When a Lottery Can Make a Contest Work Harder

Designing incentives, not just picking winners

A contest does more than distribute prizes after people perform. Its rules can influence how much effort they make beforehand. In Optimal Large-Contest Design, Giovanni Valvassori Bolgè studies that connection in a theoretical setting representing a very large population of competitors.

The paper models participants as a continuum rather than a finite list of individuals. A fixed share can receive prizes. Allocation rules specify winning probabilities and must be nondecreasing in effort—or in observed performance when measurement is noisy. Better performance therefore cannot reduce someone’s chance of winning.

The analysis distinguishes two questions: how to allocate prizes when the effort distribution is taken as given, and how to design rules when participants adjust their effort in response. That distinction matters because a selection rule that rewards existing performance need not generate the most effort.

Why simple thresholds emerge

A central author claim is that, across the settings characterized in the paper, an optimal rule can be found among simple step functions with at most two jumps. Rather than continuously adjusting winning probabilities for every performance difference, the designer can use a small number of thresholds.

One such rule excludes people below a lower threshold, gives an intermediate group a common lottery chance, and guarantees prizes above an upper threshold. Some optimal rules need only one threshold, or give everyone the same chance.

The mathematical approach uses extreme points: feasible choices that cannot be expressed as nontrivial averages of other feasible choices. When effort responds strategically, the author reformulates the problem around equilibrium effort distributions and participants’ payoffs. In the homogeneous, perfectly observed setting, an optimal equilibrium can be supported on at most two effort levels.

This is an existence result for a simple optimal design, not a claim that every conceivable optimal rule must look identical.

The objective changes the answer

When effort is treated as exogenous, a designer who values higher-performing winners more should select from the top, with randomization where necessary at the cutoff. Within the model’s monotonicity restriction, a designer whose weighting decreases with effort instead optimally gives everyone an equal chance. Nonmonotonic preferences can support the two-threshold arrangement.

With strategic effort, another distinction becomes crucial: does the designer care only about winners, or about everyone’s effort? An objective focused on winners can favor concentrating effort among a smaller group. Maximizing aggregate effort can favor broad participation instead.

For perfectly observable effort and strictly convex effort costs, the paper identifies an optimal aggregate-effort rule that sets a qualifying threshold and awards prizes randomly above it. In its attaining equilibrium, everyone chooses the same qualifying effort and receives the same winning probability.

The intuition is not that performance is impossible to rank. Convex costs make additional effort increasingly expensive. Spreading effort across participants can therefore produce more total effort than concentrating costly exertion among a subset. The lottery changes incentives before selection; it is not merely a way to break ties afterward.

Noise can reverse the recommendation

The strategic noisy-performance extension uses a more specialized model, with performance equal to log effort plus an independent noise term, and restrictions on costs and the noise distribution.

Under the stated framework, sufficiently small noise favors a lottery above a performance threshold. The author reports that this design reaches the model’s first-best aggregate effort: the upper bound implied by available prize value and effort costs.

With larger noise, deterministic selection above a performance cutoff becomes optimal. Randomization would weaken incentives too much; giving qualifying performers a guaranteed prize supplies stronger incentives. Optimal aggregate effort then falls below the first-best level. The formal optimal equilibria described in this noisy model have all participants choosing the same effort.

The boundary depends on prize availability, prize value, effort costs and the noise distribution. It is not a universal numerical standard for deciding when an admissions test or grant assessment is sufficiently accurate.

What the findings do—and do not—establish

The paper cites admissions and funding lotteries as motivating examples. It does not present an empirical evaluation showing that those institutions increased effort, improved research, or produced fairer outcomes. Its incentive-based rationale is distinct from arguments about diversity or difficulty distinguishing similarly qualified applicants.

The conclusions depend on the objective and assumptions. An extension with several effort-cost types permits more steps, while the author identifies more general utility specifications and relaxing compact effort support as future research directions. The noisy section also contains a technical wording inconsistency: it calls the density strictly increasing across the real line while listing normal and logistic examples.

The practical lesson is conditional: ranking competitors and motivating them are different design problems. A lottery can be optimal for the second without being justified by uncertainty about the first.