Marginal Subsidies in Tullock Contests

• Published date: 01 April 2017
• Author: JONATHAN P. THOMAS,ZHEWEI WANG
• DOI: http://doi.org/10.1111/jpet.12203
• Date: 01 April 2017

MARGINAL SUBSIDIES IN TULLOCK CONTESTS

JONATHAN P. THOMAS

University of Edinburgh

ZHEWEI WANG

Shandong University

Abstract

In a general Tullock contest, we examine a situation where a limited re-

source can be used to provide marginal subsidies to either player (weak

or strong), or to increase the prize directly. We show that to maximize

total effort, subsidizing the weak/strong player is preferred when the

contest is sufficiently accurate/inaccurate. This result generalizes to

n-player lottery contests. In a lottery contest (Tullock contest with

r=1), we derive the optimal scheme for a full range of resource: when

the resource is small, it is optimal to only subsidize the weak player;

when it is large, both players should be subsidized simultaneously.

1. Introduction

A contest is a situation in which players compete against each other by making irre-

versible effort, often for a prize or multiple prizes. Many situations in the real world

have been studied as contests or “contest situations.”1In practice, setting prizes has

been considered as the most important and effective way to attract potential contestants

and stimulate competition between participants. As a result, prize allocations have been

studied extensively in the contest literature.2

1Contest situations refer to a variety of interactions in reality such as sports, rent-seeking, litigation,

beauty contests, patent races, research and development (R&D), political competition, or arms races.

2For instance, Clark and Riis (1998), Moldovanu and Sela (2001), Szymanski and Valletti (2005), Fu

and Lu (2009, 2012b), Akerlof and Holden (2012), and Schweinzer and Segev (2012) among others

have studied prize allocation (or some related issues) within different settings.

Jonathan Thomas, School of Economics, University of Edinburgh, 31 Buccleuch Place, Edinburgh,

EH8 9JT, UK (jonathan.thomas@ed.ac.uk). Zhewei Wang, School of Economics, Williamson Centre

for Law, Economics, and Organization, Shandong University, 27 Shanda Nanlu, Jinan, China 250100

(zheweiwang@sdu.edu.cn).

We are very grateful to the associate editor and two anonymous reviewers for a number of construc-

tive comments and suggestions, which have considerably improved the paper. We also thank Marco

Faravelli, Qiang Fu, Jeong-Yoo Kim, Zhiyong Liu, Jingfeng Lu, James Mirrlees, Dieter Schmidtchen,

Donald Wittman, and the audience at the 8th Annual Meeting of Asian Law and Economics Associa-

tion (Jinan) and the 7th Biennial Conference of the HKEA (Hong Kong) for comments. Wang grate-

fully acknowledges financial support from Project 71501112 supported by NSFC, and from Qilu Young

Scholars and Zhongying Young Scholars of Shandong University.All remaining errors are our own.

Received June 12, 2016; Accepted June 13, 2016.

C2016 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 19 (2), 2017, pp. 511–526.

511

512 Journal of Public Economic Theory

In addition to awarding prizes, subsidizing contestants can also be a good way to

induce effort. This has not drawn much attention in the literature, although in practice

subsidies are often observed in contests or contest situations. By estimating an econo-

metric model using contractor-level data, Lichtenberg (1990) shows that the U.S. De-

partment of Defense (DoD) encourages private military R&D investment not only by

establishing prizes, but also by subsidizing expenditures (costs of making effort) dedi-

cated toward winning the prize.3,4 He concludes, “On the surface, it appears that the

marginal subsidy on the R&D investment is zero, but this is only true in the short term.

Due to the DoD’s policy of allowable-cost determination, the long-run marginal subsi-

dies are substantial.”

Similarly, within large firms in sectors where product innovation is of importance,

there may be two or more teams (or individuals) working independently on the same

project or task (e.g., designing next-generation products). The best-performing team

will be rewarded with a prize, such as a bonus or opportunity for promotion. At the

same time, the firm would typically provide resources to reduce the costs of the teams

in carrying out their tasks. The question we address is whether it is better for the firm

to provide such a marginal subsidy, or better to make the prize larger. Other possible

applications include education. For instance, in a class where students exert effort to

achieve higher degree classifications, it is common for a teacher to offer marginal help

to a student or a specific group of students: the teacher will offer more help when a

student exerts more effort.5

The contest designer may face a budget constraint on the resource that can be

used as subsidies. For instance, the DoD or the firm may have a fixed amount of money

available for providing subsidies, the teacher may have a fixed amount of time for tutor-

ing her students. Following convention in the contest theory literature where a (fixed)

prize is often assumed to have no intrinsic value to the contest designer, we assume that

the (limited) resource also has no intrinsic value to the contest designer.6Then the

problem for the contest designer is as follows: How can the limited resource be used

most efficiently to maximize total effort? For instance, the DoD, which has the objective

of encouraging military R&D in some specific field, has to decide which contractor to

subsidize, the “underdog” (the weak firm) or the “favorite” (the strong firm); similarly,

the firm has to decide which team to subsidize, the strong team or the weak team; the

teacher, who wants to improve the overall academic performance of her students, may

have to decide whether the helpdesk is mainly for helping the less able or more able

students. Moreover, if feasible, would adding the resource directly to the prize be more

efficient than providing subsidies? For example, should the DoD (firm) use the money

to subsidize a contractor (team) or add the money directly to the prize? This paper is

an attempt to answer the above questions.

3Lichtenberg (1988) shows that the DoD has conventionally sponsored numerous design competitions

to stimulate private investment in defense technology. R&D contests sponsored by the DoD remain

common. For instance, in 2007, the DoD set a prize of 1 million dollars to lessen the weight of more

than 20 pounds of batteries a soldier carries on a typical four-day mission.

4This and other examples have been discussed in Fu, Lu, and Lu (2012).

5For instance, suppose a teacher runs a helpdesk in order to help some students. A student who does

little homework (i.e., makes little effort) will gain little from the helpdesk (i.e., gets little help); while a

student who is well prepared (i.e., makes a large effort) will benefit a lot from it (i.e., gets much help).

6While this might often seem unrealistic, focusing on the question of utilizing a fixed resource most

efficiently can be regarded as the first stage in a two-stage process: at the second stage, not analyzed

here, using the cost function (of raising the resource) and the revenue function (of effort) derived

from the results we establish, the optimal amount of the resource can be determined.