Optimal Allocation Of Prizes Research Articles

Optimal contests with incomplete information and convex effort costs

I investigate the design of effort‐maximizing mechanisms when agents have both private information and convex effort costs, and the designer has a fixed prize budget. I first demonstrate that it is always optimal for the designer to utilize a contest with as many participants as possible. Further, I identify a necessary and sufficient condition for the winner‐takes‐all prize structure to be optimal. When this condition fails, the designer may prefer to award multiple prizes of descending sizes. I also provide a characterization of the optimal prize allocation rule for this case. Finally, I illustrate how the optimal prize distribution evolves as the contest size grows.

Multi-prize contests with risk-averse players

This paper studies a multi-prize imperfectly discriminatory contest with symmetric risk-averse contestants. Adopting a multi-winner nested Tullock contest model, we first establish the existence and uniqueness of a symmetric pure-strategy equilibrium under plausible conditions. We then investigate the optimal prize allocation in the contest. Our analysis provides a formal account of the incentive effects triggered by a variation in the prevailing prize structure when contestants are risk averse. We demonstrate that contestants' incentive subtly depends not only on the degree of a contestant's risk aversion but also that of his prudence. The former affects the marginal benefit of effort, while the latter affects the marginal cost. We derive sufficient conditions under which a single-(multi-)prize contest would be optimal when the contest designer aims to maximize total effort. We also discuss in depth the roles played by risk aversion and prudence in optimal prize allocation.

Correction to: Entry regulations and optimal prize allocation in parallel contests

Correction to: Entry regulations and optimal prize allocation in parallel contests

Multi-prize contests with expectation-based loss-averse players

We investigate the optimal prize allocation in a multi-winner nested Tullock contest model with symmetric contestants who exhibit expectation-based loss aversion. We show that (i) multiple prizes can be optimal when contestants are sufficiently loss averse; (ii) all prizes should be equal in the optimal contest; and (iii) the number of prizes increases as the degree of loss aversion increases.

Entry regulations and optimal prize allocation in parallel contests

In parallel contests, the contest organizer controls the entry of heterogeneous contestants by regulating access to the contests and determining the prize allocation across contests. The organizer can prevent a contestant from entering more than one contest. I show that the organizer allows entry to multiple contests and uniquely sets identical prizes across contests to maximize aggregate effort in all contests. Independent of the entry regulation, I find no sorting effects. Thus, a contest with a relatively high prize does not necessarily attract contestants with higher abilities. Furthermore, I discover interesting spillover effects of prizes between contests in the case of restricted entry regulations. For instance, the individual (aggregate) effort increases (decreases) in a contest if the prize in another contest increases. The endogeneity of contestants’ participation drives many of these results.

The optimal allocation of prizes in two-stage contests

We study two-stage contests in which the designer can award a prize for winning in each stage, and also a prize for winning in both stages. For this purpose, we analyze the optimal allocation of prizes for a designer who wishes to maximize the players’ total effort when the matches in each stage are modeled as Tullock contests. It is demonstrated that the prize for winning in both stages should be allocated in the two-stage Tullock contest with two players, but not when there are more than two players.

Tournament rewards and heavy tails

Heavy-tailed fluctuations are common in many environments, such as sales of creative and innovative products or the financial sector. We study how the presence of heavy tails in the distribution of shocks affects the optimal allocation of prizes in rank-order tournaments à la Lazear and Rosen (1981). While a winner-take-all prize schedule maximizes aggregate effort for light-tailed shocks, prize sharing becomes optimal when shocks acquire heavy tails, increasingly so following a skewness order. Extreme prize sharing—rewarding all ranks but the very last—is optimal when shocks have a decreasing failure rate, such as power laws. Hence, under heavy-tailed uncertainty, typically associated with strong inequality in the distribution of gains, providing incentives and reducing inequality go hand in hand.

Optimal prize allocations in group contests

We characterize the optimal prize allocation, namely the allocation that maximizes a group’s effectiveness, in a model of contests. The model has the following features: (i) it allows for heterogeneity between and within groups; (ii) it classifies contests as “easy” and “hard” depending on whether the marginal costs are concave or convex. Thus, we show that in an “easy” contest the optimal prize allocation assigns the entire prize to one group member, the most skilled one. Conversely, all group members receive a positive share of the prize when the contest is “hard” and players have unbounded above marginal productivities. If the contest is “hard” and the marginal productivities are bounded above, then only the most skilled group members are certain of receiving a positive share of the prize for any distribution of abilities. Finally, we study the effects of a change in the distribution of abilities within a group. Our analysis shows that if the contest is either “easy” or a particular subset of “hard”, then the more the heterogeneity within a group, the higher its probability of winning the prize.

On the optimal allocation of prizes in best-of-three all-pay auctions

We study best-of-three all-pay auctions with two players who compete in three stages with a single match per stage. The first player to win two matches wins the contest. We assume that a prize sum is given, and show that if players are symmetric, the allocation of prizes does not have any effect on the players’ expected total effort. On the other hand, if players are asymmetric, in order to maximize the players’ expected total effort, independent of the players’ types, it is not optimal to allocate a single final prize to the winner. Instead, it is optimal to allocate intermediate prizes in the first stage or/and in the second stage in addition to the final prize. When the asymmetry of the players’ types is sufficiently high, it is optimal to allocate intermediate prizes in both two first stages and a final prize to the winner.

Contest Design as a Risk Averse Bidder Problem

I investigate the optimal design of contests when contestants have both private information and convex effort costs. The designer has a fixed prize budget and her objective is to maximize the expected total effort. I first demonstrate that it is always optimal for the designer to employ a grand static contest with as many participants as possible. Further, I identify a sufficient and necessary condition for the winner-takes-all prize structure to be optimal. When this condition fails, the designer may prefer to award multiple prizes of descending sizes. I also provide a characterization of the optimal prize allocation rule for this case. Lastly, I illustrate how the optimal prize distribution evolves as contest size grows: the prize distribution first becomes more unequal until the optimal level of competition intensity is obtained and then becomes less unequal to maintain the optimal intensity. (A previous version has been circulated under the title Contest with Incomplete Information: When to Turn Up the Heat, and How?)

The optimal allocation of prizes in contests with costly entry

Contestants often need to incur an opportunity cost to participate in the competition. In this paper, we accommodate costly entry and study the effort-maximizing prize allocation rule in a contest environment of all-pay auction with incomplete information as in Moldovanu and Sela (2001). As equilibrium entry can be stochastic, our analysis allows prize allocation rule to be contingent on the number of entrants. With free entry, Moldovanu and Sela establish the optimality of winner-take-all when effort cost function is linear or concave. Costly entry introduces a new trade-off between eliciting effort from entrants and encouraging entry of contestants, which might demand a more lenient optimal prize allocation rule. Surprisingly, we find that the optimality of winner-take-all is robust to costly entry when cost function is linear or concave. On the other hand, we provide examples to show that the new trade-off due to costly entry does make a difference to the optimal design when effort cost function is convex.

The Optimal Allocation of Prizes in Contests with Costly Entry

Contestants often need to incur an opportunity cost to participate in the competition. In this paper, we accommodate costly entry and study the effort-maximizing prize allocation rule in a contest environment of all-pay auction with incomplete information as in Moldovanu and Sela (2001). As equilibrium entry can be stochastic, our analysis allows prize allocation rule to be contingent on the number of entrants. With free entry, Moldovanu and Sela establish the optimality of winner-take-all when effort cost function is linear or concave. Costly entry introduces a new trade-off between eliciting effort from entrants and encouraging entry of contestants, which might demand a more lenient optimal prize allocation rule. Surprisingly, we find that the optimality of winner-take-all is robust to costly entry when cost function is linear or concave. On the other hand, we provide examples to show that the new trade-off due to costly entry does make a difference to the optimal design when effort cost function is convex.

Optimal prize allocation in contests: The role of negative prizes

In this paper, we analyze the role of negative prizes in contest design with a fixed budget, risk-neutral contestants, and independent private abilities. The effort-maximizing prize allocation rule features a threshold. When the highest effort is above the threshold, all contestants with lower efforts receive negative prizes. These negative prizes are used to augment the prize to the contestant with the highest effort, which better incentivizes contestants with higher abilities. When no contestant's effort exceeds the threshold, all contestants equally split the initial budget (or a portion of it) to ensure their participation. We find that allowing negative prizes can increase the expected total effort dramatically. In particular, if no bound is imposed on negative prizes, the expected total effort can be arbitrarily close to the highest possible effort inducible when all contestants have the maximum ability with certainty. The above contest is shown to be the optimal mechanism for a more general class of mechanisms.

Equilibrium analysis of the all-pay contest with two nonidentical prizes: Complete results

This paper studies contests in which three or more players compete for two nonidentical prizes. The players have distinct constant marginal costs of performance or bid, which are commonly known. We show that the contests have a generically unique Nash equilibrium, and it is in mixed strategies. Moreover, we characterize the equilibrium payoffs and strategies in closed form. We also study how the equilibrium payoffs and strategies vary with the prizes. As an application, we numerically compute the optimal allocation of prizes that maximizes the total expected bid of asymmetric players.

Effective Prize Structure for Simple Crowdsourcing Contests with Participation Costs

This paper studies the use of a multi-prize compensation scheme for "simple" contests where participation is costly and the quality of participants' contributions is a priori uncertain at the time they make their decision related to participating in the contest. The equilibrium analysis provided enables demonstrating not only that a multi-prize structure is often beneficial but also that in some cases the principal's expected profit is maximized when offering a second prize greater than the first prize. This may seem somehow counter-intuitive especially given that the principal's profit is only influenced by the quality of the best submission rather than the aggregate of submissions. Special emphasis is placed on the case where the contestants are a priori homogeneous which is often the case in real-life, whenever the contestants are basically a priori alike and the quality of their submissions is determined subjectively by some referee. Here, we manage to prove that a multi-prize structure is dominated by a winner-takes-all scheme, suggesting that the benefit in the multi-prize contest scheme fully derives from the heterogeneity between prospective contestants. Finally, we show that there is a class of settings where the use of the multi-prize crowdsourcing contest model enables achieving the performance of the fully cooperative model (which is an upper bound for the performance in any type of contest), and that for settings of this class the optimal prize allocation can be extracted through a set of linear equations.

Optimal Prize Allocations in Group Contests

We characterize the optimal prize allocation, namely the allocation that maximizes a group’s effectiveness, in a model of contests. The model has the following features: i) it allows for heterogeneity between and within groups; ii) it classifies contests as “easy” and “hard” depending on whether the marginal costs are concave or convex. Thus, we show that in a “easy” contest the optimal prize allocation assigns the entire prize to one group member, the most skilled one. Conversely, all group members receive a positive share of the prize when the contest is “hard” and players have unbounded above marginal productivities. If the contest is “hard” and the marginal productivities are bounded above, then only the most skilled group members are certain of receiving a positive share of the prize for any distribution of abilities. Finally, we study the effects of a change in the distribution of abilities within a group. Our analysis shows that if the contest is either “easy” or a particular subset of “hard”, then the more the heterogeneity within a group, the higher its probability of winning the prize.

Optimal Prize Allocations in Contests

Given a fixed prize budget for a contest, what is the optimal prize allocation among contestants? The answer depends on the objective of the contest designer, which typically is either to maximize the total performance of all contestants or simply the champion's performance. We try to shed light on this question for both objectives in a standard model in which contestants are heterogeneous in skill and exert effort to win a prize. We show that weak concavity of the reduced-form cost function leads to optimality of single prize for both objectives, which generalizes the previous results in the literature. We find a dual relationship between the cost function and the principal's utility function (in particular, risk attitude), which not only helps to provide intuition for the optimality but also directly provides results for a principal with a different risk attitude. Surprisingly, with the traditional Cobb-Douglas functional form, optimality of single prize, when the number of contestants is three, continues to hold for arbitrary degree of convexity under maximal performance objective. On the contrary, if the reduced-form cost function is piecewise linear, then it may be optimal to reward the runners-up if the function is convex enough. When the number of prizes under consideration is two, there is an interesting relationship between the two objectives. In the derivation of the results, a series of simple facts about the winning probability functions are presented, which may be useful for future works in contest theory and multi-object auction theory.

Prize and incentives in double-elimination tournaments

I examine a game-theoretical model of two variants of double-elimination tournaments, and derive the equilibrium behavior of symmetric players and the optimal prize allocation assuming a designer aims to maximize total effort. I compare these theoretical properties to the well-known single-elimination tournament.

THE OPTIMAL ALLOCATION OF PRIZES IN TOURNAMENTS OF HETEROGENEOUS AGENTS

Tournaments are widely used in organizations, explicitly or implicitly, to reward the best‐performing employees, for example, by promotion or bonuses, and/or to penalize the worst‐performing employees, for example, by demotion, withholding bonuses, or unfavorable job assignments. These incentive schemes can be interpreted as various prize allocations based on the employees' relative performance. While the optimal prize allocation in tournaments of symmetric agents is relatively well understood, little is known about the impact of the allocation of prizes on the effectiveness of tournament incentive schemes for heterogeneous agents. We show that while multiple prize allocation rules are equivalent when agents are symmetric in their ability, the equivalence is broken in the presence of heterogeneity. Under a wide range of conditions, loser‐prize tournaments, that is, tournaments that award a low prize to relatively few bottom performers, are optimal for the firm. The reason is that low‐ability agents are discouraged less in such tournaments, as compared to winner‐prize tournaments awarding a high prize to few top performers, and hence can be compensated less to meet their participation constraints. (JEL M52, J33, J24)

Prize and Incentives in Double-Elimination Tournaments

I examine a game-theoretical model of two variants of double-elimination tournaments, and derive the equilibrium behavior of symmetric players and the optimal prize allocation assuming a designer aims to maximize total effort. I compare these theoretical properties to the well-known single-elimination tournament.