Strategic budgets in sequential elimination contests

• Author: Gagan Ghosh,Steven Stong
• DOI: http://doi.org/10.1111/jpet.12292
• Date: 01 August 2018
• Published date: 01 August 2018

Received: 1 September 2017 Accepted: 16 February2018

DOI: 10.1111/jpet.12292

ARTICLE

Strategic budgets in sequential elimination

contests

Gagan Ghosh1Steven Stong2

1CaliforniaState University Fullerton

2Universityof Iowa

GaganGhosh, Department of Economics, Mihaylo

Collegeof Business and Economics, California

StateUniversity Fullerton, 800 N. State College,

Blvd.SGMH-3313, Fullerton, CA 92834, USA

(gghosh@fullerton.edu).

StevenStong, Department of Economics, Univer-

sityof Iowa, 312 W PBB, Iowa City, IA 52246, USA

(ststong1.0@gmail.com).

We study a sequential elimination contest where contestants (e.g.,

campaigns) spend resources that are provided by strategic players

called backers (e.g., donors). In the unique symmetric equilibrium,

backersinitially provide small budgets, increasing their contributions

only if their contestant wins the preliminary round. If backers are

only allowed to provide budgets at the start of the game, as opposed

to before each round, spending is higher. When unspent resources

are refunded to the backer, total spending is higher than when all

resources are sunk costs. We also study an extension of the model

with asymmetric players.

1INTRODUCTION

Beginning with the seminal contribution of Tullock(1980), contests are often used to model competitions where play-

ers spend resources in order to win prizes. This framework is also used to model games such as tournaments where

players haveto compete with others sequentially, with the winner moving on to the next round. Such tournaments are

called elimination contests with a leading examplebeing political contests that take place over multiple rounds. In most

political competitions, from council elections at local municipalities to presidential elections, candidates have to win in

preliminary stages to enter a final race. In any political competition, campaigns compete with each other by spending

resources to increase their chances of winning. In many cases, the campaigns are funded by donors who are interested

in their candidate winning, but who also care about how much they spend. This suggests that there are strategic inter-

actions that take place between donors and campaigns as well as between rival donors and rival campaigns during

sequential contests. In this paper, our aim is to investigatesome of these interactions that play out through campaign

contributions.

To capture these interactions, we developa model of sequential elimination contests with the following feature:

campaigns control the timing and amount of spending while they depend on donors to providethe funds. By separating

spendingdecisions from financing ones, ouraim is to investigate the strategic nature of contributions. We believe this is

important, as donors are not passiveplayers and thus choose their contributions strategically, both in terms of amounts

andtiming. Therefore, modeling fundsreceived by campaigns as endogenous objects, as opposed to an exogenous ones,

couldhelp us to understand the reasons behind certain patterns of campaign donations, such as timing of contributions.

Furthermore, from a policy standpoint, this model allows us to study the effect of campaign finance rules on donors, as

they will react strategically to anypolicy change in the model.

Specifically,we focus on an elimination contest with two groups of contestants. The preliminary contests are within

groups. The winners of each preliminary round compete for a prize in a final round. Contestants in the elimination

Journal of Public Economic Theory.2018;20:499–524. wileyonlinelibrary.com/journal/jpet c

2018 Wiley Periodicals,Inc. 499

500 GHOSH AND STONG

contest depend on a set of strategic players, called backers, to provide the resources that will be spent in the con-

test. Each contestant's objective is to spend resources to maximize the probability of winning the final contest taking

into account the other players'strategies.Contestants'spending is constrained by the resources they receive. Backers

would like to increase the chance of their contestant winning, but the probability of winning is weighed against the

cost of providing resources. This formulation highlights the different incentives to spendof the two sets of players. We

derive the unique symmetric equilibrium of such a model under a variety of spending rules (Propositions 2.1 and 2.2).

We show that backers'spending is lower when they can provide funds at different stages of the game, as opposed

to providing everything up front (Section 2.4). Therefore, backers prefer to give contributions in a piecemeal fashion,

as opposed to all at once. This is in line with the empirical observation that candidates typically raise more money in

the general election than in the primaries. This result also provides us with a policy proposal. Spending will be lower if

a sufficient amount of time is provided between stages for contestants to raise more donations.

Wealso study the effect of a rule governing how unspent resources are used. Unspent budgets can occur in the elim-

ination contest when a contestant loses the preliminary contest and has some budget remaining. In political contests,

the way unspent campaign funds are used after the campaign is often regulated. Although donations can be returned

to donors, more often theyare spent in a way that is an imperfect substitute for a refund. For example, the funds may be

donated to a charity,another campaign, or a political party. We show that in an elimination contest in which resources

are only provided at the beginning of the game, spending increases with the fraction of unspent resources that are

returned to backers.1

Finally, we carry out an extensionof the baseline mode to include the possibility of asymmetric players. We do so

by including an incumbent in the model, who is assumed to be uncontested in the preliminary round. We also allow the

incumbent to havedifferent valuations than the other players called challengers. We compare rent dissipation and costs

between the baseline model and the extension and show that some of the results reverse. In our framework, we also

show that an incumbent is more likely to win the final round conditional on not being too weak, that is, not having too

low a valuation compared to the challengers.

This feature of the equilibrium suggests that strategic behavior of donors could be one of the sources of the incum-

bency advantage, an empirical observation that essentially states that incumbents tend to win more often.2Levitt and

Wolfram(1997) show that the ability of incumbents to deter high-quality challengers seems to account for increases in

incumbency advantage for U.S. House races from 1948 to 1990. This is precisely what occurs in our framework: even

moderately weak incumbents havea higher likelihood of winning against strong challengers. In the model, challengers

have inadequate time to raise funds between the primary and general election. Due to this, the challenger's backers

haveto commit to contribution levels before the preliminary round. In contrast, the incumbent's backercan react opti-

mally to the budget the challenger brings to the final round, giving her an advantage. This depresses the contribution

levelsfor the challengers and allows even moderately weak incumbents to have an advantage over strong challengers.3

The rest of the paper is organized as follows. In the next subsection, we discuss our paper in the context of related

works. We present the model in Section 2. In Sections 2.1 and 2.3, we discuss the unique equilibrium of the model

under two different spending rules and the associated costs and rents. In Section 3, we introduce a variation to the

baseline model where we allow players to be asymmetric. Section 4 concludes the article. All proofs are presented in

the appendix.

1.1 Previous literature

Sequential elimination contest models are a natural extension of their one shot contest counterparts and have been

studied as far back as Rosen (1986) in the context of labor promotions and sport tournaments. His work grew

1Ifthe backers are allowed to provide resources at each stage, there are no unspent resources in equilibrium.

2SeeGelman and King (1990), Cox and Katz (1996), and Ansolabehere and Snyder (2002).

3Being an incumbent is not always an advantage. We establish conditions under which a (very)weak incumbent faces a disadvantage in equilibrium. Under

someconditions, the incumbent is completely deterred from competing.