In the contest design problem, there are $n$ strategic contestants, each of whom decides an effort level. A contest designer with a fixed budget must then design a mechanism that allocates a prize $p_i$ to the $i$-th rank based on the outcome, to incentivize contestants to exert higher costly efforts and induce high-quality outcomes. In this paper, we significantly deepen our understanding of optimal mechanisms under general settings by considering nonconvex objectives in contestants' qualities. Notably, our results accommodate the following objectives: (i) any convex combination of user welfare (motivated by recommender systems) and the average quality of contestants, and (ii) arbitrary posynomials over quality, both of which may neither be convex nor concave. In particular, these subsume classic measures such as social welfare, order statistics, and (inverse) S-shaped functions, which have received little or no attention in the contest literature to the best of our knowledge. Surprisingly, across all these regimes, we show that the optimal mechanism is highly structured: it allocates potentially higher prize to the first-ranked contestant, zero to the last-ranked one, and equal prizes to the all intermediate contestants, i.e., $p_1 \ge p_2 = \ldots = p_{n-1} \ge p_n = 0$. Thanks to the structural characterization, we obtain a fully polynomial-time approximation scheme given a value oracle. Our technical results rely on Schur-convexity of Bernstein basis polynomial-weighted functions, total positivity and variation diminishing property. En route to our results, we obtain a surprising reduction from a structured high-dimensional nonconvex optimization to a single-dimensional optimization by connecting the shape of the gradient sequences of the objective function to the number of transition points in optimum, which might be of independent interest.

翻译：在竞赛设计问题中，存在n个策略性参赛者，每位参赛者决定其努力水平。预算固定的竞赛设计者需基于参赛结果设计一种机制，将奖金\(p_i\)分配给第i名参赛者，以激励参赛者付出更高代价的努力并产生高质量结果。本文通过考虑参赛者质量中的非凸目标函数，显著深化了在一般环境下对最优机制的理解。值得注意的是，我们的结果适用于以下目标函数：（i）用户福利（受推荐系统启发）与参赛者平均质量的任意凸组合，以及（ii）关于质量的任意正项式函数，这两类函数均可能既非凸也非凹。这些目标函数涵盖了经典度量指标，如社会福利、顺序统计量和（反）S形函数——据我们所知，这些指标在竞赛文献中鲜少或从未被关注。令人惊讶的是，在所有上述场景中，我们证明了最优机制具有高度结构化的形式：它向排名第一的参赛者分配可能更高的奖金，向最后一名参赛者分配零奖金，而其他中间排名的参赛者获得相等的奖金，即\(p_1 \ge p_2 = \ldots = p_{n-1} \ge p_n = 0\)。借助这一结构特征，我们在给定值函数预言机的情况下，得到了全多项式时间近似方案。我们的技术结果依赖于伯恩斯坦基多项式加权函数的Schur凸性、全正性以及变差缩减性质。在得出结果的过程中，我们通过将目标函数梯度序列的形状与最优解中的转折点数量相关联，从结构化的高维非凸优化问题意外地简化为单维优化问题，这一方法可能具有独立的研究意义。