Optimal Contest Beyond Convexity

Abstract

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.

Figures (4)

• Figure 1: Structures of optimal policy with respect to different $\alpha$ and $\beta$ values given $n = 5$ producers. We compute the optimal policy using a brute-force search over every possible policy after discretizing the policy space with granularity $0.005$, and using trapezoidal rule to approximate integral values over the grid $[0,0.001,0.002,\ldots, 1]$. For the left figure, we set $\beta = 2$ and for the right figure, we set $\alpha = 0.24$. This yields that our characterization is tight in a sense that the structures $p_1 > p_2 = \ldots = p_{n-1} \ge p_n = 0$ or $p_1 = p_2 = \ldots = p_{n-1} \ge p_n = 0$ indeed appear in the optimal policies.
• Figure 2: Structures of optimal policy with respect to different $\alpha$ and $\beta$ values given $n=5$ producers. Darker red means larger $p_1$ values, and darker blue means larger $p_2 = p_3 = \ldots = p_{n-1}$ values. Each of the optimal policy is computed in a brute force manner after discretizing policy space with granularity $0.001$ and using trapezoidal rule with number of points $200$. Parameters $\alpha$ and $\beta$ are selected from the ranges $[0.05, 1]$ and $[0.1,5]$, respectively, with $1000$ points calculated for each.
• Figure 3: Behavior of $d_i$ for $n = 5$. Since $p_5 = 0$, $d_5$ is omitted. Quasiconvexity of $d_i$ follows from the quasiconvexity of $q(x)$ shown in Figure \ref{['fig:quasi-conv-q']}.
• Figure 4: Behavior of $q(x)$ for $n = 5$ (see Appendix \ref{['apd:thm:opt-general']} for definition). The quasiconvexity of $q(x)$ transfers to $d_i$ due to variation diminishing properties of the kernel defined by $a_i(x)$.

Theorems & Definitions (90)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Proposition 6
• Theorem 7
• Proposition 8
• Lemma 9
• Theorem 10: Uniqueness of the symmetric MNE