This paper studies the optimal allocation of multiple prizes in a contest where contestants have private information about their abilities. The contest designer aims to maximize the expected sum of bids. The main findings are:
1. **Linear Cost Functions**: For any number of contestants with linear cost functions and any distribution of abilities, it is optimal for the designer to allocate the entire prize sum to a single "first" prize.
2. **Concave Cost Functions**: For any number of contestants with concave cost functions and any distribution of abilities, it is also optimal to allocate the entire prize sum to a single "first" prize.
3. **Convex Cost Functions**: For contestants with convex cost functions, the optimal prize structure may involve multiple prizes. The necessary and sufficient condition for the optimality of two prizes is given by a specific integral condition involving the distribution function, cost function, and number of contestants.
The paper provides detailed proofs and examples to support these findings, including the derivation of equilibrium bid functions and the analysis of the designer's problem. The results have implications for various economic, social, and biological contexts where contests are used to allocate resources or incentives.This paper studies the optimal allocation of multiple prizes in a contest where contestants have private information about their abilities. The contest designer aims to maximize the expected sum of bids. The main findings are:
1. **Linear Cost Functions**: For any number of contestants with linear cost functions and any distribution of abilities, it is optimal for the designer to allocate the entire prize sum to a single "first" prize.
2. **Concave Cost Functions**: For any number of contestants with concave cost functions and any distribution of abilities, it is also optimal to allocate the entire prize sum to a single "first" prize.
3. **Convex Cost Functions**: For contestants with convex cost functions, the optimal prize structure may involve multiple prizes. The necessary and sufficient condition for the optimality of two prizes is given by a specific integral condition involving the distribution function, cost function, and number of contestants.
The paper provides detailed proofs and examples to support these findings, including the derivation of equilibrium bid functions and the analysis of the designer's problem. The results have implications for various economic, social, and biological contexts where contests are used to allocate resources or incentives.