Sharing Prizes When Luck Is Hard to Predict

Designing for an unfavorable environment

A prize scheme that works well under one pattern of luck may work poorly under another. Drugov and Ryvkin’s theoretical preprint asks how an organizer should allocate rank-based rewards without knowing the distribution of performance noise. It does not report an experiment or establish that a particular reward scheme improves actual workplace performance.

The organizer’s objective is deliberately conservative: maximize the lowest equilibrium effort across the noise distributions allowed by the model. This is not the same as maximizing expected effort under a best guess about the environment. The adversarial “Nature” used in the analysis is a mathematical device for finding the least favorable distribution, not an additional real-world participant.

Some restriction on uncertainty is essential. Without one, noise could become so dispersed that the guaranteed effort approaches zero. The authors mainly impose an upper bound on differential Shannon entropy, a measure involving the average logarithm of the noise density. This limits how far noise can be spread through rescaling while leaving considerable freedom over its shape.

What the model assumes

Competitors are identical before competing and are risk neutral. Each chooses costly effort, and observed output equals effort plus an individual random shock. Effort costs rise at an increasing rate. Competitors know the environment, including the noise distribution; the organizer does not.

In the finite model, the organizer divides a fixed budget among nonnegative prizes ordered by performance. The last prize is zero. Additional restrictions on noise densities, relative to the curvature of effort costs, ensure a unique equilibrium in which all competitors choose the same effort.

These restrictions matter. The paper’s large-population analysis uses a bounded-hazard condition that excludes familiar distributions such as the normal and uniform. The finite model allows unbounded hazards, but still imposes regularity conditions. The results therefore do not cover every conceivable form of luck.

Broad rewards, unequal prizes

Under the conditions needed to implement the solution as an equilibrium, the authors find a unique robust finite-tournament prize scheme. Every rank except the last receives a positive prize, and the winner receives strictly more than the runner-up. Intermediate ranks can receive equal amounts.

Broad sharing does not mean near equality. For three competitors, the numerical solution gives approximately 90.55% of the budget to the winner, 9.45% to second place, and nothing to third. With four competitors, the shares are approximately 79.8%, 10.1%, 10.1%, and zero. Thus, even the robust allocation can resemble winner-take-all in a small contest.

The mechanism concerns improvements in rank. Extra effort is rewarded when it helps a competitor cross a performance cutoff and obtain a higher prize. Concentrating rewards at very few cutoffs leaves incentives vulnerable to an unfavorable noise distribution. Spreading incentives more broadly reduces that vulnerability.

Importantly, the authors spread budget-weighted marginal incentives across performance quantiles—not prizes equally across people. Raising rewards above a high cutoff affects fewer recipients, so larger prize increases near the top can be consistent with evenly distributed incentive weights.

A logarithmic rule for large contests

In the large-population model, with the average prize normalized to one, the robust reward at performance quantile z is −log(1 − z). Rewards become progressively steeper toward the top.

For finite contests, the authors solve an optimization problem that makes the distribution of incentive-relevant performance cutoffs as close to uniform as possible, measured by Kullback–Leibler divergence from the uniform distribution. Exact uniformity is impossible with finitely many competitors because rank remains random even conditional on an individual’s output.

As the number of competitors grows, appropriately rescaled robust prizes approach the logarithmic profile. A schedule constructed from sums of reciprocal ranks—the harmonic-number schedule—provides an asymptotic approximation.

The associated least favorable noise distributions have bounded support for finite contests. In the auxiliary optimization problem, holding the entropy bound fixed, they converge to an exponential distribution. However, ensuring equilibrium implementation at every growing contest size may require increasing the entropy bound. The prize schedule itself does not depend on that bound.

What the findings establish

Numerical results for three through 50 competitors show declining top prizes and Gini coefficients as participation grows. The limiting logarithmic schedule has a Gini coefficient of 0.5. The paper does not establish an unambiguous inequality ordering across all contest sizes.

In the large-population model, logarithmic rewards and exponential worst-case noise also arise under certain alternative uncertainty restrictions, including a bound on expected noise above its lower support endpoint.

The contribution is a conditional rationale for sharing prizes while retaining substantial inequality. As a preprint offering mathematical analysis rather than empirical validation, it does not establish a universal compensation rule. Differences in ability, risk preferences, or competitors’ own uncertainty remain outside the model studied here.