Supporter Influence on Club Governance in a Sports League; a “Utility Maximization” Model

S U P PORT ER I N F LUENCE ON CLUBGOVERNANCE I N A S PORT S L EAGUE;A “U T I L I T Y MAX IM I ZA T I ON” MODEL

Paul Madden* and Terry Robinson**

ABSTRACT

The article formalizes a seminal suggestion of Sloane (1971), studying a sports

league in which club objectives are multi-argument utility functions defined over

profits, win percentages and fan (=supporter) welfare, thus combining the three

objectives that have been addressed separately in previous models. Particular

focus is on the consequences of increasing the utility weight on fan welfare, to

capture the recent increasing supporter involvement in club governance in UK

football. Positive consequences are unambiguously higher attendances, with more

nuanced affects on ticket prices and player expenditure. A normative conse-

quence is that positive profits for club owners indicate social sub-optimality.

I INTRODUCTION

Most existing theoretical analysis of the professional sports league industry

has focused on leagues where the objective of individual clubs1 is either

profit maximization, or maximization of win percentage (equivalent to maxi-

mizing relative team quality) subject to a budget constraint2 The conven-

tional view is that profit maximization may approximate reasonably well

behaviour in the major North American sports leagues where clubs seem lar-

gely to have been run on the lines of businesses in other industries, but win

maximization may be more prevalent in European football (soccer), where

wealthy club owners have seemingly been prepared to forego profit to pro-

duce champion teams. In a recent article, Madden (2012) introduced a

third alternative, namely fan welfare maximization, whereby (again subject

to a budget constraint, and motivated by the observed members’ clubs in

*School of Social Sciences, University of Manchester**Manchester Business School, University of Manchester

1 We use the following terms solely with their sporting meanings; club, team, match,player. However, games refer to their usual meaning in economic models.

2 Coverage of the literature can be found in the major surveys of Fort and Quirk (1995)and Szymanski (2003), the textbooks by Fort (2006) and Sandy et al. (2004), and in thematerials for the increasing numbers of courses on Sports Economics being taught aroundthe world. The book by Kesenne (2007) provides a full account of existing results on bothprofit and win maximization.

Scottish Journal of Political Economy, Vol. 59, No. 4, September 2012© 2012 The Authors. Scottish Journal of Political Economy © 2012 Scottish Economic Society. Published by BlackwellPublishing Ltd, 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350 Main St, Malden, MA, 02148, USA

339

European football) the fans or supporters of a club, who have a particular

allegiance to the club and are the consumers of its products, also have direct

control over club policies. In Madden (2012), positive and normative conse-

quences of fan welfare maximization are compared and contrasted to those

of profit and win maximization, assuming that clubs have ‘pure’ objectives

of either profit, or win, or fan welfare maximization.

In reality, however, one might reasonably expect that club objectives are

more complicated, multi-dimensional objects, ‘utility functions’ to use the

term proposed in one of the early and now much-cited articles in the litera-

ture (Sloane (1971)). The first objective of this article is to provide a novel

(positive and normative) analysis of sports leagues with clubs whose objec-

tives are such utility functions, defined over profits, win percentage and fan

welfare,3 thus combining (with varying weights) the three objectives studied

separately elsewhere, capturing the original Sloane (1971) suggestion, and

generalizing the ‘pure’ analysis of Madden (2012). There is no doubt that

football club objectives do differ nationally and internationally, and have

gradually evolved over time. Our Sloane utility model is then well placed

to address the consequences of such evolution, via comparative static exer-

cises with respect to the utility function weights, the second objective of the

article.

A particular motivation stems from the emerging and growing role for

supporters’ trusts in UK football in the last 15 years (see Michie et al.

(2006)), where these associations of fans of a club are gradually acquiring

increased shareholdings and representation on club boards, and so increased

influence on the club decision processes. Indeed, the role of supporters in

football club governance has recently entered the UK political arena as an

election issue in the 2010 general election and as a focus of the resulting

parliamentary enquiry. To date, direct supporter involvement in club gover-

nance in English football has largely been restricted to lower league levels

(Football Leagues 1 and 2 – tiers 3 and 4 in the hierarchy, below Premier

League and Championship), where the extent of involvement and control is

variable – some clubs are fully owned by supporters trusts, some have a

supporters trust director on the board and some have a non-trivial support-

ers trust shareholding (greater than 10%), as the following table (Source:

Supporters Direct) indicates. Swansea City’s promotion to the Premier Lea-

gue in 2011 means that they are the first club to have meaningful supporter

representation at this level.

The general tendency towards increasing power of supporters in club gover-

nance triggers questions, both positive and normative, about the likely affects,

questions which we aim to address by investigating, in particular, the conse-

quences of increased weight on fan welfare in clubs’ utility functions.

3 Although some authors have used the term utility maximization to refer to the univariatewin maximization objective, the only article that has previously addressed a multi-variateobjective is Rascher (1997), where profits and win percentage were the two arguments. Weadd fan welfare, and argue later that this provides a close approximation to Sloane’s (1971)original suggestion.

340 PAUL MADDEN AND TERRY ROBINSON

Scottish Journal of Political Economy© 2012 The Authors. Scottish Journal of Political Economy © 2012 Scottish Economic Society

We take the simplified, basic framework used by Madden (2012) for pure

club objectives of profit, win or fan welfare maximization and add the utility

maximization analysis. With European football in mind and its relatively

fierce inter-league competition for players, this framework assumes a per-

fectly elastic supply of playing talent to the league, which consists of two

clubs that play each other twice over the season, once at home and once

away in stadiums of given large capacity. Clubs earn revenue from atten-

dance by their fans at their home game and incur the costs of hiring playing

talent, making decisions on ticket prices for entry to their home game and

on their expenditure on playing talent. As suggested above, the eventual

focus on the positive economics side will be on the affects of changing the

weights on the arguments of the utility function on ticket prices, player

expenditures and the resulting match attendances. Normatively, the question

to be answered will be whether these utility weight changes are a good thing

in terms of welfare (aggregate surplus).

Section II sets out the model of a league with utility maximizing clubs, Sec-

tion III analyses individual club decisions in this framework and Section IV

looks at some welfare issues associated with league (Nash) equilibria. Section

V concludes.

II UTILITY MAXIMIZING CLUBS

We present a simple economic model of a league with utility maximizing clubs

in the Sloane (1971) sense; the general framework is similar to that found in

Madden (2012).

Two clubs and their teams comprise the league (but see Remark 1 at the

end of this section); each team plays the other twice over the season, once

at home and once away. Club i = 1, 2 has a stadium where its team plays

Table 1

English football league clubs with supporter involvement

Club name

Owned by

supporters trust

Supporters trust director

on Club’s board

Supporters trust

shareholding

Brentford Yes 2006 Yes 60%

Bournemouth No Yes since 1997 No

Bristol City No Yes since 2006 No

Bury No Yes since 2002 11% since 2002

Carlisle United No Yes since 2003 25% since 2003

Chesterfield No Yes since 2001 No

Exeter Yes 2007 Yes since 2001 Majority since 2007

Lincoln City No Yes since 2002 26% since 2002

Luton Town No Yes since 2003 No

Northampton Town No Yes since 1992 No

Oldham Athletic No Yes since 2004 No

Swansea City No Yes since 2002 17% since 2002

Wycombe Wanderers No Yes since 2005 No

Sheffield Wednesday No Yes since 2005 10% since 2002

SUPPORTER INFLUENCE ON CLUB GOVERNANCE 341


its home match; the stadium has a given capacity, sufficiently large so as to

be never binding on match attendance.4 Clubs hire players and Qi � 0

denotes the expenditure on playing talent by team i. Following the estab-

lished treatment for a European football league, talent is in perfectly elastic

supply at a wage normalized to 1, so Qi is also the quantity of playing tal-

ent (and alternatively a measure of the quality of team i). Player expendi-

tures are the only club costs, abstracting (e.g.) from stadium costs, as is

usual in the literature.

Club i sets the ticket price pi for admission to its home match and

receives all gate revenue from this match; implicitly admission is a homoge-

neous good for fans (offering equal quality match views and spectator facil-

ities – see Remark 2 at the end of this section), and no price

discrimination is possible. There are disjoint sets of fans of each club i,

who feel an (exogenously given) affinity to club i. In a terminology used in

the literature, our fans are core rather than floating fans – a fan of club i

could not switch allegiance to the rival club.5 To simplify, it is assumed

that fans do not attend away matches.6 Fans of i are heterogeneous in

their willingness to pay for tickets, denoted v(Qi, Qj) � x where the hetero-

geneity parameter is x � 0 and v(Qi, Qj) is the maximum willingness to

pay. It is assumed that x is uniformly distributed over [0,1], and li denotesthe number of fans (the ‘fanbase’ of club i), assumed to exceed stadium

capacity (in turn assumed to exceed attendance). It is also assumed that v

(Qi, Qj) is C2 and strictly increasing in both arguments, reflecting the desire

of fans to see better quality matches. As v(Qi, Qj)2 appears in the objective

function of many of the subsequent optimization problems, we assume that

it (and hence v(Qi, Qj) itself) is strictly concave and satisfies the Inada con-

ditions. If v(Qi, Qj) is symmetrical, fans are non-partisan and would divide

a given amount of talent equally between the two teams for their optimal

match. In our context, with core fans in mind with their wish to see their

team win, an asymmetry leading to more talent going to the home team

for a fan’s optimal match is appropriate. This fan bias is captured by the

assumption that v(Qi, Qj) > v(Qj, Qi) if Qi > Qj; in the extreme limiting case

of completely home partisan fans v(Qi, Qj) depends only on Qi, with sym-

metric v(Qi, Qj) and non-partisan fans at the other extreme.7

4 In the context of Football Leagues 1 and 2, this assumption is entirely appropriate,where attendances are almost always significantly below capacity. The average attendance asa percentage of capacity for League One (3rd tier) was 53% for the 2006–2007 season and54% for the 2007–2008 season. The same Figures for League Two (4th tier) are 39% and38% respectively (Deloitte and Touche, 2007–2009).

5 Partisanship is an often assumed characteristic of football supporters. However, there isevidence that certainly not all fans are core in this sense – see Robinson (2012).

6 Given that away fan attendance is always a small fraction of attendance, the assumptionis plausible.

7 The fan utility microfoundation essentially generalizes that of Falconieri et al. (2004),who assume the specific Cobb-Douglas form vðQi;QjÞ ¼ Qa

i Qbj with non-partisan fans for

their TV audience (a = b); for our fan bias, a would exceed b.



A fan with heterogeneity parameter x will demand a ticket if x � v(Qi,

Qj) � pi, so that i’s (linear in price) ticket demand or match attendance

(given large stadium capacity) is Ai(Qi, Qj, pi) = li[v(Qi, Qj) � pi] yielding

revenues8 piAi(Qi, Qj, pi), and profits Πi(Qi, Qj, pi) = piAi(Qi, Qj, pi) � Qi,

which is the first club utility function argument.

Once talent has been hired and tickets sold, matches are played and a win-

ner emerges. Ex ante, before the play of matches, the probability that i is the

winner is some C2 contest success function W(Qi, Qj), increasing and strictly

concave in Qi and decreasing in Qj with values in [0,1], whose exact specifica-

tion is irrelevant for most of our purposes. Following established usage, W

(Qi, Qj) is referred to as the win percentage and is the second component of

the club utility function.

The final club utility function argument is their fan welfare, defined to be

the following aggregate surplus accruing to their fans:

FiðQi;Qj; piÞ ¼Z vðQiQjÞ�pi

0

li½vðQi;QjÞ � pi � x�dx ¼ li½vðQi;QjÞ � pi�2=2:

Notice that fan welfare is a monotone increasing transformation of atten-

dance.

Formalizing Sloane (1971), a club’s utility function is assumed to be a

weighted average of profits, win percentage and fan welfare, where kiX,X = Π, W, F are the non-negative weights:

UiðQi; Qj; piÞ ¼ kiPPiðQi;Qj; piÞ þ kiWWðQi;QjÞ þ kiFFiðQi;Qj; piÞ ð2:1Þ

We thus effectively pick up three of the utility function arguments suggested

originally by Sloane (1971, p. 136). In his terminology, these are playing suc-

cess (equates to win percentage), profits and attendance (equivalent to fan

welfare, as noted above).9 The previous literature has attempted to capture

the Sloane idea less completely than here. Kesenne (2007) and others equate

utility maximization to win maximization, whereas Rascher (1997) uses a

weighted average of just profits and win percentage.

The clubs make independent decisions about ticket prices (pi) and player

expenditure (Qi) to maximize utility subject to a budget constraint, which is

taken for simplicity to be the non-negative profit requirement. So, the decision

problem for a utility maximizing club is:

maxpi;Qi

UiðQi;Qj; piÞ subject to PiðQi;Qj; piÞ� 0 ðUMAXÞ

8 Home gate revenues are the only revenue source in the model – merchandizing andbroadcasting provide relatively little extra in Football Leagues 1 and 2.

9 The fourth argument suggested by Sloane is the health of the league; “utility is derivedfrom the health of the league because it is better to win a keenly fought competition than towin easily” (Sloane 1971, p. 136). We think this is probably already picked up in our specifi-cation, in trade-offs between win percentage and profits.



Before proceeding to the analysis of (UMAX), a couple of remarks about

two of our assumptions are in order.

Remark 1: The assumption of a two-club league is clearly a simplification.

Some generalization is possible. Suppose, there are n clubs with talent

expenditures Q1, Q2, …, Qn and a maximum valuation function v(Qi, Q�i),

where Q�i is the n � 1 vector of Qj0s; j 6¼ i. Suppose also that the ticket

offered at price pi is a season ticket, allowing entry to all n � 1 home games

of club i. Most of the statements and results that follow do immediately

generalize, simply replacing Qj by Q�i. The exceptions are in Section IV;

Assumptions 1 and 2 and the remaining results do draw on the n = 2

restriction.

Remark 2: An interesting topic for further research10 is a model where

clubs offer a variety of tickets to its home match, the tickets being differen-

tiated by allowing entrance to different sections of the stadium, with

differing quality of match views and spectator facilities. Clubs will then be

selling multiple differentiated goods, possibly with both horizontal and

vertical differentiation. Such a situation is indeed the case in reality,

traditionally with fans separated into seated areas and standing areas, and

more recently with hospitality packages bundled with a stadium seat to pro-

vide (perhaps) a higher quality product. Such an extension is a challenge to

the entire literature on sports leagues, where, to the best of our knowledge,

our assumption of clubs providing a single, homogeneous product is

ubiquitous. Questions regarding optimal product differentiation, distribution

of fan welfare between fans selecting the different products, and cross-

subsidization between products occur immediately on what might be a rich

agenda.

III ANALYSIS OF CLUB DECISIONS

We consider the utility maximizing ticket price (pi) and player expenditure

(Qi) of a club with fanbase li (given the pj and Qj chosen by the other team in

the league), and how these (and the resulting match attendance) vary with the

club’s utility function weights. The other team’s price (pj) does not in fact

affect any of the answers (pj has no affect on i’s utility or constraints – each

club sells tickets only to its own fans), so solutions are denoted as piU(Qj),

QiU(Qj), and the resulting attendances are AiU(Qj).

The case where club objectives are ‘pure’ (i.e. with positive weight on only

one argument) was analysed in Madden (2012), as follows: First, the budget

constraint delimiting feasible ticket price and player expenditure choices corre-

sponds to the bubble-shaped region in Figure 4.1, bounded above by high

break-even prices (denoted as piH(Qi, Qj)) along H, below by low break-even

10 We are grateful to a referee for bringing up this point.



prices (piL(Qi, Qj)) along L, with monopoly prices (pM(Qi, Qj)) along M in

between.11

To find optimal ticket prices and player expenditures proceed in two stages,

solving first for optimal prices given both Qj and Qi (the optimal pricing rule),

and then solving for the optimal Qi. For pure objectives, the optimal pricing

rules are obvious – monopoly pricing for the pure profit-maximizer, low

break-even pricing for the pure fan welfare-maximizer, with indifference to all

prices between high and low break-even levels for the pure win-maximizer

(where price does not affect the win percentage objective).

The optimal player expenditure for a win-maximizer is then obviously QiðQjÞat W in Figure 1. Madden (2012) shows that the other optimal player expendi-

tures lead to a point Π in Figure 1 under pure profit maximization, and to F

under pure fan welfare maximization where dotted curves are fan welfare (or

attendance) contours. Unambiguous comparisons are that player expenditure

(or team quality) is highest for a win-maximizer and lowest for a profit-maxi-

mizer; match attendance will be highest for fan welfare-maximizers and lowest

for profit-maximizers; win-maximizers charge the highest ticket prices.

For the more general (UMAX) problem with positive weights on more than

one argument, it is again helpful to proceed in two stages, solving first for

optimal pricing rules (given Qj and Qi), and then solving for the optimal Qi.

For a utility maximizer with positive weights on only win percentage and

profits (i.e. kiW > 0 and kiΠ > 0 but kiF = 0), the optimal pricing rule is again

pi

Π

M W

L

H

∂Πi / ∂Qi = 0

Qi

F

(Qj)Qi

Figure 1. The bubble-shaped non-negative profit region.

11 The bubble boundary corresponds to zero profits, where p2i � pivðQi;QjÞ þQi=li ¼ 0.Solving the quadratic gives formulae for these high and low prices as:

piHðQi;QjÞ ¼ 1

2vðQi;QjÞ þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðQi;QjÞ2 � 4Qi=li

q;

piLðQi;QjÞ ¼ 1

2vðQi;QjÞ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðQi;QjÞ2 � 4Qi=li

q;

with monopoly prices pMðQi;QjÞ ¼ 12vðQi;QjÞ: The roots are real if Qi 2 ½0;QiðQjÞ�, and QiðQjÞ is

the unique positive solution in Qi (given the strict concavity and Inada properties of v) to v(Qi,

Qj)2 = 4Qi/li.



monopoly pricing, along M in Figure 1 – any other feasible choice leaves win

percentage unchanged, but lowers profit. The optimal expenditure will then be

at a point on M between Π (as kiW/kiΠ ? 0) and W (as kiW/kiΠ ? ∞),increasing along M as kiW/kiΠ increases.12

Now, introduce into the objective a small positive weight on fan welfare

also, kiF ∊ (0, kiΠ). There is now a potential benefit to pricing below the

monopoly level, as this will increase attendance, and so fan welfare, to be

traded off against the lower profits. The outcome of the trade-off is illustrated

in Figure 2, denoted piU(Qi, Qj).

What happens is that when player expenditure (Qi) is low, the optimal

price is a fraction of the monopoly price (namely piUðQi;QjÞ ¼2kiP�2kiF2kiP�kiF

pMðQi;QjÞ) producing positive profits, but eventually (for

Qi [ Q̂i

�Qj;

kiFkiP

��this price is less than the low break-even price, at which

point low break-even pricing, and zero profits, takes over. As kiF increases,

the monopoly price fraction�2kiP�2kiF2kiP�kiF

�and the switch point

�Q̂i

�Qj;

kiFkiP

��decrease, until kiF = kiΠ when the switch point reaches zero and universal

low break-even pricing takes over for kiF � kΠ. The following is a formal

statement, proved in the appendix, where (a) relates to positive profit out-

comes and (b) to zero profits:

Lemma 3.1: Assume a given Qi, Qj where Qi 2 ½0;QiðQjÞ�, and assume

kiΠ > 0. Defining Q̂i Qj;kiFkiP

� �by

�2� kiF

kiP

�2

Qi ¼ lið1� kiFkiPÞvðQi;QjÞ2, the utility

maximizing ticket price is given by:

(a) piUðQi;QjÞ ¼ 2kiP�2kiF2kiP�kiF

pMðQi;QjÞ if kiF\kiP and Qi\Q̂i

�Qj;

kiFkiP

�;

(b) piUðQi;QjÞ ¼ piLðQi;QjÞ if kiF � kiP; or if kiF\kiP and Qi � Q̂i

�Qj;

kiFkiP

�:

pi

Qi

piU (Qi ,Qj)

)(Qj ,ˆΠi

iFQi λλ

M

H

L

Figure 2. Typical utility maximizing pricing rule.

12 As the fan welfare/attendance contours in Figure 1 show, one would expect to see fanslobbying the board of such a club (where there is no supporter involvement in governance)for greater expenditure on players, as is often the case. We are grateful to a referee for thispoint.



A natural and intuitive consequence of the argument is that as the weight on

fan welfare in the club objective increases from 0, the ticket price for a match of

given quality (i.e. for given Qi, Qj) will fall from the monopoly level, until the

price reaches its low break-even level. Conversely, as the weight on profits

increases from 0, price will eventually start to rise from the low break-even level,

converging to the monopoly level as the profit weight approaches infinity.

For the second stage Qi solution with kiΠ > 0, substitution of the pricing

rule from Lemma 3.1 into utility (2.1) gives the following, where the top

branch involves positive profits and the lower branch zero profits:

Ui ¼12livðQi;QjÞ2 k2iP

2kiP�kiF� kiPQi þ kiWWðQi;QjÞ; Qi\Q̂i

�Qj;

kiFkiP

�; kiF\kiP

12likiFpiHðQi;QjÞ2 þ kiWWðQi;QjÞ; kiF � kiP orQi � Q̂i

�Qj;

kiFkiP

�; kiF\kiP

8<:

ð3:1ÞA first point about (3.1) is that, despite the piecewise nature of the

definition, it does in fact define Ui as a globally differentiable function of

Qi.13

Secondly, consider the case of a pure profit-maximizer, where

0 = kiF = kiW < kiΠ. As in Madden (2012), let QiΠ(Qj) denote the optimal

player expenditure for the pure profit-maximizer, which [from the top branch

of (3.1)], occurs where marginal revenue�¼ 1

2livðQi;QjÞviðQi;QjÞ�

equals

marginal cost (=1). In relation to this special case, the following helps to iden-

tify optimal player expenditures for general utility functions:

Lemma 3.2: Suppose kiΠ > 0. Then Ui is strictly increasing in Qi if

Qi < QiΠ(Qj), and strictly concave in Qi if Qi � QiΠ(Qj).

Proof: See appendix. h

An immediate comparative static result about the effect of changing utility

function weights on optimal player expenditures follows from the increasing

nature of Ui below QiΠ(Qj) in Lemma 3.2:

Theorem 3.1: Suppose kiΠ > 0. Then for all utility functions QiU(Qj) �QiΠ(Qj).

Thus, the pure profit-maximizing objective leads to lower player expendi-

ture than any other utility function – the team quality produced by pure

profit-maximizers is less than that produced by any other objective, including

the pure fan welfare and pure win-maximizer cases (thus generalizing Madden

13 If kiF � kiΠ the formula in the bottom branch of (3.1) certainly has this global differen-tiability. If kiF < kiΠ, then the definition of Ui switches from the top branch of (3.1) to thebottom branch at Qi ¼ Q̂i

�Qj;

kiFkiP

�. However, after some tedious calculations (details omit-

ted), derivatives of both branches are the same at the switch point, and Ui is globallydifferentiable.



(2012)). It also follows, from the strictly concave nature of Ui above QiΠ(Qj)

in Lemma 3.2, that the optimal player expenditures are characterized by station-

ary points of Ui. Performing the required differentiation and rearranging gives:

Lemma 3.3: Assume a given Qj, and assume kiΠ > 0. Then the utility maxi-

mizing player expenditure is given by:

(a) livðQi;QjÞviðQi;QjÞ=�2� kiF

kiP

�þ kiW

kiP@W@Qi

ðQi;QjÞ ¼ 1 with strictly positive

profits iff kiF\kiP and Qi\Q̂i

�Qj;

kiFkiP

�[with ticket prices given by

Lemma 3.1(a)];

(b) 12kiFkiPli

@p2iH@Q2

i

ðQi;QjÞ þ kiWkiP

@W@Qi

ðQi;QjÞ ¼ 0 with zero profits iff either kiF < kiΠ

and Qi � Q̂i

�Qj;

kiFkiP

�, or kiF � kiΠ [with ticket prices given by Lemma

3.1(b)].

The characterizations of utility maximizing ticket prices (Lemma 3.1) and

player expenditures (Lemma 3.3) allow systematic investigation of local compar-

ative statics relating to changes in utility function weights. How do optimal

ticket prices, player expenditures (and the implied match attendances) change as

the utility function weights change? Answers are now provided. Let piU(kiΠ, kiF,kiW) and QiU(kiΠ, kiF, kiW) now denote utility maximizing ticket prices and player

expenditures when utility function weights are (kiΠ, kiF, kiW) and Qj is given (and

suppressed), and let AiU(kiΠ, kiF, kiW) denote the resulting match attendance.

Theorem 3.2: Suppose kiΠ > 0 and suppose profits are strictly positive at the

utility maximizing choice. Then;

(a) @QiU=@kiP\0; @QiU=@kiF [ 0; @QiU=@kiW [ 0;

(b) @AiU=@kiP\0; @AiU=@kiF [ 0; @AiU=@kiW [ 0;

(c) @piU=@kiW [ 0:


Thus, for a profitable club, increases in the weight on fan welfare or on win

percentage will lead to increased player expenditure and attendances, the exact

reverse of the consequences of an increase in the profit weight. Ticket prices defi-

nitely go up with increases in the win percentage weight, but the effect of changes

in the weights on profits or fan welfare on price is ambiguous. For instance, in

the fan welfare case, the ticket price would definitely fall if there were no changes

in player expenditure, but the increase in player expenditure allows some increase

in price consistent with enhanced fan welfare, and such an increase could be opti-

mal; analogously for the case of changes in the weight on profits.

Theorem 3.3: Suppose kiΠ > 0 and suppose profits are zero at the utility max-

imizing choice. Then;



(a) @QiU=@kiP ¼ 0; @QiU=@kiF\0; @QiU=@kiW [ 0;

(b) @AiU=@kiP ¼ 0; @AiU=@kiF [ 0; @AiU=@kiW\0;

(c) @piU=@kiP ¼ 0; @piU=@kiF\0; @piU=@kiW [ 0:


Hence, if a club is already earning zero profits despite some utility weight

on profits, then the optimum is a corner solution and a small change in the

profits weight will produce no change in behaviour at all. But, increasing the

weight on fan welfare or win percentage does create changes, increased

emphasis on fan welfare leading to lower ticket prices and lower player expen-

diture, but increased attendances again, all reversed for win percentage weight

increases.

Given the supporters trust motivation in the introduction, Figure 3 illus-

trates possible consequences of Theorems 3.2 and 3.3 for the affect of increas-

ing the weight on fan welfare from zero to infinity, with constant kiW, kiΠ > 0.

With a zero weight on fan welfare, the utility maximizing solution for ticket

prices and player expenditure will be at a point like A shown, where profits are

positive. As kiF increases player expenditure increases, and the solution follows

the bold curve from A until profits reach zero at B (prices are shown as

decreasing, but this need not be the case, as noted above). Further increases in

kiF leave profits at zero, and now do definitely reduce price, and player expen-

diture as well, until the pure fan welfare maximizer’s optimum is reached at F.

Throughout – see dashed attendance contours – attendance increases.

To compare these findings with the stylized facts, we present some data

relating to two Football League clubs that have seen an increase in supporter

involvement in club governance over the last decade, and hence are viewed as

having acquired increased weight on fan welfare in the club objective.

Tables 2 and 3 provide data on league position, real wage expenditure and

attendances for Brentford F.C. and Lincoln City F.C for four seasons before

and after the transition to effective supporter involvement.

pi

Qi

A WΠ

H B

MF

L

Figure 3. Utility maximizing price-expenditure locus as the weight on fan welfare increases.



These clubs were chosen because league position was relatively stable

around the period, in which they made the transition to supporter involve-

ment. This enables us to assess the extent to which the wage expenditures and

attendances of these clubs changed over time once supporter welfare became

Table 2

Brentford F.C: league position, real wage expenditure and attendances

Season League position Real wage expenditure (£000s) Average attendance

2001–2002 49 1291 6713

2002–2003 62 918 5759

2003–2004 63 746 5541

2004–2005 50 1057 5477

2005–2006 49 1175 6774

2006–2007 70 1010 5599

2007–2008 84 919 4465

2008–2009 71 1000 5707

2009–2010 57 NA 6141

Pre-SI mean 1174 6095

Post-SI mean 1087 6457

Table 3

Lincoln City F.C: league position, real wage expenditure and attendances

Season League position Real wage expenditure (£000s) Average attendance

1997–1998 73 881 3968

1998–1999 69 1051 4654

1999–2000 84 NA 3405

2000–2001 87 933 3194

2001–2002 91 758 3223

2002–2003 76 734 3923

2003–2004 77 814 4910

2004–2005 76 840 4483

2005–2006 77 861 4739

Pre-SI mean 907 3581

Post-SI mean 809 3981

Notes: League Position is calculated as position in entire league hierarchy (top of premier league = 1),Real wages were deflated using RPI: 1987 = 100.

Table 4

Comparison of admission prices between si teams and non-SI teams in leagues one and two

(2009–2010 prices)

Average ticket price (£)

SI Clubs 19.18

Non-SI clubs 19.74

Source: Clubs’ official websites.



more important. As can be seen from pre- and post-SI mean values, real wage

expenditure fell and average seasonal attendances rose after the weight on

supporter welfare increased.

We also compare the current ticket prices of two groups of Football league

clubs – clubs with supporters involvement (SI Clubs) and some conventionally

owned clubs (Non-SI clubs); the appendix identifies the clubs in each group.

Table 4 displays their average ticket prices for the season 2009–2010.14 It can

be seen that the mean ticket price is lower for SI clubs than for non-SI clubs

for that season, although this was not statistically significant even at the 10%

level.

IV SOME WELFARE ANALYSIS OF LEAGUE EQUILIBRIA

Madden (2012) provided some results on the welfare economics associated

with three pure leagues: the Π-league (two pure profit-maximizers), the F-league

(two pure fan welfare-maximizers) and the W-league (two pure win-maximiz-

ers). Stemming largely from the elementary inefficiency of monopoly pricing

(in ticket sales for the Π-league), the main general result was that the Π-lea-gue (Nash) equilibrium was dominated from the welfare viewpoint (lower

aggregate surplus) by the F-league, with more nuanced results comparing the

F-league and the W-league. Given our current, more general framework, the

natural object of study is a U-League, consisting of two utility maximizing

clubs, with possibly differing utility function weights, for which the results

of Section III provide the information on best responses. Our question is:

do changes in utility function weights affect league equilibria in a desirable

way? In particular, given our supporters trust motivation, do increases in

fan welfare weights increase aggregate surplus? Two results are provided,

and again the driving force behind them is the inefficiency of monopoly pric-

ing. The message from the results, loose at the moment, is that positive

profits in a U-League equilibrium are not a good thing from the aggregate

surplus viewpoint, and some increase in fan welfare utility weights can be

improving.

The relevant aggregate surplus measure is as follows: For each club i = 1, 2

(with j 6¼ i), define the sum of consumer (fan) and producer surplus for that

club to be:

SiðQi;Qj; piÞ ¼Z vðQi;QjÞ�pi

0

li½vðQi;QjÞ � pi � x�dxþ pili½vðQi;QjÞ � pi� �Qi

ð4:1ÞThe usual sum of these surpluses is our welfare measure. This aggregate

surplus is denoted as S(Q1, Q2, p1, p2) = S1(Q1, Q2, p1)+S2(Q2, Q1, p2), where

14 It may be instructive to make an inter-season comparison of ticket prices, but historicaldata for individual English clubs is notoriously difficult to obtain, unlike current prices,which are publicly available.



pi ¼ piUðQi;QjÞ; i = 1, 2.15 S1U; S2U and SU ¼ S1U þ S2U will denote surplus

values in a U-league equilibrium.

A first precise (and surprisingly general) result is quite immediate. Suppose,

there is a U-league equilibrium in which for some club Πi > 0 and kiW > 0.

Then, from part (a) of Lemmas 3.1 and 3.3, kiF < kiΠ and;

piUðQi;QjÞ ¼ 2kiP � 2kiF2kiP � kiF

pMðQi;QjÞ ð4:2Þ

livðQi;QjÞviðQi;QjÞ=�2� kiF

kiP

�þ kiW

kiP

@W

@QiðQi;QjÞ ¼ 1 ð4:3Þ

But, with (Qi, Qj) unchanged (4.3) can be maintained with equality at a

slightly lower kiW and slightly higher kiF. Thus, Qi remains a best response

to Qj at these adjusted weights, and from (4.2) pi goes down. Leaving utility

weights unchanged for club j means Qj remains a best response to Qi with

unchanged pj, and (Qi, Qj) is still a Nash equilibrium. The lower pi with

unchanged (Qi, Qj) means a higher attendance at club i’s home match and

an increase in SiU, whereas SjU is unchanged. Thus, aggregate surplus

increases.

Theorem 4.1: Suppose, there is a U-league equilibrium in which for some club

Πi > 0 and kiW > 0. Then, some small decrease in kiW and increase in kiF,other utility weights unchanged, produces a U-league equilibrium with higher

aggregate surplus.

If one thinks of an aggregate surplus maximizing planner with the power

to influence the nature of club governance via the utility weights, then the

optimum for this planner will imply that for each club either the weight on

win percentage is zero, or profits are zero in the resulting U-league equilib-

rium.

To go beyond this first and general result requires more structure, which

takes the form of the following two assumptions:

Assumption 1: vij(Qi, Qj) � 0.

This seems a natural assumption on fan preferences – increases in rival

team quality increase the amount a fan is willing to pay for an increase in the

quality of their team.

15 Notice that our assumption of a perfectly elastic supply of playing talent means thatplayers gain no extra surplus from playing in our league, and so do not enter the social wel-fare evaluation. Given the supply assumption, this seems appropriate, but differs from thesocial welfare specifications analysed in Falconieri et al. (2004) who also have the perfectlyelastic supply assumption. Also, with quasi-linear utility for fans (footnote 11), maximizationof this aggregate surplus equates in the usual way to Pareto efficiency, legitimizing the use ofS1(Q1, Q2, p1) + S2(Q2, Q1, p2) as the appropriate welfare criterion.



We also assume:

Assumption 2: o2W(Qi, Qj)/oQi oQj � 0.

An example of a contest success function with the desired properties is a

difference form (Skaperdas 1996), with WðQi;QjÞ ¼ gðQiÞ � gðQjÞ þ 12 where

g : <þ ! ½0; 12� is increasing and concave with g(0) = 0 and gðQiÞ ! 12

as Qi ! 1:

The best response problems faced by utility maximizing clubs always gener-

ate continuous reaction functions under the general assumptions made here.

Assumptions 1 and 2 ensure that these functions are (weakly) upward sloping

whenever they are differentiable, which ensures (weak) global strategic com-

plementarity;

Lemma 4.1: Assume Assumptions 1 and 2. Then dQiU/dQj � 0 whenever

QiU(Qj) is differentiable.


The welfare conclusion is now:

Theorem 4.2: Assume Assumptions 1 and 2, and suppose there is equilibrium

in which Πi > 0 for some club. Then some small change in clubs’ utility

weights, including an increase in kiF, produces a U-league equilibrium with

higher aggregate surplus.


If our planner observed positive profits being taken from a club, then this

indicates a social sub-optimality; increasing the weight on fan welfare in such

a club’s objective (perhaps by increasing supporter representation on the

board) is needed to effect an improvement. The lack of positive profits in

Football Leagues 1 and 2 is not an indication of a poorly performing industry

– possibly quite the opposite.

V CONCLUSIONS

Formalizing one of the seminal ideas in the literature on the economics of

professional sports leagues (Sloane (1971)), the article has provided a model

of club decisions on player expenditure and match ticket prices in such a lea-

gue, where club objectives (utility functions in Sloane’s terminology) are a

weighted average of profits, win percentages and the welfare of the club’s fans.

The effects of changes in utility function weights have been traced, for player

expenditure, match ticket prices and attendances, and for the resulting aggre-

gate surplus. Particular attention has been given to the effects of changing

the weight on fan welfare, to capture the recent and increasing influence of



supporters’ trusts on club governance in the English Football Leagues 1 and 2

(tiers 3 and 4 in the hierarchy).

The local comparative static effects of an increase in the fan welfare utility

weight depend in general on club profitability. If a club is making positive

profits, the increase will definitely also increase player expenditure (equated

here to team quality), although the effect on ticket prices is more ambiguous –holding team quality fixed would certainly produce a fall in the match ticket

price, but an increase in ticket price to accompany the increased team quality

is also a possibility. For an unprofitable club (zero profit in the model), the

effects are a (now definite) fall in ticket price, but team quality typically also

decreases. However, irrespective of profitability, the effects on match atten-

dances are clear and unambiguous – increase in the fan welfare utility weight

will increase attendances.

The emerging influence of supporters’ trusts on club governance in

the Football League is too short-lived to provide definitive data and

tests. However, the theoretical results are consistent with the limited

data, we have been able to collect for clubs in Football Leagues 1 and

2, namely that increased supporter trust involvement in club governance

(interpreted as an increase in the fan welfare utility weight) has led to

an increase in attendances. The data also indicate zero (rather than

positive) profits, and lower ticket prices and player expenditure when

there is supporter involvement in governance, again consistent with the

comparative static results.

Normatively, the theoretical results point clearly towards the conclusion

that the observation of positive profits accruing to owners would not be a

good thing, and increased supporter involvement in governance (e.g. increased

supporter trust representation on club boards) is needed to improve perfor-

mance. For such leagues, zero rather than positive profits are a better indica-

tion of a socially healthy league.

ACKNOWLEDGEMENTS

We are thankful to the editor and referees for some very helpful comments

and suggestions. The authors thank Simon Hearne and Dan Jones of Deloitte

& Touche, Kevin Rye of Supporters Direct and Babatunde Buraimo for their

assistance in providing data. These colleagues should not be held responsible

for any views expressed in the article.

APPENDIX

CLUBS USED FOR TABLE 4

SI clubs: Brentford, Carlisle, Exeter, Oldham, Wycombe, Bournemouth, Bury,

Chesterfield, Lincoln, Northampton.

Non-SI clubs: Brighton, Bristol Rovers, Charlton, Colchester, Gillingham,

Hartlepool, Huddersfield, Leeds, Leyton Orient, Millwall, MK Dons, Norwich,



Southampton, Southend, Stockport, Swindon, Tranmere, Walsall, Yeovil, Acc-

rington, Aldershot, Barnet, Bradford, Burton, Cheltenham, Crewe, Dag & Red,

Darlington, Grimsby, Hereford, Macclesfield, Morecambe, Notts Co, Port Vale,

Rochdale, Rotherham, Shrewsbury, Torquay.

In the rest of the appendix, we use the shorthand notation fi ¼ kiFkiP

and

wi ¼ kiWkiP.

Proof of Lemma 3.1: Note that the constraint in (UMAX) is alternatively

written as pi ∊ [piL(Qi, Qj), piH(Qi, Qj)].

(a) When kiF < kiΠ, the utility function is strictly concave (quadratic) in pi

for Qi, Qj where Qi 2 ½0;QiðQjÞ�, with stationary point pi ¼ kiP�kiF2kiP�kiF

vðQi;QjÞ. As

the stationary point involves a price less than the monopoly (and so the

high break-even) price, it will be the utility maximizing price if it exceeds

piL(Qi, Qj), an inequality which says (2 � fi)2Qi < li(1 � fi)v(Qi)

2, or equiva-

lently Qi\Q̂i

�Qj;

kiFkiP

�.

(b) When kiF < kiΠ and the stationary point price in (a) does not exceed

piL(Qi, Qj), the utility function concavity ensures that piL(Qi, Qj) is the utility

maximizing price.

When kiF � kiΠ, so kiF > 0, Ui(Qi, Qj, pi) is decreasing in pi for all Qi, Qj

where Qi 2 ½0;QiðQjÞ�. Hence, piU(Qi, Qj) = piL(Qi, Qj) again.

The following Lemma is useful subsequently.

Lemma A.1: For Qi 2 ½0;QiðQjÞ�, the function piH(Qi, Qj)2 is strictly increas-

ing in Qi if12livðQi;QjÞviðQi;QjÞ[ 1 (i.e. Qi < QiΠ(Qj)), and strictly concave in

Qi if12livðQi;QjÞviðQi;QjÞ� 1 (i.e. Qi � QiΠ(Qj)).

Proof of Lemma A.1: For notation convenience, we omit arguments of func-

tions (Qi, Qj) and write qi¼ 2piH

2 ¼ v2 � 2Qi

liþ v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 � 4Qi

li

qand A ¼


li

q;

@qi@Qi

¼ 2vvi � 2liþ viAþ v

A

�vvi � 2

li

�which has the sign of B ¼

�vvi � 1

li

�ðvþ AÞ � 1

livþ 1

liðv� 2QiviÞ whose last term is positive as v2 is strictly concave.

The remaining terms are also positive if vðvvi � 2liÞ þ A

�vvi � 1

li

�[ 0 which is

true if vvi � 2li[ 0, establishing the strictly increasing claim for piH(Qi, Qj)

2.

@2qi@Q2

i

¼ 2vvii þ 2v2i þ viiAþ viA

vvi � 2

li

� �þ 2vv2i þ v2vii

A

� v2vivvi � 2

li

A3� 2viliA

þ 2

livvvi � 2

li

A3

¼ 2

livvi � 2

li

� �v2 � 2Qivvi

A3

� �þ 2vvii þ 2v2i þ viiA

þ 2vv2iA

þ v2viiA

� 2v

liA



The first term on the right hand side is negative as vvi > 2/li is assumed,

and v2/Qi > 2vvi as v2 is strictly concave. Also because v2 is strictly concave

vvii \�v2i . The result follows if B ¼ �2v2i þ 2v2i � v2iv Aþ 2

vv2iA � vv2i

A � 2viliA

\0. But

B ¼ viA

h� vi

v

�v2 � 4Qi

li

�þ vvi � 2

li

i¼ � 2vi

liA

�1� 2Qivi

v

�\0, as v/Qi > 2vi as v2 is

strictly concave, establishing the strictly concave claim for piH(Qi, Qj)2.

Proof of Lemma 3.2: It follows immediately from Lemma A.1 that the bot-

tom branch formula for Ui in (3.1) has the properties claimed for Ui, given

the increasing, strictly concave assumptions on W(Qi, Qj). The top branch for-

mula is globally strictly concave under our assumptions, and is increasing if1

2�filivvi [ 1. But 1

2�filivvi [ 1

2livvi [ 1 when Qi < QiΠ(Qj), and the top branch

formula in (3.1) also has the properties claimed in Lemma 3.2, establishing

the Lemma.

Proof of Theorem 3.2: (a) From Lemma 3.3(a), positive profit optimal player

expenditure is characterized by the condition;

livðQi;QjÞviðQi;QjÞ=ð2� fiÞ þ wi@W

@QiðQi;QjÞ ¼ 1 ðA1Þ

Differentiating with respect to fi and treating Qi as a function of fi (Qj

fixed) gives, suppressing function arguments;

@QiU

@filiðv2i þ vviiÞ þ wið2� fiÞ@

2W

@Q2i

� �¼ �1� wi

@W

@Qi¼ � livvi

2� fi\0

As v2 is strictly concave and @2W@Q2

i

\0, the square bracket on the left hand

side is negative, which ensures oQiU/ ofi > 0. Differentiating (A1) similarly

with respect to wi gives: @QiU

@wiliðv2i þ vviiÞ þ wið2� fiÞ@2W@Q2

i

h i¼ �ð2� fiÞ@W@Qi

\0

and oQiU/owi > 0 follows. The sign of the effect of a change in kiΠ is that of

��

@QiU

@fiþ @QiU

@wi

�, and so oQiU/okiΠ < 0.

(b) Given the optimal ticket price in Lemma 3.1 (a), match attendance is;

AiU ¼ li½v� piU� ¼ liv=ð2� fiÞ ) @Ai

@fi¼ liv

ð2� fiÞ2þ livið2� fiÞ

@QiU

@fi[ 0:

Similarly, @Ai

@wi¼ livi

ð2�fiÞ@QiU

@wi[ 0, and oAiU/okiΠ < 0 follows analogous to (a).

(c) Again using the optimal ticket price in Lemma 3.1 (a), @piU@wi

¼ ð1�fiÞvi2�fi

@QiU

@wi[ 0.

Proof of Theorem 3.3: (a) From Lemma 3.3 (b) zero profit optimal player

expenditure is characterized by the condition;

1

2fili

@p2iH@Qi

ðQi;QjÞ þ wi@W




kiΠ has no influence locally, giving the zero derivatives in parts (a), (b) and

(c). Differentiating (A2) with respect to fi and treating Qi as a function of fi(Qj fixed) gives, suppressing function arguments:

@QiU

@filifi

@2p2iH@Q2

i

þ 2wi@2W@Q2

i

h i¼ �li

@p2iH@Qi

[ 0, where the inequality follows from

(A2). The square bracket is negative as @2W@Q2

i

\0 and as@2p2iH@Q2

i

\0 from Lemma

A.1. Thus, oQiU/ofi < 0.

Differentiating (A2) similarly with respect to wi gives; @QiU

@wi

hlifi

@2p2iH@Q2

i

þ2wi

@2W@Q2

i

i¼ �@W

@Qi\0, which implies oQiU/owi > 0, and completes (a).

Using optimal prices in Lemma 3.1 (b), match attendance is; AiU ¼li½v� piU� ¼ lipiH ) @Ai

@fi¼ li

@piH@Qi

@QiU

@fi[ 0, where the inequality follows from

part (a) and as Assumption 2 implies @piH@Qi

\0. Similarly @Ai

@wi¼ li

@piH@Qi

@QiU

@wi\0,

completing (b).

Again using Lemma 3.1 (b), @piL@Qi

[ 0, @piU@fi

¼ @piL@Qi

@QiU

@fi\0 and @piU

@wi¼

@piL@Qi

@QiU

@wi[ 0, completing (c).

The following Lemma is also helpful subsequently.

Lemma A.2: Assume Assumption 1 and suppose Qi 2 ½0;QiðQjÞ�, 12livðQi;QjÞ

viðQi;QjÞ[ 1. Then o2piH(Qi, Qj)2/oQi oQj > 0.

Proof of Lemma A.2: For notation convenience, we omit arguments of func-

tions (Qi, Qj) and write qi¼ 2piH

2 ¼ v2 � 2Qi

liþ v


li

qand A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2 � 4Qi

li

q; @qi

@Qi¼ 2vvi � 2

liþ viAþ v

A

�vvi � 2

li

�and @2qi

@Qi@Qj¼ 2vivj þ 2vvijþ

vijAþ C, where C ¼ vvivjA þ 2vvivjþv2vij

A � v3vivjA3 � 2vj

liAþ 2

liv2

vjA3. Hence,

A3C ¼ vvivj v2 � 4Qi

li

� �þ ð2vvivj þ v2vijÞ v2 � 4

Qi

li

� �� v3vivj � 2

livj v2 � 4

Qi

li

� �

þ 2

liv2vj ¼ 2vj 4

Qi

l2i� 6

Qi

livvi þ v3vi

� �þ v2vij v2 � 4

Qi

li

� �

As vvi > 2/li, the square bracket exceeds v3vi � 4Qi

li¼ vvi

�v2 � 4Qi

li

�[ 0.

Using Assumption 1, C > 0 and so @2qi@Qi@Qj

[ 0, as required.

Proof of Lemma 4.1: Suppose first that Πi > 0 at QiU(Qj). Then from Lemma

3.3(a), fi < 1, Qi\Q̂iðQj; fiÞ and:


@QiðQi;QjÞ ¼ 1:

Differentiating with respect to Qj, treating Qi as a function of Qj, gives:

dQiU

dQjliðv2i þ vviiÞ þ wið2� fiÞ@

2W

@Q2i

� �¼ �liðvvij þ vivjÞ � wið2� fiÞ @2W

@Qi @Qj



The right hand side is non-positive from Assumptions 1 and 2. The square

bracket on the left hand side is strictly negative from the strict concavity of

v2and W. Thus, dQiU

dQj� 0.

Suppose now that Πi = 0 at QiU(Qj). Then from Lemma 3.3(b):

1

2fili

@p2iH@Qi

ðQi;QjÞ þ wi@W

@QiðQi;QjÞ ¼ 0

Differentiating with respect to Qj, treating Qi as a function of Qj, gives:

dQiU

dQj

1

2lifi

@2p2iH@Q2

i

þ wi@2W

@Q2i

� �¼ �1

2fili

@p2iH@Qi@Qj

� wi@2W

@Qi@Qj

The right hand side is non-positive from Lemma A.2 and Assumption 2.

The square bracket on the left hand side is strictly negative as @2W@Q2

i

\0 and as

@2p2iH@Q2

i

\0 from Lemma A.1. Thus, dQiU

dQj� 0.

The following formulae, found by substituting utility maximizing prices

from Lemma 3.1 into (4.1), will be useful;

SiðQi;Qj; piUðQi;QjÞÞ ¼ 3� 2fi

2ð2� fiÞ2livðQi;QjÞ2 �Qi if Pi [ 0 ðA3Þ

SiðQi;Qj; piUðQi;QjÞÞ ¼ 1

2lipiHðQi;QjÞ2 if Pi ¼ 0 ðA4Þ

Proof of Theorem 4.2: Suppose, there is an equilibrium in which Π1 > 0 and

Π2 > 0.

From Lemma 3.3 and from (A3), for i = 1, 2, fi < 1, Qi\Q̂iðQj; fiÞ and;



SiðQi;Qj; piUðQi;QjÞÞ ¼ 3� 2fi

2ð2� fiÞ2livðQi;QjÞ2 �Qi ðA6Þ

If w1 > 0 or w2 > 0, Theorem 4.1 ensures the result, so assume that

w1 = w2 = 0. A small enough increase in f1 (say), other weights unchanged,

will continue to produce an equilibrium characterized by the same (A5), (A6)

conditions, and;

@S1

@f1¼ 1� f1

ð2� f1Þ3l1v

2 þ @Q1

@f1

3� 2fi

ð2� fiÞ2l1vv1 � 1

" #þ 3� 2f1

2ð2� f1Þ2l1vv2

@Q2

@f1ðA7Þ

As w1 = 0, l1vv1 = 2 � f1, and the square bracket above is positive. From

the strategic complementarity assumptions, @Q1

@f1[ 0; @Q2

@f1� 0, and so @S1

@f1[ 0.

Also, @S2

@f1¼ @Q2

@f1

h3�2f2ð2�f2Þ2l2vv2 � 1

iþ 3�2f2

2ð2�f2Þ2l2vv1@Q1

@f1[ 0 for analogous reasons to

above. Thus, the change in f1 increases aggregate surplus.



Suppose, there is an equilibrium in which one club makes strictly positive

profits and the other zero profits, say Π1 > 0 and Π2 = 0. Again, we assume

w1 = 0, otherwise Theorem 4.1 completes the proof. From Lemma 3.3 for club

2 we know: either f2 < 1 and Q2 � Q̂2ðQ1; f2Þ or f2 � 1 and, in both cases;

1

2f2l2

@p22H@Q2

ðQ2;Q1Þ þ w2@W

@Q2ðQ2;Q1Þ ¼ 0 ðA8Þ

S2ðQ2;Q1; p2UðQ2;Q1ÞÞ ¼ 1

2l2p2HðQ2;Q1Þ2 ðA9Þ

Suppose first that w2 = 0 also. If either f2 < 1 and Q2 [ Q̂2ðQ1; f2Þ or

f2 � 1, then (A8) continues to define the best response and (A9) continues to

define the surplus generated by club 2 after small enough changes in Q1.

Then:

@S2

@f1¼ l2

@p22H@Q2

@Q2

@f1þ @p22H

@Q1

@Q1

@f1

� �¼ l2

@p22H@Q1

@Q1

@f1[ 0 ðA10Þ

where the sign follows as generally@p2

2H

@Q1[ 0, and @Q1

@f1[ 0 from strategic com-

plementarity. As in the first part of the proof, @S1

@f1[ 0, and again aggregate

surplus increases after a small increase in f1. If f2 < 1 and Q2 ¼ Q̂2ðQ1; f2Þthen small increases in f1 and hence Q1 imply Q2\Q̂2ðQ1; f2Þ, as

@Q̂2=@Q1 [ 0, and club 2’s best response and contribution to surplus are now

described by (A5) and (A6). The argument of the first part of the proof then

ensures that aggregate surplus increases after a small increase in f1, again.

Now, suppose that w2 > 0. We now construct a nearby dominating equilib-

rium by increasing f1 again, but also by changing f2 and w2 so that Q2

remains unchanged. The effect of such a change on Q1 is described locally by

(A5), equivalent locally to a function Q1(f1) say with dQ1/df1 > 0. The effect

of such a change on S1 is given by the right hand side of (A7), deleting the

final term and replacing oQ1/ of1 by dQ1/df1; the effect is certainly positive again.For club 2, f2/w2 is adjusted to maintain (A8) with an unchanged Q2 and withQ1 = Q1(f1); to do this, it follows from Assumptions 1 and 2 that f2/w2 increases,and so Q̂2ðQ1; f2Þ decreases and club 2’s best response continues to be describedby (A8), even when originally Q2 ¼ Q̂2ðQ1; f2Þ. The effect on S2 is therefore givenby l2

@p22H

@Q1

dQ1

df1[ 0, as in (A10), now as Q2 is unchanged. Thus, the changes in f1, f2

and w2 lead to an increase in aggregate surplus, completing the proof.

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Date of receipt of final manuscript: 8 March 2012



Supporter Influence on Club Governance in a Sports League; a “Utility Maximization” Model

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