In the present paper a model of competition between sports clubs in a sports league is presented. Clubs are endowed with initial players but at a cost clubs are able to sell their initial players and buy new players. The results are that: if the quality of players is one-dimensional, then equilibria in pure strategies exist, and; if the quality of players is multi-dimensional, then there need not exist equilibria in pure strategies, but equilibria in mixed strategies exist. Equilibria in mixed strategies resemblance signings on deadline day in european soccer.competition between sports clubs; dimension of quality of players; equilibrium in pure strategies; equilibrium in mixed strategies

Bodil Olai Hansen

Mich Tvede

English

Research Papers in Economics

Discussion Papers 
Department of Economics 
University of Copenhagen 
 
 
 
No. 07-31 
 
 
Competition in Soccer Leagues 
 
 
Bodil Olai Hansen 
Mich Tvede 
 
 
 
 
 
 
 
Studiestræde 6, DK-1455 Copenhagen K., Denmark 
Tel.: +45 35 32 30 82 – Fax: +45 35 32 30 00 
http://www.econ.ku.dk 
 
 
ISSN: 1601-2461 (online) 
 
 Competition in Soccer Leagues
Bodil Olai Hansen
Copenhagen Business School
Mich Tvede
HEC School of Management
University of Copenhagen ∗
Abstract
In the present paper a model of competition between sports clubs
in a sports league is presented. Clubs are endowed with initial
players but at a cost clubs are able to sell their initial players and
buy new players. The results are that: if the quality of players
is one-dimensional, then equilibria in pure strategies exist, and;
if the quality of players is multi-dimensional, then there need not
exist equilibria in pure strategies, but equilibria in mixed strate-
gies exist. Equilibria in mixed strategies resemblance signings on
deadline day in european soccer.
Keywords: competition between sports clubs, dimension of quality of play-
ers, equilibrium in pure strategies, equilibrium in mixed strategies.
JEL-classiﬁcation: C72, D21, L83.
∗Department of Economics, Studiestraede 6, DK-1455 Copenhagen K, Denmark; Tel:
+45 35 32 30 92; Fax: 45 35 32 30 85; mich.tvede@econ.ku.dk.
11 Introduction
Competition between sports clubs in a sports league has been studied for
ﬁfty years at least and many papers have pointed out several peculiarities of
sports leagues. A peculiarity of sport clubs in a sports league is that there are
production externalities between clubs as pointed out by El-Hodiri & Quirk
(1971), Neale (1964) and Rottenberg (1956). Indeed a game takes two clubs
and the quality of a game depends on the quality as well as the tactic of
both clubs, but also the competitive balance in the league is of importance.
Therefore without regulation the outcome of competition between sports
clubs need not eﬀecient.
Sport clubs have many players, for soccer clubs a team consists of eleven
players and ﬁve substitutes, but typically clubs have more players. We
stoically assume that a club has a single player to make the analysis sim-
ple. Sport clubs seem to have diﬀerent objectives: proﬁt( L i v e r p o o la n d
Manchester United in UK and FCK in Denmark), utility of the owner (Chelsea
in UK and AC Milan and Inter Milan in Italy) and welfare of majority of club
members (Barcelona and Real Madrid in Spain) just to mention a few pos-
sible objectives. We assume that clubs maximize their proﬁts, but revenues
may be interpreted as utilities of owners, welfare of majorities of members
or something else.
In the present paper a model of competition between sport clubs in a
league is presented. The outcome of the competition depends on the distri-
bution of players between clubs, so the strategic variable for a club is the
quality of its player and the performance of a club depends on the quality
of its player as well as the quality of the players in the other clubs. Clubs
are endowed with initial players, but there is a market for players, so clubs
are able to sell their initial players and buy new players. Since there are
externalities between clubs the decision to go on the market or not depends
on the decisions of the other clubs. The cost of an initial player consists of
a salary while the cost of selling an initial player and buying a new player
consists of a transaction cost and a salary to the new player. Therefore there
2is a build-in discontinuity in the cost function of a club at the quality of the
initial player.
In the paper we show that if the quality of players is one-dimensional and
the revenues of clubs have increasing diﬀerences, so the change in revenue
for a club of going from a player of low quality to a player of high quality is
increasing in the quality of the players of the other clubs, then there exists
a Nash equilibrium in pure strategies (Theorem 1). Depending on the initial
players, no clubs, some clubs or all clubs sell their initial players and buy new
players. More realistically, if the quality of players is multi-dimensional, then
there need not exist an equilibrium in pure strategies. However we show that
if the quality of players is multi-dimensional, then there exist equilibria in
mixed strategies and strategies in these equilibria are rather simple as every
club mixes over their initial players and one other player (Theorem 2 and
Corollary 1). For a club a mixed strategy corresponds to trading players just
before the transfer window closes so other clubs do not know whether the
club keeps its initial player or signs a new player and therefore cannot react
to the actual action of the club.
The paper is organized as follows: in Section 2 the model is introduced;
in Section 3 existence of Nash equilibria in pure strategies is studied, and;
in Section 4 existence and structure of Nash equilibria in mixed strategies is
studied.
2 Setup and assumptions
There is a ﬁnite number n of clubs j ∈ N = {1,...,n} a n de a c hc l u bh a so n e
player. Clubs compete so their proﬁts depend on the quality level of their
own players as well as the players of the other clubs. Players are described
by a quality vector q ∈ Rm
+, where each coordinate corresponds to some
capability, skill or talent.
Outside the league there is a market for players where clubs can sell and
buy players. Club j is characterized by its initial player ωj ∈ Rm
+ where
3ωj  = 0, its revenue function rj :( Rm
+)n → R+ which depends on its own
player as well as the players of the other clubs, and its cost function cj :
Rm
+ × Rm
+ → R+ which depends on its own initial player and its new player.
T h ec o s to fan e wp l a y e ro fq u a l i t yqj consists of a cost of having the player
(salary) and a transaction cost of selling the initial player and buying the
new player (transfer fee).
Let sj(qj) be the cost of having a player of quality qj and let tj(qj,ωj)b e
the transaction cost of selling a player of quality ωj and buying a player of
quality qj, then the cost function takes the following form:
cj(qj,ωj)=
⎧
⎨
⎩
sj(qj)f o r qj = ωj
sj(qj)+tj(qj,ωj)f o r qj  = ωj.
The cost function has a build-in discontinuity at qj = ωj because of the
transaction cost.
The following assumptions are supposed to be satisﬁed
(A.1) rj is continuous, bounded from above and strictly concave in qj.
(A.2) sj is continuous, convex and monotone.
(A.3) tj is continuous, convex and monotone in qj.
(A.4) tj is strictly positive on the strictly positive part of the diagonal, so
tj(qj,ωj) > 0f o ra l l( qj,ωj) such that qj = ωj and qj,ωj > 0.
(A.5) sj + tj is unbounded, so lim qj →∞ sj(qj)+tj(qj,ωj)=∞.
(A.1)-(A.3) are quite natural. (A.4) implies that there are transaction costs
because the cost of selling a player and buying a player of identical quality is
strictly positive. (A.5) implies that costs increases without bound as quality
increases without bound.
43 Equilibrium in pure strategies
Let Πj(qj,q −j)=rj(qj,q −j) − cj(qj,ωj) be the proﬁto fc l u bj.T h e n t h e
problem of club j is to choose a player in order to maximize its proﬁtg i v e n
the players of the other clubs q−j =( q1,...,q j−1,q j+1,...,q n)
max
qj
Πj(qj,q −j)=rj(qj,q −j) − cj(qj,ωj) (1)
Deﬁnition 1 A collection of players q∗ =( q∗
1,...,q∗
n),w h e r eq∗
j ∈ Rm
+ and
q∗
j is a solution to Problem (1) given q∗
−j, is an equilibrium in pure strategies.
Suppose that the quality of players is 1-dimensional, so the quality of a
player is described by a number (or alternatively only co-linear qualities of
players are available) and revenue functions has increasing diﬀerences. Then
an equilibrium in pure strategies exists
Theorem 1 Suppose that the quality of players is 1-dimensional, so m =1 .
If rj(qj,q −j) has increasing diﬀerences in (qj,q −j) (so if qj ≥ q 
j and q−j ≥
q 
−j, then rj(qj,q −j) − rj(qj,q 
−j) ≥ rj(q 
j,q −j) − rj(q 
j,q 
−j)), then there exists
an equilibrium in pure strategies.
Proof: Firstly it is shown that the game (N,(Sj,Πj)j)w h e r eSj = R+ and
Πj : Rn
+ → R is the proﬁt, is a supermodular game. Secondly Theorem
4.2.1. in Topkis (1998) on existence of equilibrium in pure strategies in
s u p e r m o d u l a rg a m e si sa p p l i e d .
T h eg a m e( N,(Sj,Πj)j)i ss u p e r m o d u l a r : 1 . Sj is a lattice; 2. Πj is
supermodular in qj, because all functions from R to R are supermodular,
and; 3. q−j ≥ q 
−j implies that Πj(qj,q −j) − Πj(qj,q 
−j)i sn o n - d e c r e a s i n gi n
qj,b e c a u s eq−j ≥ q 
−j implies that rj(qj,q −j) − rj(qj,q 
−j)i sn o n - d e c r e a s i n g
in qj by assumption. Moreover, there exists ¯ q ∈ R+,s u c ht h a tf o ra l lj and
q−j,i fqj is a solution to Problem (1), then qj ≤ ¯ q because rj is bounded
from above and sj + tj is unbounded. Therefore the set of equilibria in the
game (N,(Sj,Πj)j) and set of equilibria in the game (N,(qj,Πj)j)w h e r e
5qj =[ 0 , ¯ q], coincide. According to Theorem 4.2.1. in Topkis (1998), the
game (N,(Sj,Πj)j) has an equilibrium in pure strategies.
2
Let πj(qj,q −j)=rj(qj,q −j) − sj(qj) − tj(qj,ωj)b et h ep r o ﬁto fc l u bj
g i v e nt h et r a n s a c t i o nc o s th a st ob ep a id even if the club keeps its initial
player. Then the artiﬁcial problem of club j is
max
qj
πj(qj,q −j)=rj(qj,q −j) − sj(qj) − tj(qj,ωj). (2)
Problem (2) has a solution because the revenue is bounded from above and
the cost is unbounded and the solution is unique because the proﬁti ss t r i c t l y
concave. Therefore let the map Fj : R
n−1
+ → R+ be the solution map, so
Fj(q−j) is the solution to Problem (2) given q−j,t h e nFj is continuous ac-
cording to Berge’s Maximum Theorem and Fj is bounded from above because
the revenue is bounded from above and the cost is unbounded.
Clearly for all q−j ∈ R
n−1
+ t h e r ee x i s tap a i ro ff u n c t i o n sF L
j ,FU
j : R
n−1
+ →
R+,s u c ht h a ti fωj <F L
j (q−j)o rωj >F U
j (q−j), then Fj(q−j)i st h es o l u t i o n
to Problem (1), and if FL
j (q−j) < ωj <F U
j (q−j), then ωj is the solution to
Problem (1).
Suppose that πj(qj,q −j)=rj(qj,q −j) − sj(qj) − tj(qj,ωj) is twice diﬀer-
entiable and diﬀerentiable strictly concave, so D2
qjqjπj(qj,q −j) < 0, then if qj
and qk where k  = j are strategic complementarities for all k,s oD2
qjqkπj(qj,q −j) >
0f o ra l lk  = j,t h e nπj satisﬁes increasing diﬀerences. Indeed according to
the Implicit Function Theorem
DqkFj(q−j)=−
D2
qjqkπj(qj,q −j)
D2
qjqjπj(qj,q −j)
so if πj is twice diﬀerentiable strictly concave and diﬀerentiable strictly
concave and qj and qk are strategic complementarities, then Fj is strictly
monotone as illustrated in the ﬁrst diagram in Figure 1. However if qj and
qk are not strategic complementarities, then Fj need not be monotone as
illustrated in the second diagram in Figure 1.
6q1
q2
Increasing diﬀerences
r
ω   ¡
¡
¡
μ
r ω  -
r
ω
F U
1 F1 FL
1
F U
2
F2
F L
2
q1
q2
Not increasing diﬀerences
r
ω
¢
¢¢
FU
1 F1 FL
1 FU
2
F2
FL
2
-
6
-
6
Figure 1: The importance of increasing diﬀerences.
In the ﬁrst diagram: 1. if the initial players are ω,t h e nω is an equilib-
rium, because the clubs’ initial players are suﬃciently close to the optimal
players; 2. if the initial players are ω , then in equilibrium club 1 trade play-
ers, while club 2 keeps its player, and; 3. if the initial players are ω  ,t h e n
in equilibrium both clubs trade players, because both clubs’ initial players
are too far away from the optimal players. In the second diagram: if the
initial players are ω, then: 1. ω is not an equilibrium; 2. if only one club
trades players, then the other club also wants to trade players, and; 3. if
both clubs trade players, then club 1 regrets that it traded players. There-
fore in the second diagram where the proﬁt function of club 1 does not have
increasing diﬀerences, if the initial players are ω, then there does not exist
an equilibrium in pure strategies.
4 Equilibrium in mixed strategies
Unfortunately Theorem 1 does not generalize to multi-dimensional players,
because if qj is multi-dimensional, then the function Πj is not supermodular:
For qj,q  
j ∈ Rm
+,l e tqj∨q 
j (qj∧q 
j) be their join (meet), so the i’th coordinate
of qj ∨ q 
j (qj ∧ q 
j) is the maximum (minimum) of the i’th coordinate of qj
7and the i’th coordinate of q 
j. Πj is supermodular in qj if and only if
Πj(qj,q −j)+Πj(q
 
j,q −j) ≤ Πj(qj ∨ q
 
j,q −j)+Πj(qj ∧ q
 
j,q −j).
Suppose that qj = ωj and that q 
j converges to qj such that qj∨q 
j,q j∧q 
j  = qj
1.
Then Πj(qj,q −j)=Πj(ωj,q −j) > πj(ωj,q −j)a n dΠj(q 
j,q −j), Πj(qj ∨ q 
j,q −j)
and Πj(qj ∧ q 
j,q −j) converges to πj(ωj,q −j), so Πj is not supermodular in
qj. Note that for one-dimensional players qj ∨q 
j =m a x {qj,q 
j} and qj ∧q 
j =
min{qj,q 
j},s oqj ∨ q 
j = qj and qj ∧ q 
j = q 
j or qj ∨ q 
j = q 
j and qj ∧ q 
j = qj.
Therefore it is impossible that both qj ∨q 
j  = qj and qj ∧q 
j  = qj are satisﬁed
for one-dimensional players.
Since an equilibrium in pure strategies need not exist for multi-dimensional
players, let P be the set of probability measures on Rm
+ and let μ ∈ P be de-
noted a random player. The problem of club j is to choose a (random) player
in order to maximize its expected proﬁt given the random players of the other
clubs μ−j =( μ1,...,μj−1,μj+1,...,μn)
max
μj
 
qj
  
q−j
Πj(qj,q −j)dμ−j(q−j)
 
dμj(qj)( 3 )
Deﬁnition 2 A collection of random players μ∗ =( μ∗
1,...,μ∗
n) where μ∗
j ∈
P, such that μ∗
j is a solution to Problem (3) given μ∗
−j is an equilibrium in
mixed strategies.
Theorem 2 There exists an equilibrium in mixed strategies.
Proof: For all μ−j ∈ Pn−1, there exists a solution to Problem (3). Let
Γ : Pn → Pn be deﬁned by μ 
j ∈ Γj(μ) if and only if μ 
j is a solution to
Problem (3) given μ−j,t h e nΓ is convex valued and upper hemi-continuous
according to Berge’s Maximum Theorem. Moreover, there exists ¯ q ∈ Rnm
+ ,
such that for all μ if μ  ∈ Γ(μ), then μ ([0, ¯ q]) = 1. Therefore there exists μ∗
such that μ∗ ∈ Γ(μ∗) according to Kakutani’s Fixed Point Theorem.
1If qj = ωj =( 1 ,1) and q 
j =( 1+ε,1 − ε)w h e r eε > 0, then qj ∨ q 
j =( 1+ε,1) and
qj ∧ q 
j =( 1 ,1 − ε).
82
The random player version of the artiﬁcial problem of club j is to maxi-
mize its proﬁt given the transaction cost has to be paid even if the club keeps
its initial player
max
qj
πj(qj,μ−j), (4)
Problem (4) has a unique solution. Therefore let the map Fj : Pn−1 → Rm
+ be
the solution map, so Fj(μ−j) is the solution to Problem (4) given μ−j,t h e n
Fj is continuous according to Berge’s Maximum Theorem and Fj is bounded
from above according to (A.2).
Corollary 1 Suppose that μ∗ is an equilibrium in mixed strategies. Then for
all j
μ
∗
j(ωj)+μ
∗
j(Fj(μ
∗
−j)) = 1.
Proof: Clearly for all μ−j Problem (3) has either one pure solution: ωj or
Fj(μ−j), or two pure solutions: ωj and Fj(μ−j). Let the correspondence
Gj : Pn−1 → Rm
+ be the solution correspondence, so qj ∈ Gj(μ−j)i fa n do n l y
if qj is a solution to Problem (3).
Let the correspondence Γ : Pn → Pn be deﬁned by μ 
j ∈ Γj(μ)i fa n do n l y
if μ 
j has support on Gj(μ−j), then μ 
j ∈ Γ(μ)i fa n do n l yi fμ 
j is a solution
to Problem (3) given μ−j. Therefore there exists μ∗ such that μ∗ ∈ Γ(μ∗)
according to Theorem 2 and for all μ 
j in Γ(μ)
μ
 
j(ωj)+μ
 
j(Fj(μ−j)) = 1
by construction.
2
According to Corollary 1 if a club uses a mixed strategy then it mixes
between keeping its initial player and selling its initial player and buying the
optimal player given the transaction cost has to be paid even if the club keeps
its initial player.
9Signings well before the transfer window closes resemblance pure strate-
gies as other clubs are able to react to the signings. In the real signings
just before the transfer window closes world resemblance mixed strategies as
other clubs are not able to react because they do not know about the sign-
ings before after the transfer window closes. Deadline day is usually one the
busiest days in the the transfer window: it attracts a lot of attention from
t h em e d i aa n ds o m ev e r yb i gm o v e sh a v eo c c u r r e do nt h i sd a y .
In the present paper every club is a ﬁrm and there are production ex-
ternalities between ﬁrms. Therefore the outcome should not be expected
to be eﬃcient. Indeed generally if μ∗ =( μ∗
1,...,μ∗
n) is an equilibrium col-
lection of random players, then there exists another collection of players
q∗ =( q∗
1,...,q∗
n)s u c ht h a tt h ea g g r e g a t ep r o ﬁti sl a r g e r
 
i(π(q∗)−ci(q∗
i)) >
   
i(πi(q) − ci(qi)) dμ∗(q).
References
El-Hodiri, M., & J. Quirk, An economic model of a professional sports league,
Journal of Political Economy 79 (1971), 1302-19.
Neale, W., The peculiar economics of professional sports: a contribution to
the theory of the ﬁrm in sporting competition and market competition,
Quarterly Journal of Economics 78 (1964), 1-14.
Rottenberg, S., The base ball players’ labor market, Journal of Political
Economy 64 (1956), 242-258.
Topkis, R., Supermodularity and Complementarity, Princeton University Press
(1998).
10

An economic model of a professional sports league,

Supermodularity and Complementarity,

The base ball players’ labor market,

The peculiar economics of professional sports: a contribution to the theory of the ﬁrm in sporting competition and market competition,

Competition in Soccer Leagues

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Competition in Soccer Leagues

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