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.

• The paper establishes that optimal contest design under nonconvex objectives necessitates a novel two-threshold rank-based prize structure that outperforms traditional mechanisms.
• It leverages topological arguments and Schur-theoretic methods to prove the existence and uniqueness of a symmetric mixed Nash equilibrium with a reduction to a single-parameter optimization.
• Algorithmically, the approach enables fully polynomial-time approximations for complex contests, offering practical solutions for online platforms and economic mechanism design.

Optimal Contest Mechanism Design Beyond Convexity: Structural and Algorithmic Contributions

Introduction and Context

This paper addresses the fundamental problem of optimal contest design, extending the classic Lazear and Rosen framework to high-dimensional, nonconvex, and application-rich objective functions. In the standard paradigm, a designer allocates a fixed budget among $n$ strategic contestants, each choosing an effort level, leading to observable qualities. The designer seeks to structure rank-based prizes $(p_1, ..., p_n)$ as incentive mechanisms to induce effort that optimally aligns with platform-level objectives (e.g., content quality, user welfare). While prior literature provides sharp characterizations for specific instances (notably average quality maximization [Glazer and Hassin, 1988], social welfare, and some forms of convex/concave objectives), general mechanisms for high-dimensional nonconvex objectives have remained elusive.

Generalization to Nonconvex Objectives

The paper rigorously extends the contest design problem to settings where the objective may be:

• An arbitrary convex combination of user welfare and average quality (neither convex nor concave),
• General posynomials (encompassing order statistics, classic social welfare, S-shaped/exponential utility, etc.), and
• Other structurally complex forms arising in modern online platforms and economic applications.

This generalization is motivated by and directly applicable to settings such as recommender systems, where both the mean quality and its distribution (user-facing utility, diversity, fairness) are relevant, and their trade-offs reflect nonconvex properties.

Main Results: Structural Characterization

Existence and Uniqueness of Symmetric Mixed Nash Equilibrium

The induced game among contestants does not admit a pure Nash equilibrium for any nontrivial (i.e., non-constant) policy. Nevertheless, leveraging topological arguments (via Reny's existence theorem for discontinuous games), the existence and uniqueness of a symmetric mixed Nash equilibrium (MNE) are established for arbitrary rank-based policies. Key technical points include:

• The strategic action space is metrized using the Lévy-Prokhorov topology.
• The symmetric MNE is atomless and strictly increasing on its support, enabling tractable characterization of equilibrium payoffs and objective values.
• The reduction to a unique symmetric MNE is achieved via analysis of revenue monotonicity and payoff continuity.

Policy Structure: Beyond HARDMAX and UNIFORMBUTLAST

Previous literature revealed the optimality of two extremal mechanisms:

• HARDMAX: winner-take-all (all reward to top rank),
• UNIFORMBUTLAST: uniform allocation among all but the last rank.

This work demonstrates neither is generally optimal beyond convex/concave objectives. For a broad class of realistic objectives, the optimal mechanism has a highly structured form:

• The top-ranked contestant receives a maximal or higher-than-average prize, with all intermediate ranks receiving an equal (lower) prize, and the last-ranked always receiving zero:

$p_1 \geq p_2 = \cdots = p_{n-1} > p_n = 0$

• There is a parameterized phase transition between HARDMAX and UNIFORMBUTLAST, determined by specific properties of the objective and the cost function. For example, with strictly convex costs, UNIFORMBUTLAST is optimal for average-quality maximization; with strictly concave costs, HARDMAX is optimal for user welfare; for in-between regimes (and mixed/complex objectives), the interpolated structure above is optimal.
• Strong claim: HARDMAX can be arbitrarily suboptimal as $n\to\infty$ under highly convex costs, thus incentivizing designs which allocate nonzero prize mass beyond the winner.

Reduction to Tractable Optimization

A major methodological advance is a general reduction from high-dimensional nonconvex policy design to a one-dimensional (single-parameter) optimization, based on:

• The structure of the equilibrium facilitates expressing platform objectives (expected social welfare, mean quality, etc.) as functionals over Bernstein basis polynomials weighted by the prize vector.
• Via Schur-convexity/concavity and total positivity, the derivative sequence of the objective w.r.t. $(p_i)$ is proven to be quasiconvex, meaning it is monotonic except at a single inflection. This constrains the optimal prize structure to require (at most) a single threshold point separating distinct prize levels.
• The general argument applies to objectives with partial derivatives representable as integrals over a totally positive kernel with a quasiconvex operand—encompassing all posynomials and many other functions of practical interest.

Thus, for a wide class, optimal contest design admits a fully polynomial-time approximation scheme (FPTAS), using a branch-and-bound search on the reduced policy space.

Algorithmic Implications

• The policy search is efficiently computable: the complex, possibly nonconvex high-dimensional problem in $n$ variables is reduced to a single-variable search with explicit upper and lower bounds, and standard numerical integration techniques suffice for practical implementation.
• Additive or multiplicative $\epsilon$-approximate policies are achievable in time polynomial in $n$ and $1/\epsilon$ for fixed-cost parameters.

Generality: Beyond Additive Quality Metrics

The approach admits significant further generality:

• General Posynomials: Any positive linear combination of monomials in ranked qualities (possibly with non-integer exponents), order statistics, S-shaped utility, and exponential objectives are included.
• If the sign pattern of the partial derivatives of the objective (as a function of the prize vector) changes at most once (from negative to positive), the resulting policy inherits the two-threshold structure.
• Tractable extensions to social welfare maximization/combination with contest designer’s own utility (as a posynomial in contestant efforts) are formally covered.

Implications and Future Directions

Theoretical

• Provides a unifying analysis subsuming all previous classic results as special cases, with strict generalization and improved structural clarity (via Schur-theoretic arguments and variation-diminishing properties).
• Establishes a template for contest analysis in settings where the objective is not nicely convex/concave, broadening the toolkit for both economics and algorithmic game theory.
• Suggests possibilities for further generalization: characterizing structural optimality when objective gradients exhibit more complicated (non-quasiconvex) patterns and extending to incomplete information with heterogeneous agents.

Practical

• Direct applicability to contest-like mechanism design for online platforms, crowdsourcing, research funding agencies, innovation tournaments, etc., in domains where both individual and collective performance, as well as fairness and incentivization, must be balanced.
• Impacts fairness-aware recommender systems: the result that intermediate (non-winner-take-all) exposure allocations arise endogenously from optimal objectives suggests that explicit fairness constraints may be less necessary if objectives are well specified.
• Implies that simplistic winner-take-all mechanisms may perform very poorly in environments with strongly convex effort costs and complex platform objectives.

Conclusion

This work provides a comprehensive structural, equilibrium, and computational analysis of incentive-optimal contests for a broad range of nonconvex objectives. By exploiting deep connections between game-theoretic equilibria, majorization theory, and the variation-diminishing property of totally positive kernels, it resolves several long-standing open questions in contest design. It delivers new tools for practical platform incentivization, and opens directions for contest design under more complex objective landscapes and agent heterogeneity.

Reference:

For the full formalism, technical proofs, and empirical validation, see "Optimal Contest Beyond Convexity" (2604.04844).