The Impact of Revenue Sharing on Club Objectives in
Professional Sports
Shumei Hirai ∗Graduate School of Economics
Chuo University

Yasuo Kawashima †Faculty of Economics
Chuo University

August 10, 2009

Abstract
This paper presents an n-club sporting contest model in which each club consistsof an owner and its manager. It is demonstrated that sports clubs behave as if they are
maximizing a utility function which is a linear combination of proﬁts and wins, while
keeping the proﬁt maximizing objective at the level of owners. It is also demonstrated
that revenue sharing arrangements by the leagues play an essential role in formulating the club objectives. The results imply that club objectives and revenue sharing
arrangements are endogenous.

Acknowledgement
Financial support from Chuo University, the Japan Ministry of Education, Culture, Sports,
Science and Technology (Grand-in-Aid for Scientiﬁc Research, 20604001) and Hakumon
Scholarship is highly appreciated. The earlier version of this paper was presented at theAnnual Meeting of Japanese Association for Applied Economics, Kanazawa University,
Japan, 22-23 November 2008. The authors are grateful to Professor Junich Watanabe of
Fukuoka University for his comments and suggestions. We would like to thank Professor
Toichiro Asada, Hiroshige Tanaka and Akio Matsumoto of Chuo University, Professor
Masakazu Fukuzumi of Tsukuba University for their kind advice. All remaining errors are
our own.
JEL Classiﬁcation: D72, L83, D43, C72
Keywords: club objective; revenue sharing; strategic delegation∗†e-mail: l3112013@grad.tamacc.chuo-u.ac.jp
e-mail: yasuokaw@tamacc.chuo-u.ac.jp1

1

Introduction

The purpose of this paper is to show that the objectives of sports clubs can be explainedby revenue sharing arrangements which a league uses. We present an n-club sporting
contest model which each club consists of an owner and its manager, which is closely
related to the strategic delegation literature (Fershtman and Judd 1987; Sklibas, 1987):
Proﬁt maximizing owners set goals for managers to pursue, and managers compete forthe given goals. The present model will ﬁrst show that sports clubs will behave as if
they are maximizing a utility function which is a linear combination of proﬁts and wins
(Sloane, 1971; Rascher, 1997). Then, we will show that as the league intensiﬁes revenue
sharing, club objectives become more proﬁt oriented; on the other hand, if the league lacks
a redistribution mechanism, club objectives are more win oriented. This result is quiteconsistent with the relation between existing empirical observations of club objectives
described below and a system of revenue sharing in real-world sports leagues.
Since pioneering works of Rottenberg (1956) and Sloane (1971), the general perception is that professional sports clubs in the U.S. traditional leagues, such as Major League
Baseball (MLB) and the National Football League (NFL), are more commercially oriented
and have become more businesslike companies than in European football. Some empiricalevidence from the U.S. traditional leagues conﬁrms the hypothesis that clubs’ main objective is proﬁt maximization (Ferguson et al., 1991; Alexander, 2001).1 On the other hand,
it has been widely held that European football clubs are primarily interested in sporting
objectives, such as win maximization subject to a budget constraint; see Késenne (1996,
2000). However, Andreﬀ and Staudohar (2000) argue that European football clubs are
evolving toward the American sports clubs’ ﬁnancial structure because of recent revenue
growth, especially from broadcasting, and the conversion of many clubs to joint stockcompanies. This might be expected to lead to a great emphasis on proﬁts and thus behavior more akin to the North American model (Hoehn and Szymanski, 1999; Fort, 2000).
Yet, Garcia-del-Barrio and Szymanski (2009) empirically ﬁnd that behavior of clubs in
English and Spanish leagues over the period 1994-2004 seem to closely approximate win
maximization subject to a ﬁnancial constraint. They suggest that football club executives
may ﬁnd it in their best interest to maximize wins.
This paper is motivated by these observations above: Although, in the past few years,
ﬁnancial models in European professional sports came closer to the U.S. professional sports
model, club objectives in Europe and the U.S. leagues are diﬀerent. Our investigation
demonstrates that objectives of clubs may be aﬀected by the organization of sports leagues,
while keeping the proﬁt maximizing objective at the level of owners. In the past, mostowners were individuals; more recently, the form of sports club ownership has become
diversiﬁed, such as corporations and limited partnerships, due to the costs of purchasing
and operating a club. These owners usually invest a sum of money, hoping for a proﬁtable1

However, for several reasons, these results are not entirely convincing. See Zimbalist (2003) for details.

2return, but do not take an active part in the management of the club. Thus, when we say
an “owner”, we mean a decision-maker whose objective is to maximize the proﬁts of the
club. A “manager” refers to an agent that the owner hires to delegate all player personnel
decisions and decides on an investment level in playing talent. What has to be emphasizedis that we deﬁne club objective as an objective function for the manager.
Although there are substantial diﬀerences in the organizational framework of North
American and European sports leagues, the principal diﬀerence that this paper focuses
on is the revenue sharing rules used by the leagues.2 Revenue sharing arrangements are
typically more extensive in the U.S. sports than in European football. In Europe, there
are almost no gate-sharing arrangements. There are no arrangements for the sharing ofincome from merchandising such as exist in the U.S. sports leagues. While collective sellingof media rights exists in most European football leagues, the distribution formulas tend
to favor the strong clubs; this is known as performance-based reward schemes. So, withthese institutional diﬀerences of revenue sharing arrangements in mind, the present paper
shows why professional sports clubs in European football are more win oriented than in
the U.S. sports leagues.
The intuition behind our results can be explained using the club-objective function
which is a linear combination of proﬁts and wins. If revenue sharing arrangements arelimited and/or the league authorities choose a performance-based reward scheme, thenpotential revenue of the club would be closely correlated with the success of the club.
Under such organizational framework of sports leagues, the more weight a club puts on
wins, the more it invests in playing talent; consequently, the club will be more successful
and attractive to fans or broadcasters. Thus, proﬁt maximizing owners give an incentive toand put pressure on managers of clubs to win. On the other hand, if the leagues engage inthe leveling of club revenues via revenue sharing arrangements, then the relation between
improvements in club performance and increases in club revenue is weak; hence, owners
will force managers to strive for proﬁt maximization. As mentioned above, there is a much
lower incidence of revenue sharing in Europe. Then, in view of our results, this meansthat professional sports clubs in European football are more win oriented than in the U.S.
sports leagues.
Our analysis also can shed some light on the eﬀect of revenue sharing policy. In
theory, there seems to be an emerging consensus that the nature of clubs in a league
as proﬁt or win maximizers may aﬀect the success of policies to promote competitivebalance, such as revenue sharing and salary caps (see e.g., Zimbalist, 2003; Késenne, 2007;
Vrooman, 2007).3 However, our results suggest that club objectives may alter after a2

Other diﬀerences are league system (e.g. promotion and relegation), league functions (e.g. centralized
marketing), player market (e.g. rookie draft) and so on.
3
To put it more concretely, revenue sharing has no positive impact on competitive balance in a proﬁt
maximization league where clubs are proﬁt maximizers (e.g., U.S. sports leagues); on the contrary, if all
clubs are win maximizers (e.g., European football leagues), revenue sharing is eﬀective in establishing amore balanced distribution of talent among clubs.

3

change of revenue sharing arrangements. Therefore, league authorities should recognizethat competitive-balance rules may aﬀect not only clubs’ investment level in playing talent
but also their objectives.
The remainder of the paper is organized as follows. Section 2 presents an n-club
sporting contest model. In Section 3, we present our main results and show some importantfeatures of our model. Concluding remarks are given in Section 4.

2

Sporting Contest with Revenue Sharing

We consider a professional sports league. A league is made up of n(n ≥ 2) clubs in which
each club consists of an owner and its manager who are both risk neutral. Our analysis of
the sports league is formulated as a dynamic game with complete and perfect informationand the concept of equilibrium is deﬁned as the subgame perfect equilibrium. At the ﬁrst
stage, each owner i simultaneously has to decide on an incentive scheme for his manager
i. At the second stage, each manager i observes all chosen incentive schemes. After that,the managers simultaneously choose their clubs’ investment level in playing talent.
To simplify our expositions and to proceed with the analysis, we specify the revenue
function of a club as follows:
Ri (si ) = si R,

i = 1, · · · , n,(1)where R > 0 is the total market volume of a given league sport, which is exogenously
determined (e.g., by sales of tickets, TV rights revenues, merchandising, revenues fromadvertising and sponsorship etc.)4 ; si is the share of talent of the club or the relative
quality of the club. We assume that the relative quality of the club is characterized by thecontests success function (CSF) and depends on the proportion of playing talent hired byeach club. We use the typical logit CSF, which is the most widely used functional form ofa CSF in sporting contests.5 The share of talent of the club can be deﬁned as

n
 nti
if
j=1 tj > 0
t
j
j=1
(2)
si (t1 , · · · , tn ) = 1 otherwise, i = 1, · · · , n,nP

where ti ≥ 0 is investment in playing talents of a club. In view of (1) and (2), the revenue
function of a club means that with an increase in a club’s share of playing talent, the clubs
will be more successful and attractive to fans or broadcasters, and thus generate a higher
revenue through the sale of tickets, broadcasting rights, sponsorship money etc.
We now present the simplest revenue sharing arrangement in order to investigate theimpact of interventions by a league on club objectives. In real-world sports leagues, there
4

Exogeneity of R is assumed for simplicity. If the total market size R is endogenously determined,
diﬀerent results may be obtained. However, here it is left to future research to investigate these extensions.
5
See the excellent survey by Szymanski (2003) and the references therein.

4are several arrangements for sharing revenue: Gate revenue sharing, pool revenue sharing,
sharing broadcasting revenues and a complex combination of these.
In theory, it has been recognized that diﬀerent revenue sharing arrangements can havea diﬀerent impact on competitive balance (see e.g. Késenne, 2007). However, apart fromany impact on competitive balance, another consequence of revenue sharing is the levelingof club ﬁnances. So, from this point of view, a model of revenue sharing is formulated asfollows:6
1−α
1 − α
R = R αsi +, 0 < α ≤ 1,
i = 1, · · · , n,(3)
Ri (si ; α) = αsi R +nnwhere α measures the level of revenue sharing chosen by sports leagues. This parameter
gives the share of a club’s revenue that depends on the relative quality of the club. Thenequation (3) shows that club i’s revenue can be decomposed into two components. The
ﬁrst revenue component is the variable part si R. It follows from (1) and (2) that thisterm describes club i’s revenue under the condition of no revenue sharing arrangement(α = 1). The second component is the ﬁxed part ( 1−αn )R. Independent of on-ﬁeld success,this part will always bring in a part of club i’s revenue as long as revenue sharing takes
place (α < 1). Therefore, as α is close to zero, an egalitarian sharing rule is introduced
or intensiﬁed by the league; consequently, the relationship between improvements in therelative quality of the club and increases in club revenue becomes weak.7 Note that theassumption of a positive real number for α reﬂects the fact that no European footballleagues and the U.S. traditional leagues share all revenues.8
On the cost side, in line with most of the existing literature, we assume that talent
can be hired in the players’ labor market at a constant marginal cost c > 0, so that thecost function can be written as:
Ci (ti ) = cti ,i = 1, 2, · · · , n.

(4)Under the conventional hypothesis of a perfectly competitive labor market, clubs are wagetakers, so that the unit cost of talent is the same for every club.
It follows from (2),(3) and (4) that with revenue sharing, the proﬁt function for club6

Typically, revenue sharing as a distribution or redistribution means that revenue that is earned by one
club is given to other clubs. For the purpose of this paper and analytical tractability, here we omit the tax
on clubs’ local-revenue from revenue sharing system.
7
A similar after-sharing revenue function can be found in Dietl et al. (2008). Note that as in Tullock(1980), they have used the logit CSF for n clubs as the probability of winning the championship in orderto model revenue sharing arrangements. However, the logit-form approach is also used to describe market
share so that we have no uncertainty and no winner-take-all situation in our model (see e.g. Schmalensee
1976; Kräkel and Sliwka, 2006).
8
Single-entity leagues, such as Major League Soccer (MLS), the Women’s National Basketball Association (WNBA) and the Women’s United Soccer Association (WUSA), share all revenues and expenses
equally, regardless of the size of the market. In this paper, however, we are not concerned with thesingle-entity league.

5

i is described by
1 − α
ti
1 − α
−cti = R α n−cti ,+
πi (t1 , · · · , tn ; α) = R αsi +nn
j=1 tji = 1, · · · , n. (5)The owner’s objective is to maximize these proﬁts.
Next, in the spirit of Fershtman and Judd (1987), Sklibas(1987) and Rascher (1997), it
is assumed that club i’s manager is motivated to choose ti to maximize a linear combinationof proﬁts and wins, which can be written as:9
Oi (t1 , · · · , tn ; θi , α) = πi + θi wi ,

θi ≥ 0,i = 1, · · · , n.

(6)where θi is the incentive parameter chosen by owner i in order to maximize his proﬁts;
following the existing literature, wi represents the winning percentage for each club andis written as:10
wi (t1 , · · · , tn ) =

t
n
n i
,
2 j=1 tji = 1, · · · , n.

(7)As already mentioned in the Introduction, we deﬁne club objective as an objective function
for the manager in equation (6). Then, it allows clubs to be more proﬁt oriented or more
win oriented because the weight parameter θi can be diﬀerent for every club. Note that
θi = 0 means that owner i forces the manager to maximize proﬁt only.11
As in Fershtman and Judd (1987), the managers’ compensation contracts are publicly
observable and linear in the objective function:
Ui = Ai + Bi Oi ≥ Ū ,where Ai and Bi are constants, and Ui and Ū are the manager’s wage and reservation
utility respectively. It is assumed that this reservation utility is zero so that it will always
pay for an owner to delegate decisions. Note that in equilibrium, the owner will earn thewhole surplus πi as a manager’s participation constraint will always be binding. Hence,in the following we only have to care about the eﬀect of θi on the managers’ choice of ti .
Finally, in the analysis of sports league, it is conventionally assumed that the total
supply of talent is ﬁxed. Authors who have made this assumption have used a non-Nash
∂ti
−1
= n−1
= 0 for all i = j to reﬂect this scarcity in each club’s ﬁrst-orderconjecture ∂tjcondition (Fort and Quirk, 1995; Vrooman, 1995): The loss of playing talent, caused byan increase of playing talent in one club, is spread evenly among the rival clubs. However,9

Fershtman and Judd (1987) and Skilvas (1987) use a combination of proﬁts and revenues. On theother hand, Rascher (1997) assumed that sports clubs are maximizing a linear combination of proﬁts andwins. However, note that in Rascher (1997), an owner and a manager were modeled as a single entity.
10
As long as every club plays the same number of games, it can be shown that the sum of win percentagesof all clubs is not equal to 100% but equals half the number of clubs in the league (see e.g. Fort and Quirk,
1995; Késenne, 2007).
11
We shall take the owners’ delegation decisions as exogenous, though it can be interpreted another way:
There is no delegation at all and the owner also serves as the manager of the club.

6

in view of game theoretic approach and a distinctive feature of European football players’∂tlabor market, Szymanski and Késenne (2004) choose to apply a Nash conjecture ∂tji = 0
in the determination of theoretical sports league equilibrium.12 Indeed, opinion is divided
among sports economists on this subject, but we use the Nash conjecture in this paper.13
The main reason is that we would like to compare the North American and European sports
leagues on the basis of the same theoretical assumptions. Moreover, as Szymanski (2004)
acutely pointed out, as far as modeling the game is concerned, there is no inconsistency
between the use of Nash conjecture and the assumption that supply of talent is ﬁxed . A
game involving a ﬁxed supply is simply a type of zero-sum game, and zero-sum games canhave Nash equilibria.

3

Club Objectives and Revenue Sharing

To solve our two-stage game, we rely on backward induction. Consider ﬁrst the gameplayed among the managers at the second stage given their respective objective functions(6). For simplicity, the following deﬁnitions are useful:
T =


j

tj ,

T−i =



tj ,

j=i

zi = αR +nθi,2where T and T−i are total investments in playing talent within a league and the aggregate
investments in playing talent by all competitors of club i respectively. It follows from (5),(6), (7) and the above deﬁnitions that we can rewrite the objective function for manager
i as


n

t
ti
1 − α
ti
1 − α
n i−cti + θi wi = R α n−cti + θi++
Oi = R α nnn
2 j=1 tj
j=1 tj
j=1 tj

nθi
t
1−α
ti
1−α
n i
R+ αR +
R + zi − cti , i = 1, · · · , n.
− cti =
(8)
=
n
2
n
T
j=1 tjThe objective of each manager is to maximize Oi subject to ti ≥ 0. By using the KuhnTucker conditions for a maximum, subject to inequality constraints, we obtain:

ti = 0 if zi ≤ cT−i ,T−i− c = 0, if zi > cT−i .
zi(ti + T−i )212(9)

A theoretical reason is that if each club makes a single choice, simultaneously and independently of
the choices of others, its rivals will have no opportunity to respond; hence it makes no sense to talk of any
conjectural variation other than zero. Another reason is that after the liberalisation of football players’
labor market by the Bosman verdict (1995), an extra talent in one club does not necessarily imply a lossof talent in another club in the same league.
13
See Szymanski (2004), Eckard’s (2006) comment and Szymanski’s (2006) reply for details.

7

We can solve these equalities for ti to obtain manager i’s reaction function:ti = maxzi
T−i − T−i , 0c

i = 1, · · · , n.

To ﬁnd an equilibrium we must solve this system of n equations. Note that not all
managers invest strictly positive amounts in playing talent. Let H denote the set of active
clubs whose managers invest strictly positive amounts in playing talent. For notational
simplicity, we now use the new deﬁnitions: 1  1Θ≡, Θ−i ≡
.
zi
zj
i∈H

j∈H\{i}For a given vector of incentive parameters (θ1 , · · · , θn ) the following result characterizes
the Nash equilibrium of the second stage.
Lemma 1. A unique Nash equilibrium exists at the second stage, which has the following
properties: There is a subset H ⊆ {1, · · · , n} of all managers who invest positive amounts
in playing talent. The managers contained in this subset are those that have the m = Hhighest θi values. The investment level of each manager in the subset is(m − 1) Θ−i − (m − 2) z1i
.
(10)
ti =
cΘ2
The managers with the n − m lowest θi do not spend any resources on playing talent.
For the θi s of the managers contributing positive amounts on playing talent the following
condition holds:
Θ−i >

m−2
.
zi

(11)

All managers with Θ−i ≤ m−2
zi choose ti = 0. The managers with the two highest θi always
contribute a strictly positive amount on playing talent.
Proof. First, we will show uniqueness of the Nash equilibrium at the second stage byfollowing the approach in Cornes and Hartley (2005). Suppose we have an equilibrium(t1 , · · · , tn ). Recall that T ≡ j tj . If ti > 0, we must have from (9) thatzicT 2(T − ti )
−
c
=
0,
⇒
t
=
T
−
.
i
T2
zi

Then we deﬁneti (T ) = max T −

cT 2
, 0 ,
zi

(12)

which is referred to by Cornes and Hartley as the manager i’s replacement function. The
replacement function gives the unique value (possibly an equilibrium) of ti given that the
total investments in playing talent is T . Furthermore, rather than use the replacement8

function directly, it proves convenient to divide both sides of (12) by T > 0. The resulting
functionsi (T ) = max 1 −

cT
, 0 ,
zi

(13)

which is called the manager i’s share function by Cornes and Hartley. Nash equilibriumvalues of T occur where the aggregate share function, i si (T ), equals unity. In our model,
it follows from (13) that this gives T satisfyingnsi (T ) =

i=1

n 


max 1 −i=1

cT
, 0 = 1.
zi

The aggregate share function is monotonically decreasing, and has a value n(> 1) if
T = 0 but tends to 0 if T becomes large. Hence, a unique T and therefore a uniqueequilibrium must exist. In fact, solving this equation for active clubs, Nash equilibrium
total investments in playing talent level yields
T =

m−1
.
cΘ

(14)

Substituting this into active clubs’ replacement function (12), we havec  m − 1 2
cT 2
m−1
−
=
zi
cΘ
zi
cΘ

(m − 1) zi Θ − (n − 1).
czi Θ2ti = T −=Using Θ−i ≡1j∈H\{i} ( zj ) and zi Θ = 1 + zi Θ−i , this can be rearranged asti =(m − 1) Θ−i − (m − 2) z1icΘ2,

which is the Nash equilibrium investment of each manager in the subset H. Note thatthe existence of the equilibrium is guaranteed because the manager’s objective function in(8) satisﬁes the second-order conditions. In fact, it follows from (8) that the second-ordercondition of the ith manager is given by

2T−i
nθi
2T−i
∂ 2 Oi= −zi= − αR +< 0, for θi ≥ 0,
2
3
(ti + T−i )2 (ti + T−i )3∂tiwhere the inequality comes from the assumption that α is a positive real number.
To obtain the condition
for positive investment level for club i just check that ti > 0
is equivalent to(m−1) Θ−i −(m−2) z1
cΘ2

i

Θ−i >

> 0 and this is equivalent to
m−2
2(m − 2)=.
zi
2αR + nθi

If this condition does not hold, the club’s manager will stay out of the players’ labor market
and choose ti = 0. Now we suppose without loss of generality that θ1 ≥ θ2 ≥ · · · ≥ θn ≥ 0
9holds. It follows from the inequality condition above that clubs with higher values of
θi (i ∈ H) will make positive expenditures whereas those with suﬃciently low values will
choose to incur no expenditure.
To see that in equilibrium at least the two managers with the two highest θi s will spend
positive amounts on playing talent, simply note that if the manager with the second highest
θi does not choose a positive ti , the manager with the highest θi can always reduce his own
investment level and be better oﬀ. Hence, there is no equilibrium with only one manager
investing a positive amount in playing talent.
The results of Lemma 1 show that the owners can not only directly inﬂuence the behavior of their managers, but also the behavior of the other clubs’ managers. In particular,a weight on winning percentage has an eﬀect of increasing the investment in talent. In
fact, the derivative of (10) with respect to a change in θi can be written as∂tin(m − 1) m − 2
=
−
mΘ
> 0,
−i
∂θi
zi
2czi2 Θ3(15)

where the inequality sign comes from (11). Thus, delegation opens up to the option to
future actions. However, as will be shown later, whether or not clubs with larger weights
on wins via delegation earn more proﬁt than those with smaller weights is a diﬀerent
question.
Now we can go back to the ﬁrst stage of the game where the owners simultaneously
choose incentive schemes for their respective managers. In view of (5), owner i’s proﬁts
are given byti
πi = R α  n

j=1 tj+ t
1 − α
1 − α
i−cti = R α +−cti ,nTni = 1, · · · , n.

where the Nash equilibrium level (t1 , · · · , tn ) of talent investments are determined by the
results of Lemma 1 for given incentive parameters (θ1 , · · · , θn ). Each owner i chooses θi tomaximize his proﬁts by taking into account the incentive parameters set by his competitors
and the resulting Nash equilibrium in the second stage. Indeed, in the following result weshow that a symmetric subgame perfect equilibrium exists. Note that an asterisk denotes
the equilibrium outcome.
Lemma 2. For n ≥ 2, a unique subgame perfect equilibrium exists which is symmetric.
At the ﬁrst stage, each owner chooses an incentive variable:θ∗ =2(n − 2)αR.
(n2 − 2n + 2)nAt the second stage, each manager invests t∗ in playing talent:t∗ =(n − 1)2 αR.
(n2 − 2n + 2)cn

10

Each owner’s equilibrium proﬁts from this investment are given by 2(n − 2n + 2) − α(n2 − 2n + 1) R.
π∗ =(n2 − 2n + 2)nProof. The proof proceeds as follows: In the ﬁrst step, we analyze necessary conditions
for the existence of a subgame perfect equilibrium. After this, we show existence of thecharacterized equilibrium in step 2.
Step 1: Necessary conditions
Suppose there is some equilibrium with m active clubs in the set H such that m managers
choose ti > 0 according to (10). In view of (5), (10) and (14), owner i’s proﬁts are given
by




(m − 1) Θ−i − (m − 2) z1i 
1
cΘ
1−α
1−α
R+ αR − c ti =
R+
−
c
αR
πi =
n
T
n
cΘ2
m−1


1
Θ−i − (m − 2) zi αRΘ − (m − 1)
1−α
=
R+
.
(16)
n
Θ2
Each owner i is to maximize proﬁts (16) with incentive parameters. It follows from (16)that
Θ−i − (m − 2) z1i αRΘ − (m − 1) ∂ 1 − α
∂πi
R+
=
= 0,
∂θi
∂θi
n
Θ2
which can be rearranged so that we obtain
θi (αRΘ − m) =4(1 − m) 2αR(1 + m − αRΘ)
+
.
nΘ
n

(17)

Then, we encounter two cases depending upon whether (αRΘ − m) is a zero or not:m
. Then, we get from
1. First, suppose that αRΘ − m = 0 which is equivalent to Θ = αR2(2−m)αR= 0 which can only hold if m = 2, i.e., there are only two active(17) thatnm
clubs. Hence, in this case we have a continuum of equilibria as follows:

−

αRθj
2
+ αR(3 − αRΘ) = 0 ⇒ θi = −,Θ2θj + αRi, j = 1, 2; i = j.

However, since θi (i = 1, 2) is restricted to being non-negative in this paper, we musthave θ1 = θ2 = 0 in this case.
2. Second, we must consider the case whereαRΘ − m = 0.

(18)

Note that (17) here directly implies that the incentive parameter θi must be thesame for all active clubs. Hence
θi = θ =(1 − m + αRΘ)(2 − αRΘ).
(αRΘ − m)nΘ11m
As Θ = αR+nθ we can conclude that2
2 (m − 2)αR − nΘ 2αR + (1 − m)nΘ2(m − 2)αR∗⇒ θ = θm
=
θ=
2
2
m n θ
2 + (m − 2)m n

when m clubs are active. It is easy to check that (18) indeed then holds for m > 2.
Hence, if there is any pure-strategy perfect equilibrium when m > 2 then it will be∗ .
the one where m active clubs all choose θm
We now have to show that there is no equilibrium in which less than n clubs areactive. To do this, ﬁrst suppose that there is an equilibrium with m(< n) active clubs.
In that case, there must be at least one club j with tj = 0 which can only be the case if
condition (11) is violated for j, i.e., Θ−j ≤ m−2
zj . Note that Θ−j = Θ as club j is inactive.

Furthermore, in view of (5), this club j earns proﬁts πj = (1−α)Rin such an equilibrium.
n
Again we have to diﬀerentiate two cases. If m > 2 then all active clubs in equilibrium∗∗ and we must have that θ ≤ (m−2)(αR+θm ) − αR for club j to be inactive.
choose θmjm∗ . Note that any manager
Suppose now that, j deviates and chooses a small value θj = θmof a “previously” inactive club k ∈/ H ∪ {j} will stay inactive after such a deviation. In∗ . The investment levelthat case, stage 2 is reached with m + 1 clubs which have chosen θmin playing talent of the m + 1 active managers can then be computed from (10) as


(m−1)
m
−
m
∗
∗
nθ
nθ
∗ )
m(2αR + nθm
αR+ 2mαR+ 2m=.
t=22c(1 + m)2
m+1
c
∗
nθm
αR+

2

Then, in view of (5), each active club’s proﬁts are t
1 − α
1
1 − α
−ct = R α+−ct
π = R α +
T
n
1+m
n
∗
(1 − α)R 2αR − mnθm+.
=n2(1 + m)2
2

∗ < 2(1−α)(1+m) R + 2αR . It follows from θ ∗ above that this
These proﬁts are positive if θmmmn
mn2
will be the case if

2(1 − α)(1 + m)2 R 2αR
2(m − 2)αR
+
<
mn2
mn
2 + (m − 2)m n
 2 2⇔ (1 − α) m (m − 1) + 2m + 2 +2nα > 0,

which is always the case. Therefore, the diﬀerence between each active club’s proﬁts andthe inactive club j’s proﬁts is
π − πj =∗∗2αR − mnθm
(1 − α)R 2αR − mnθm
(1 − α)R
+
=
−
,
2
2
n
2(1 + m)n2(1 + m)which will be positive if
∗
<
θm

2αR
.
mn
12

∗ =  2(m−2)αR , this will be the case ifAs θm2+(m−2)m n2(m − 2)αR
2αR

<
mn
2 + (m − 2)m n
4αR
⇔
>0
2
(m − 2m + 2)mn
which is always the case, as the left-hand side is always positive. If only m(< n) clubs areactive and m > 2, then it is always beneﬁcial for an inactive club to deviate to a smaller
value of θ and get into the players’ labor market.
2
under the conditionIf, however, m = 2 and n > 2 we know already that Θ = αR
θi (i = 1, 2) = 0. If there is an inactive club k, we must have that Θ = Θ−k ≤ m−2
zk which
is equivalent to Θ ≤ 0, yielding a contradiction.
To sum up, if n > 2, at most a unique subgame perfect equilibrium in pure strategies
exists in which all owners choose2(n − 2)αR.
θ ∗ = θn∗ = 2(n − 2n + 2)nNote that for n = 2 the value for the incentive parameter θ corresponds to θ ∗ given above.
Step 2: Suﬃcient conditions for existence
To prove that this equilibrium indeed exists, we need consider only two cases, namelyn > 2 and n = 2.
1. For n > 2, we can proceed analogously to the computation of (17) to ensure thatequilibrium investments (θ1∗ , · · · , θn∗ ) in playing talent satisfy the second-order conditions. It follows from (17) that the suﬃcient second-order conditions are∂Θ 
4(1 − n) 2α2 R2 
∂ 2 πi, i = 1, · · · , n.
=
αRΘ
−
n
+
+
+
αRθ
i
∂θi
nΘ2
n
∂θi2
∂Θ
are exSince the equilibrium investment level for club i is symmetric, Θ and ∂θ
ipressed as 1  
∂Θ
2n2
1
n
,
=
−
,
≡
=
Θ≡
∗
zi
∂θi
(2αR + nθ ∗ )2
αR + nθ2
αR + nθ2 iiifrom which the second-order conditions reduce ton2 − 2n + 2∂ 2 πi=−2n∂θiand are obviously negative.
2. If n = 2, in view of (17), αRΘ − m = 0 and m = n = 2, the second-order conditions
are expressed as∂ 2∂ 2 πi
2
−
3αR
+
(αR)
=
Θ
∂θi Θ
∂θi2
=

2θi2 θj2 − 2α4 R4 + 4αRθi θj (θi + θj ) + α2 R2 (θi2 + 6θi θj + θj2 ), i, j = 1, 2; i = j.
(αR + θi )2 (2αR + θi + θj )2
13

αRθ

j
Substituting θi = − 2θj +αR
into equation above, the second-order conditions reduceto∂ 2 πi(2θj + αR)2=−
< 0.
2(θj + αR)2∂θi2Therefore, the suﬃcient second-order conditions are satisﬁed.
As the next step, we can apply the incentive parameter θ ∗ for all values of n(≥ 2)
in the derivation of the equilibrium proﬁts and investment level for each club. First, it
follows from θ ∗ and (10) that the equilibrium investment level for each club is given by(m − 1) Θ∗−i − (m − 2) z1∗(n − 1)2 αR
i
.
=
t∗ =
c(Θ∗ )2(n2 − 2n + 2)cn
Finally, it follows from θ ∗ and (16) that ∗
Θ−i − (m − 2) z1∗ αRΘ∗ − (m − 1)
1
−
α
i
π∗ =
R+
n
(Θ∗ )2 2(n − 2n + 2) − α(n2 − 2n + 1) R,=n(n2 − 2n + 2)
which is positive for α, R, and n within their permissible ranges.
It follows from Lemma 2 that the number of clubs plays an essential role in the determination of the optimal choice of incentive parameters by the owners. It follows from theproperties of θ ∗ that when n > 2 the equilibrium incentive variable is positive, while whenn = 2 it is equal to zero. Thus, each owner puts more weight on winning percentage rather
than proﬁt in sporting contests with more than two clubs, while in two-club contests eachowner forces the manager to maximize proﬁt only. The logic behind this result is roughly
as follows: When n = 2 there may be no advantage in precommitment, while when n > 2
there is an advantage. In view of (15), delegating decisions to managers in a strategic
situation may be useful for owners as a kind of self-commitment device. Therefore, if
each club’s objective is to maximize proﬁt only, then no owners gain from the opportunity
for precommitment to set non-proﬁt objectives, such as the winning percentage, for their
managers.
We now turn to the formal statement of these arguments about precommitment in
our model. The following lemma is a particular case of a result in Dixit (1987) whereasymmetric contests are also allowed.
Lemma 3. In symmetric n-player sporting contests with payoﬀ function (5), if one club
can commit to an observable action before other clubs,
1. When n = 2, there is no strategic advantage in precommitment to aﬀect futurecompetition.

14

2. When n > 2, there is a strategic advantage in overcommitment to aﬀect futurecompetition.
Proof. Without loss of generality, suppose that club 1 can commit to an action t1 , observable by all other clubs who then choose their action simultaneously. Then, given t1 ,club i(i = 2, · · · , n) wants to maximize its proﬁts (5). The ﬁrst-order condition for thisrequirement is simply∂πi (t1 , · · · , tn )
= 0,
∂ti

i = 2, · · · , n.

(19)

We can use these equations to derive the reaction function of club i, ri (t1 ). Then club 1’s
calculation of the rate of change of proﬁt with respect to precommitment is∂π1  ∂π1 dr2∂π1 drn dπ1=++ ··· +
(20)
dt1
∂t1
∂t2 ∂t1∂tn dt1
At the equilibrium, reaction functions ri (t1 ) for i = 1 are given implicitly by the ﬁrst-orderconditions (19). Hence, to ﬁnd the right-hand side of equation of (20), diﬀerentiate theﬁrst-order conditions (19) with respect to t1 to obtain∂ 2 πi dr2∂ 2 πi drn∂ 2 πi++ ··· += 0,∂ti ∂t1 ∂ti ∂t2 dt1∂ti ∂tn dt1

i = 2, · · · , n.

(21)

At the symmetric interior Nash equilibrium (tn1 , tn2 · · · , tnn ), it follows from (5) that thederivatives of the proﬁt function are:∂ 2 πi∂t2i∂ 2 πi∂ti ∂tj=−2αRT−i−2(n − 1)αR=,3Tn3 (tn )2

i = 2, · · · , n=

T − 2T−i(2 − n)αR
αR =
,
3
T
n3 (tn )2

i, j = 1, · · · , n; i = j,Using equations above, and summing up the n − 1 equations (21) gives∂ 2 πi  dr2drn ∂ 2 πi  ∂ 2 πi
+
+
(n
−
2)
+
·
·
·
+
=0
(n − 1)∂ti ∂tj∂ti ∂tj
dt1
dt1
∂t2i
 dr
drn 
(n − 1)(n − 2)(n − 1)∂ 2 πi /∂ti ∂tj
2+ ··· +
=− 2
=− 2
⇒
2
2
dt1
dt1
n − 2n + 2∂ πi /∂ti + (n − 2)∂ πi /∂ti ∂tj
∂π1
−αR
1
Moreover, since at symmetric Nash equilibrium ∂π∂ti = n2 tn < 0 for all i = 1, and ∂t1 = 0,
equation (20) evaluated at the symmetric Nash equilibrium (tn1 , tn2 , · · · , tnn ) is given by∂π1  dr2
drn  αR  (n − 1)(n − 2) dπ1=
+ ··· +
= 2 n
dt1 ti =tn∂ti dt1
dt1
n t
n2 − 2n + 2

It then follows from this that if n = 2, the right-hand side is zero at symmetric Nashequilibrium. Thus, in this case, neither club gains from the opportunity for club to precommit itself. On the other hand, if n > 2, the right-hand side is positive. Therefore,
the symmetric Nash equilibrium cannot be a part of subgame perfect equilibrium whenclub 1 can commit to an action. That is, there is strategic advantage in commitment tooverexertion when n > 2.
This completes the proof.
15

It follows from (15) and Lemma 3 that with the exception of the two-club case the clubi with positive weights on wins earns more proﬁt than the one with proﬁt maximization
objective provided that all clubs but i are proﬁt maximizers. However, as shown in Lemma
2, when all clubs have equal opportunities to commit themselves to managerial contracts,
each owner puts a positive weight on winning percentage in sporting contests with morethan two clubs; consequently, it leads to more competition and lower proﬁts. In fact, inview of (10), if each club’s objective is to maximize proﬁt only (i.e., θi = 0 for all i), theinvestment level in playing talent of each club is given byt̂ =(n − 1)αR.
cn2

Substituting t̂ into (5), the proﬁts of each club are given byn − α(n − 1) R
.
π̂ =
n2
Therefore, it follows from Lemma 2 that the diﬀerence between proﬁt π̂ and proﬁt π ∗ is(n − 1)(n − 2)αR
> 0,
π̂ − π ∗ = 
2 + (n − 2)n n2

for n > 2.

Hence, if all clubs jointly adopt the proﬁt maximizing objective, then the clubs would bebetter oﬀ. However, in view of (15) and Lemma 3, when n > 2 the proﬁt maximization
objective is not incentive compatible from each club’s point of view.
Thus, we can summarize these arguments as follows.
Proposition 1. Suppose that the league is made up of n(n ≥ 2) clubs in which each clubconsists of an owner and its manager who are both risk neutral. Then the proﬁt maximizing
owners set sporting objectives other than proﬁts, such as the winning percentage, for their
managers in the league with more than two clubs, while in the two-club case the proﬁt
maximizing owners set the proﬁt maximization objective for their managers.
It follows from Proposition 1 that sports clubs are maximizing a linear combination of
proﬁts and wins in order to maximize clubs’ proﬁts for n > 2; on the other hand for n = 2,
they maximize proﬁt only. Thus, our discussion of this proposition shows that sports clubs
will behave as utility maximizers, while keeping the proﬁt maximizing objective at the levelof owners. Put diﬀerently, owners may ﬁnd that the best way to maximize proﬁt is to set
sporting objectives at the club level. To give a further argument about the characteristicof club objectives, we concentrate on not two clubs but n(> 2) clubs in what follows.
In the theoretical literature that addresses the performance of sporting leagues andthe impact of diﬀerent regulations, most studies start from a model with each club’s
hypothesized mode of behavior being constant: Clubs are supposed to maximize either
their proﬁts, winning percentage or a linear combination of proﬁts and wins. However, it
follows from Lemma 2 that each club’s optimal mode of behavior will depend on revenue
sharing arrangements, α, chosen by the league, because varying the revenue sharing system16

inﬂuences the objectives of sports clubs. In view of (3), an increase in α means that revenue
sharing arrangements are limited and/or the league authorities choose a performancebased reward scheme, such as sharing broadcasting revenues, in the league. In contrast,
as α decreases, the leagues engage in the leveling of club revenues via revenue sharing
arrangements.
From the optimal incentive parameters given by Lemma 2, it is easily shown that∂θ∗∂α > 0 for n > 2. Therefore, as α increases, owners give an incentive to and put pressure
on managers of clubs to win; hence club objectives will be more win oriented. On theother hand, reducing α via revenue sharing leads clubs to be more proﬁt oriented. The
intuition behind this result is the following. If the leagues lack a redistribution mechanismthat might lead to equalization of club ﬁnances, club’s playing success and club’s revenues
are much more closely related. Then, owners may ﬁnd that the best way to maximize
proﬁts is to win at the club level. In contrast, as a league engages in revenue sharing, therelationship between improvements in club performance and increases in club revenue is
weak; consequently, club owners do not have direct ﬁnancial incentive to win.
Thus, we can summarize these arguments as follows:
Proposition 2. Suppose that the league is made up of n(n > 2) clubs in which eachclub consists of an owner and its manager who are both risk neutral. Revenue sharing
arrangements used by the leagues have the impact on club objectives or club manager’s
incentives as follows:
1. As the league intensiﬁes the leveling of club ﬁnances via revenue sharing, club owners
will force managers to strive for proﬁts in order to maximize proﬁts.
2. As the league strengthens the correlation between the success of clubs and club revenues by weakening the revenue sharing and/or implementing performance-based
reward schemes, club owners will force managers to strive for wins in order to maximize proﬁts.
It follows from Proposition 2 that revenue sharing arrangements by the leagues play
an essential role in the club objectives. As already mentioned in the Introduction, revenue
sharing arrangements are typically more extensive in the U.S. sports than in European
football. For example, broadcasting revenues typically include a performance-related element and a ﬁxed share in Europe. In contrast, in the U.S. traditional leagues, national
TV rights are evenly split among the clubs in the leagues without regards to the performance of particular clubs. Therefore, in view of Proposition 2, professional sports clubs
in European football are more win oriented than in the U.S. sports leagues. This result isquite consistent with empirical observations by Garcua-del-Barrio and Szymanski (2009);
that is, behavior of European football clubs seem to closely approximate win maximization. Moreover, this proposition shows why European football clubs are win oriented, eventhough recent TV revenue growth and the entry of corporations into the sports business17

might be expected to lead to a great emphasis on proﬁts. It is reasonable to conjecturethat objectives of a club may be aﬀected by the organization of sports leagues, while
keeping the proﬁt maximizing objective at the level of owners.
It is interesting to note that, in view of Lemma 2, as the league strengthens the correlation between the success of clubs and club revenues by removing the arrangements forrevenue sharing and/or implementing performance-based reward schemes, the clubs have∗a strong incentive for higher talent investments, that is, ∂t∂α > 0; on the contrary, the∗equilibrium proﬁts for each club decline, that is, ∂π∂α < 0. This result explains why European football clubs still struggle to balance the books, despite the inﬂux of broadcasting
income.
It follows from these arguments that our results seem to have validated the conventional
sports clubs’ behavioral assumptions: Professional sports clubs in the U.S. major leagues
are more proﬁt oriented than in European football. However, we should note that the useof revenue sharing schemes by the league can aﬀect the objectives of sports clubs. Thus,
if the league changes revenue sharing arrangements, then club objectives may also alter.

4

Conclusions

This paper is intented as a preliminary step in the analysis of the impact of diﬀerences in
the organization of sports leagues on objectives of professional sports clubs. It presents an
n-club sporting contest model in which each club consists of an owner and its manager.
We have shown that sports clubs behave as if they are maximizing a utility function which
is a linear combination of proﬁts and wins, while keeping the proﬁt maximizing objectiveat the level of owners. It is also demonstrated that revenue sharing arrangements by
the leagues may aﬀect the club objectives. If a league engages in revenue sharing (e.g.
U.S. sports leagues), then club objectives are more proﬁt oriented. On the other hand,
if a league lacks a redistribution mechanism (e.g. European football leagues), then club
objectives are more win oriented even if club owners are proﬁt maximizers. These results
seem to be quite consistent with existing empirical observations of club objectives.
Moreover, our analysis can shed some light on the eﬀects of revenue sharing policy.
The relation between revenue sharing and competitive balance has been the subject ofcontroversy in the economics of team sports. In theory, it has been recognized that club
objectives may aﬀect the success of revenue sharing policy to promote competitive balance.
However, in view of our results, league authorities should recognize that revenue sharing
arrangements also have an inﬂuence on club objectives.
Further research may take the following directions. First, our model has assumed
that given the separation of management from ownership in professional sports clubs, theproﬁt maximizing owners set goals for managers to pursue, and its managers pursue thegiven goal by choosing their clubs’ investment level in playing talent. These enable us tosimplify our analysis and to derive interesting results. As mentioned in the Introduction,18

proﬁt maximization hypothesis will be plausible as a ﬁrst approximation of the objectiveof club owner in recent years. However, the ownership of football clubs is more complex.
For example, Real Madrid in Spain is mutually owned by 50, 000 members, most of themordinary supporters who elect a president to run the club. Under this structure of ownership, it may be reasonable to suppose that regardless of the revenue sharing scheme, thecommittee of a club is willing to sacriﬁce proﬁt for the prestige of winning. Hence, ﬁrstly
we would like to extend our analysis in the presence of this diﬀerent type of ownership
structure of clubs. Second, the present model does not consider the diﬀerence in the sizeof the market in which a club operates. When we try to extend our model to include
the diﬀerence of the size of the market among the big clubs and the small clubs, diﬀerent
results may be obtained. Although this extension will be of interest, an analysis of thisproblem will form the basis of our future research.

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21