimal reward design (optimal contract) for a principal who can choose how to distribute a given reward budget over the ranks and aims to maximize the performance (i.e., stopping level) at a given rank, for instance the median performance among the players. An analogous problem was studied for the n-player case in [32], but only a partial characterization of the optimal design is available. Here, taking the mean field limit enables a clear-cut answer.…”

“…In the n-player setting, the Seel-Strack model has been generalized and varied in several directions: more general diffusion processes [19], random initial laws [20], heterogeneous loss constraints [35], behavioral players [21]. See also [18,32] for references to other models on risk-taking under relative performance pay, and [38] for an introduction to rank-order prize allocation. The novelty in the present work is to analyze a mean field model along the lines of Seel-Strack and its optimal design problem; we focus on the original Brownian dynamics.…”

“…Optimal Reward Design. In the n-player game, [32] studied the design problem for a principal maximizing the performance at the k-th rank; for example, maximizing the revenue in a second-best auction, or the median performance among employees, customers, students, etc. To consider the analogue in the mean field limit, we replace the k-th rank by the quantile α = k/n, for instance α = 0.5 for the median player.…”

“…(A full characterization of the optimal reward is only available for zero drift; cf. [32].) Again, the mean field limit proves useful in allowing for a fuller description and a clearer result.…”