On the optimality of small research tournaments

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Highlights

  • Fixed-prize tournaments with asymmetric contestants and entry fees.

  • A sharp bound on the worst-case cost-effectiveness of two-player tournaments.

  • New insights about when larger tournaments are optimal.

Abstract

A central result in Fullerton and McAfee’s (1999) analysis of fixed-prize research tournaments shows that if firms’ heterogeneous marginal effort costs are publicly known and the procurer can charge non-discriminatory entry fees, restricting entry to the two most efficient firms is optimal under a (fairly restrictive) sufficient condition on the form of heterogeneity. This note provides a complementary result. I prove a sharp, worst-case bound (across all linear cost structures) for the ratio between the cost of procuring a given total effort from the optimal number of contestants and the corresponding cost for a tournament featuring only the two most efficient firms. The analysis confirms the attractiveness of the smallest possible tournament, with some notable exceptions.

Introduction

Research tournaments are important mechanisms for procuring innovations. They may mitigate many of the problems that plague traditional procurement contracts in this case, such as non-verifiable quality of the innovation, or difficulties to monitor and verify the efforts and costs of suppliers.1 In an influential paper, Fullerton and McAfee (1999, henceforth FM) studied how to design a fixed-prize research tournament in a setting with n potential firms/suppliers that are heterogeneous with respect to their marginal effort costs. FM focused on how such heterogeneity affects the answers to two major design questions for the procurer. How many firms should be admitted to the tournament? How should these contestants be selected when costs are private information prior to the tournament?

FM characterized equilibrium efforts and expected profits in a simultaneous-move, fixed-prize tournament with prize P, when winning probabilities are determined by a standard Tullock success function and (constant) marginal effort costs are common knowledge among the mn contestants. Having additional contestants (weakly) increases total effort, which determines the distribution of the quality of the best innovation, but (weakly) decreases firms’ profits and hence their willingness to pay for entering the tournament.

For the case where all firms’ costs are publicly known and the procurer can choose P and set a non-discriminatory entry fee to select a restricted number of contestants, a central result in FM provides a condition which ensures that the total cost of “procuring” a given level of total effort is minimized by the smallest possible tournament, between the two most efficient firms. FM’s sufficient condition, which concerns the differences between firms’ marginal costs, covers many interesting cases but it is violated whenever one of the n firms is a (sufficiently) close competitor for another firm.

The purpose of this note is to provide a complementary result that quantifies the worst-case performance of the tournament with m=2 contestants: I derive a sharp bound for the ratio between the minimum cost of procuring a given level of total effort (the cost for the optimal value of m) and the corresponding cost for m=2, across all (linear) cost structures and all values of n. As this bound is quite close to 1, compared to how costly it can be (for some cost structures) to let more than two firms enter, the result generally confirms the attractiveness of the two-player tournament. However, the analysis also provides intuitive insights about when implementing a larger tournament may significantly lower costs for the procurer. Roughly speaking, this is the case if the asymmetry between the two most efficient firms is substantial but not too extreme and if there are other firms that are close competitors for the “second-best” firm.

For cases where marginal costs are i.i.d. and privately known prior to the tournament (but become common knowledge among the contestants before they choose their efforts), FM showed that the contestant selection auction, an all-pay auction with a small interim prize for entry, can be used to select any desired number of lowest-cost contestants. Comparing the “cost-effectiveness” of tournaments with different numbers of contestants in this incomplete-information model is rather involved, and is beyond the scope of this note.2

The two papers that are most closely related to the present study are FM and a seminal contribution by Taylor (1995), who studied a model with symmetric firms and also found that limiting the number of contestants is beneficial. Fullerton et al. (2002) and Schöttner (2008) compared fixed-prize tournaments to first-price auctions (where firms bid a combination of quality and price after having conducted research) in two different models with stochastic research technology and symmetric firms.3 Che and Gale (2003) studied a model with a deterministic technology, complete information and (possibly) asymmetric firms. They showed that inviting the two most efficient firms to participate in a first-price auction with a price cap for the more efficient firm is optimal within a much broader class of possible contests.4

Section snippets

The basic tournament model

This section reviews the relevant facts about FM’s basic tournament model with heterogeneous contestants.5 There are n2 risk-neutral firms, indexed by i{1,,n}, that might participate in a simultaneous-move, fixed-prize research tournament with prize P>0. If it enters, firm i’s cost of exerting an effort ziR+ is equal to γ+cizi if zi>0, where γ0 and ci>0, and is equal to 0 if zi=0. The choice of effort zi results in an innovation of random quality xi[0,x̄],

Complete information and entry fees: a sharp worst-case bound for general cost structures

If all firms’ costs are commonly known prior to the tournament, and if the procurer can charge non-discriminatory entry fees, an entry fee of (slightly below) E=P1cm(m1)j=1mcj2>0can be used to induce a tournament featuring (only) the mm̄ lowest-cost firms. Moreover, to induce a total effort of Z in this case, the prize must be P=Zj=1mcjm1. Thus, the total cost of “procuring” effort Z using a tournament with mm̄ contestants is given by TCm=PmE=Zj=1mcjm11m1cm(m1)j=1mcj2=Zj=1mcj1+2Δ

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Cited by (1)

This paper is a revised version of a chapter of my dissertation at the University of Bonn. I am thankful to the Editor, the associate editor and two anonymous referees for their thoughtful comments and suggestions. I would also like to thank Alex Gershkov and Benny Moldovanu for their comments. Financial support from the German Science Foundation, Canada is gratefully acknowledged.

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