Optimal Contest Beyond Convexity

MohammadTaghi Hajiaghayi; Negin Golrezaei; Suho Shin

arxiv: 2604.04844 · v1 · submitted 2026-04-06 · 💻 cs.GT · cs.DS· econ.TH· math.OC

Optimal Contest Beyond Convexity

Negin Golrezaei , MohammadTaghi Hajiaghayi , Suho Shin This is my paper

Pith reviewed 2026-05-10 19:35 UTC · model grok-4.3

classification 💻 cs.GT cs.DSecon.THmath.OC

keywords contest designmechanism designnonconvex optimizationprize allocationapproximation schemesSchur-convexitytotal positivity

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The pith

For nonconvex objectives in contest design, the optimal prize allocation gives more to the winner, equal shares to all intermediates, and nothing to the last place.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how a contest designer should set rank-based prizes to maximize nonconvex functions of contestant quality when participants choose efforts strategically. It covers objectives that blend user welfare with average quality or take the form of posynomials, both of which can lack convexity or concavity. The central finding is that the optimum nevertheless follows a simple stepped pattern. This pattern reduces the design problem to a tractable search and supports an efficient approximation algorithm that needs only black-box access to the objective value.

Core claim

For objectives that are any convex combination of user welfare and average contestant quality, or arbitrary posynomials over quality, the optimal prize vector satisfies p1 ≥ p2 = … = p_{n-1} ≥ pn = 0. The proof relies on Schur-convexity of Bernstein basis polynomial-weighted functions together with total positivity and the variation-diminishing property. The structure further yields a reduction from the high-dimensional nonconvex program to a single-dimensional optimization, producing a fully polynomial-time approximation scheme given a value oracle.

What carries the argument

The prize allocation structure p1 ≥ p2 = … = p_{n-1} ≥ pn = 0, which is shown to be optimal for the stated objective families and which links the number of gradient transitions to the number of changes in the prize sequence.

If this is right

The same stepped prize pattern holds for social welfare, order statistics, and inverse S-shaped functions.
A value oracle for the objective is sufficient to obtain a fully polynomial-time approximation to the optimal prizes.
The high-dimensional problem reduces to a single-dimensional search once the gradient transition count is known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The gradient-transition counting technique may simplify other high-dimensional nonconvex mechanism-design problems that possess similar monotonicity properties.
Contest platforms with complex evaluation metrics, such as recommender systems, could adopt the three-tier prize rule without custom convexification.
The structural result invites checking whether analogous tiered allocations appear in related settings such as all-pay auctions or procurement contests.

Load-bearing premise

The designer's objective must belong to one of the two families: convex combinations of user welfare and average contestant quality, or arbitrary posynomials over quality.

What would settle it

An explicit objective in one of the two families together with a contest instance in which every optimal prize vector violates p1 ≥ p2 = … = p_{n-1} ≥ pn = 0.

Figures

Figures reproduced from arXiv: 2604.04844 by MohammadTaghi Hajiaghayi, Negin Golrezaei, Suho Shin.

**Figure 2.** Figure 2: Structures of optimal policy with respect to different [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Behavior of di for n = 5. Since p5 = 0, d5 is omitted. Quasiconvexity of di follows from the quasiconvexity of q(x) shown in [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

**Figure 4.** Figure 4: Behavior of q(x) for n = 5 (see Appendix F.3 for definition). The quasiconvexity of q(x) transfers to di due to variation diminishing properties of the kernel defined by ai(x). 7.1.2 Step 2: Applying KKT Conditions With Lemma 19 in hand, we analyze the KKT conditions for the optimization problem Opt. These involve the derivatives di , the probabilities pi , and associated dual variables. Using the structur… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that for two families of nonconvex objectives in contest design, the optimal prize vector always takes the simple form p1 ≥ p2 = … = p_{n-1} ≥ pn = 0, which reduces the problem to a 1D search and yields an FPTAS.

read the letter

The main point is that even with nonconvex objectives the optimal prize allocation stays highly structured: the top contestant can get more, the bottom gets zero, and everyone in between gets exactly the same amount. This holds for convex combinations of welfare and quality as well as arbitrary posynomials, and it lets them turn a high-dimensional nonconvex problem into a single-dimensional search plus an FPTAS given a value oracle. That reduction, tied to the shape of the gradient sequence, is the cleanest new piece here and could matter beyond contests. They also correctly note that these objective families cover order statistics and S-shaped functions that prior contest work largely ignored. The technical route through Schur-convexity of Bernstein polynomials and the variation-diminishing property of totally positive kernels is standard but applied to a setting where convexity was previously required, so the extension is real. The soft spot is that the abstract gives no proof sketches or error bounds, so it is impossible to check how tightly the total-positivity argument controls the number of transitions or whether the FPTAS approximation ratio degrades for certain posynomials. If the full paper supplies the details without hidden boundedness assumptions, this is fine; otherwise the central claim rests on unverified steps. This work is for algorithmic mechanism designers who need tractable mechanisms when the designer's goal mixes welfare with quality or uses posynomial forms. A reader focused on crowdsourcing or recommender incentives will find the structure and algorithm useful. It deserves peer review because the claims are concrete, the toolkit is appropriate, and the reduction from high-dimensional nonconvex optimization is potentially reusable.

Referee Report

0 major / 2 minor

Summary. The paper studies optimal contest design where a designer with fixed budget allocates rank-based prizes to n strategic contestants to maximize non-convex objectives over contestant qualities. It focuses on two families: convex combinations of user welfare and average contestant quality, and arbitrary posynomials over quality (encompassing social welfare, order statistics, and S-shaped functions). The central claim is that the optimal prize vector always satisfies the structure p1 ≥ p2 = ⋯ = p_{n-1} ≥ pn = 0. This characterization reduces the problem to single-dimensional optimization and yields an FPTAS given a value oracle. The proofs rely on Schur-convexity of Bernstein-basis-weighted functions, total positivity, and the variation-diminishing property.

Significance. If the structural result and reduction hold, the work meaningfully extends contest theory to non-convex objectives that arise in recommender systems and related domains, where prior literature has largely restricted attention to convex cases. The reduction from high-dimensional nonconvex optimization to a 1D search via properties of gradient sequences is technically interesting and potentially reusable. The FPTAS is a concrete algorithmic contribution. The application of total-positivity tools to mechanism design is a strength.

minor comments (2)

[Abstract] Abstract: the high-level claim that the result follows from Schur-convexity and total positivity would be strengthened by a one-sentence intuition on how these properties bound sign changes in the gradient sequence.
The manuscript should explicitly state the approximation ratio and running-time dependence on the oracle and on n and the approximation parameter in the main theorem statement for the FPTAS.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work, accurate summary of the contributions, and recommendation for minor revision. The referee correctly identifies the key structural result on optimal prize allocations for non-convex objectives and the resulting FPTAS. We will make minor revisions to enhance presentation and clarity as appropriate.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external analytic properties

full rationale

The paper derives the structural characterization of the optimal prize vector (p1 ≥ p2 = … = p_{n-1} ≥ pn = 0) for the two families of possibly non-convex objectives by applying Schur-convexity to Bernstein-basis-weighted functions, together with total positivity and the variation-diminishing property to control sign changes in the gradient sequence. This yields the reduction to a single-dimensional search without any step that defines the claimed optimum in terms of itself, fits a parameter to a subset of data and renames the fit as a prediction, or invokes a load-bearing uniqueness result from the authors' own prior work. The technical toolkit consists of standard external mathematical facts applied to the stated objective families; the FPTAS follows directly from the structure once the number of transition points is bounded. No equation or claim reduces the result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard mathematical properties without introducing free parameters or new entities; the central claim rests on the assumption that the objective lies in the stated families and that the cited analytic properties apply.

axioms (2)

standard math Schur-convexity of Bernstein basis polynomial-weighted functions
Invoked to obtain the structural characterization of the optimal prize vector
standard math Total positivity and variation diminishing property
Used to control the shape of the optimum and the number of transition points

pith-pipeline@v0.9.0 · 5641 in / 1531 out tokens · 50760 ms · 2026-05-10T19:35:17.430307+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

We refer to surveys by Li et al

and combinatorial optimization (Golrezaei et al., 2024) in the past few years. We refer to surveys by Li et al. (2023); Wang et al. (2023); Deldjoo et al. (2024) for more details in fairness- aware recommender systems. Chen et al. (2024) study the two-sided fairness for both the item and user. Golrezaei et al. (2024) propose a general framework to incorpo...

work page 2024
[2]

nX i=1 1{I=i} ·q i # =E q∼µ

Thus, we can write the produceri’s payoff as follows: Ui(q,µ −i,p) =R i(q,µ −i,p)−c(q) =E    X i∈[n] 1{I=i}   −c(q)   =  X i∈[n] Pr [ Provideri’s content is recommended to useri]   −c(q) = nX i=1 pi n−1 i−1 F(q) n−i(1−F(q)) i−1 ! −c(q).(C.3) For the user welfare, letE i be the random event such that the chosen content has thei-th highest quali...

work page 2024
[3]

, xk andi 1,

Since this holds for arbitrary choice ofx 1, . . . , xk andi 1, . . . , ik, this concludes thatK(x, i) is STPn−1. Now we will prove that the diminishing property of totally positive matrixK(x, i) allows us to show the quasi-convexity of the sequence of partial derivatives. Let us defineq(x) =f(x) +g(x), where f(x) =nα(1 + 1/β)h(x,p) 1/β g(x) = (1−α)/βh(x,...

work page 1918

[1] [1]

We refer to surveys by Li et al

and combinatorial optimization (Golrezaei et al., 2024) in the past few years. We refer to surveys by Li et al. (2023); Wang et al. (2023); Deldjoo et al. (2024) for more details in fairness- aware recommender systems. Chen et al. (2024) study the two-sided fairness for both the item and user. Golrezaei et al. (2024) propose a general framework to incorpo...

work page 2024

[2] [2]

nX i=1 1{I=i} ·q i # =E q∼µ

Thus, we can write the produceri’s payoff as follows: Ui(q,µ −i,p) =R i(q,µ −i,p)−c(q) =E    X i∈[n] 1{I=i}   −c(q)   =  X i∈[n] Pr [ Provideri’s content is recommended to useri]   −c(q) = nX i=1 pi n−1 i−1 F(q) n−i(1−F(q)) i−1 ! −c(q).(C.3) For the user welfare, letE i be the random event such that the chosen content has thei-th highest quali...

work page 2024

[3] [3]

, xk andi 1,

Since this holds for arbitrary choice ofx 1, . . . , xk andi 1, . . . , ik, this concludes thatK(x, i) is STPn−1. Now we will prove that the diminishing property of totally positive matrixK(x, i) allows us to show the quasi-convexity of the sequence of partial derivatives. Let us defineq(x) =f(x) +g(x), where f(x) =nα(1 + 1/β)h(x,p) 1/β g(x) = (1−α)/βh(x,...

work page 1918