When a prize for last place can strengthen a contest

A sales competition or research award can motivate extra work, but it can also discourage people who anticipate the disappointment of losing. Should organizers concentrate rewards at the top, spread them widely, or offer something even to last place?

In Robust prizes in all-pay contests with loss aversion, Dmitry Ryvkin and Qin Wu derive a striking answer: divide the rewards equally across every rank except the bottom, and soften the disadvantage of finishing last when loss aversion is strong. Their paper is a theoretical arXiv preprint submitted on November 5, 2026; the supplied version lists no journal publication. It presents mathematical results, not a workplace experiment.

A contest where everyone pays in effort

The model features identical potential contestants and a fixed prize budget. People first decide whether to enter. Those who enter incur effort costs whether or not they earn a high prize: this is the meaning of an “all-pay” contest. Rankings follow effort directly, without performance noise, and ties are broken randomly. Staying out yields zero utility.

Contestants value money linearly, but also experience psychological gains and losses relative to the prizes they expect they might receive. Disappointment about a lower prize weighs more heavily than the pleasure of an equivalent favorable surprise. This is expectation-based loss aversion, rather than ordinary dislike of financial risk.

An important modeling choice concerns when expectations form. Contestants randomize over effort, then adjust their reference expectations to the prize lottery associated with the effort actually chosen. The authors use this modified equilibrium concept throughout; their conclusions are conditional on it.

The organizer knows that effort costs rise and are convex, meaning additional effort becomes no cheaper at the margin. But the precise cost function is unknown. Instead, the organizer has a lower bound on average productivity: across a hypothetical uniform spread of cost expenditures from zero to the normalized prize budget, average effort must exceed a specified floor. This benchmark is a restriction on possible cost functions, not a claim about actual contestant behavior.

Protecting the bottom without erasing the incentive

The organizer chooses prizes to maximize expected effort under the least favorable admissible cost function. For each cost function, however, the calculation uses the most productive symmetric equilibrium available—one in which identical contestants follow the same randomized strategy. It is conservative about costs, not about which equilibrium will occur.

Under moderate loss aversion, the optimal schedule gives nothing to last place and divides the budget equally among all other ranks. The equilibrium has full participation. There is no extra prize for finishing first rather than second: the incentive is to avoid the bottom.

Under strong loss aversion, that same two-level structure remains optimal, but last place receives a positive prize. As loss aversion rises further, the bottom prize increases and the common higher prize falls. “Punishing the bottom” here means awarding a smaller prize, not imposing a fine.

Why cushion losing? A contestant with a small chance of the higher prize may find the prospect of disappointment so painful that the prize lottery is unattractive. A higher bottom prize raises the monetary benefit of entering and narrows the gap that generates psychological losses. Yet financing it from a fixed budget also weakens the reward for avoiding last place.

The optimal balance does not eliminate withdrawal. Under strong loss aversion, an equilibrium attaining the robust effort guarantee has partial participation. Its entry probability decreases as loss aversion increases, despite the rising bottom prize. The best worst-case expected effort also declines. A more equal prize schedule therefore does not mean the contest has become more productive.

What makes the result robust—and what does not

The proof separates effort from the cost incurred to supply it. Because costs increase strictly, ranking contestants by effort is equivalent to ranking them by cost expenditure. The authors first bound expected expenditure across the relevant equilibria and prize schedules. They then show that a linear cost function is least favorable at their proposed optimum, translating the expenditure bound into a worst-case effort guarantee.

This robustness is specific. It does not cover arbitrary productivity, differences in contestants’ abilities or preferences, or noisy performance rankings. Nor does the paper establish that a manager can reliably select the most productive equilibrium when several exist.

The practical lesson is consequently conditional, not a universal recommendation to reward losing. Within this model, maximizing entry and maximizing effort are different objectives. A well-designed contest can cushion disappointment while still accepting that some people will rationally stay out.