Endogenous Group Formation in Contests: Unobservable Sharing Rules

• DOI: http://doi.org/10.1111/jems.12150
• Published date: 01 April 2016
• Date: 01 April 2016

Endogenous Group Formation in Contests:

Unobservable Sharing Rules

KYUNG HWAN BAIK

Department of Economics

Sungkyunkwan University

Seoul 03063, South Korea

khbaik@skku.edu

We study contests in which players compete by expending irreversible effort to win a prize, the

prize is awarded to one of the players, the winner shares the prize with other players in his group,

if any, and each group’s sharing rule is unobservable to the other groups and the singletons, if

any, when the players expend their effort. The number of groups, their sizes, and the number of

singletons are exogenous in the first model, whereas they are endogenous in the second model.

We show that group formation occurs if the number of players is four or smaller, but does not

occur otherwise. We examine the effect of endogenous group formation on total effort level and the

profitability of endogenous group formation. In each of the two models, comparing the outcomes

of the case of unobservable sharing rules with those of the case of observable sharing rules, we

show that the two cases yield quite different outcomes.

1. Introduction

Contests are common in which players, each as a member of a group or as a singleton

(or, equivalently, a nonmember), compete with one another by expending irreversible

effort or resources to win a prize.1Examples include various types of rent-seeking

contests, patent competition among consortiums, competition among political parties

or candidates for office, class action litigation, and sporting contests. In some contests,

group formation may occur because players want to better cope with uncertainties

including the possibility of failure in winning the prize. In others, group formation may

occur because players try to gain strategic advantage through group formation.

In such contests, the players in each group jointly decide how to share the prize

among themselves if they or one of them wins it—that is, they make a binding agreement

on their sharing rule. One example for such an agreement comes from patent compe-

tition, where the firms in each R&D joint venture decide how to share the benefits of

I am grateful to Eric Bahel, Mike Baye, Francis Bloch, Hans Haller, Hanjoon Michael Jung, Amoz Kats, Amy

Lee, Dongryul Lee, WooyoungLim, Hyeran Moon, Eric Rasmusen, Michael Rauh, Nathan Smith, Nic Tideman,

two anonymous referees, and seminar participants at the Hong Kong University of Science and Technology,

VirginiaPolytechnic Institute and State University, WesternIllinois University,and Indiana University for their

helpful comments and suggestions. I am also grateful to Daehong Min and Chang-Geun Song for excellent

research assistance. Earlier versions of this paper werepresented at the 2012 Annual Conference of the Korean

Econometric Society,Seoul, Korea, February 2012; and the 2013 Annual Meetings of the Allied Social Science

Associations, San Diego, CA, January 2013. This work was supported by the National Research Foundation

of Korea (NRF) Grant funded by the Korean Government (MEST) (No. 2012S1A5A2A01017808).

1. The literature on the theory of contests is enormous and growing. Important work in this literature

includes Tullock (1980), Rosen (1986), Dixit (1987), Hillman and Riley (1989), Katz et al. (1990), Nitzan

(1991b), Baye et al. (1993), Nitzan (1994), Skaperdas (1996), Clark and Riis (1998), Hurley and Shogren (1998),

Moldovanu and Sela (2001), Hvide (2002), Che and Gale (2003), Szymanski (2003), Corch´

on (2007), Epstein

and Nitzan (2007), Congleton et al. (2008), Konrad (2009), and Siegel (2010).

C2015 Wiley Periodicals, Inc.

Journal of Economics & Management Strategy, Volume25, Number 2, Summer 2016, 400–419

Group Formation in Contests 401

the innovation if they or one of them wins the patent. Another example comes from

competition for presidential office, where the candidates in each political party may

decide, before they campaign, how they will share the office benefits if one of them wins

the election. Yet another example comes from college football, where the teams in each

conference decide how to share or divide the prize among themselves if one of them

wins a bowl game.

Often, each group’s sharing rule is unobservable to the other groups and the

singletons. For example, in patent competition, the firms in each R&D joint venture

and the independent firms may not know the sharing rules of the other R&D joint

ventures when they compete to win the patent. In competition for presidential office, the

candidates in one political party and the independent candidates may not be sure, when

they campaign, how the candidates in another party will share office benefits, policy

influence, and the political posts among themselves in case of one of them winning

the election. This situation of unobservable sharing rules may occur simply because the

groups do not announce their sharing rules. Furthermore, it may occur, even though

they announce publicly their sharing rules, because their announced sharing rules are

unverifiable. Nitzan and Ueda (2011) say that, without restrictive assumptions that

decisions made within a group are transparent and detection of changes is easy, a model

of group contests with observable sharing rules is questionable.

In addition, in such contests, we may well expect that the number of groups, their

sizes, and the number of singletons are endogenously determined.

Accordingly, it is those ideas, unobservable sharing rules and endogenous group

formation, that motivate this paper. This paper studies contests in which risk-neutral

players compete by expending irreversible effort to win a prize, the prize is awarded to

one of the players, the winner shares the prize with other players in his group, if any,

and each group’s sharing rule is unobservable to the other groups and the singletons,

if any, when the players expend their effort. It considers two models: the model with

exogenous groups and the endogenous group formation model.

In the model with exogenous groups, the number of groups, their sizes, and the

number of singletons are exogenous. We formally consider the following game. First,

the players in each group jointly decide how to share the prize among themselves if

one of them wins it—that is, they make a binding agreement on their sharing rule (or,

equivalently, their winner’s fractional share). Next, all the players in the contest choose

their effort levels simultaneously and independently. When the players choose their

effort levels, each group’s sharing rule is unobservable to the other groups and the

singletons. Finally, the winning player is determined, and the players in his group, if

any, share the prize according to their sharing rule on which they agreed.

We show that each group’sequilibrium effort level and the equilibrium total effort

level are independent of the number of players in the contest and the sizes of the groups,

as long as changes in these do not change the number of groups or the number of

singletons. We show also that each player in a larger group expends less effort, and has

a less equilibrium expected payoff, than each player in a smaller one. Finally, we show

that the equilibrium expected payoff for each singleton is greater than that for each

player in any actual group.

We compare the outcomes of the case of unobservable sharing rules—the game

presented above—with those of the case of observable sharing rules, the game which is

the same as the one presented above with the exception that each group’s sharing rule

is now observable to all the players in the game. By the situation of “observable sharing

rules,” we mean a situation in which the groups announce publicly their sharing rules