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Modelling Association Football Scores and Inefficiencies in the Football Betting Market

Author(s): Mark J. Dixon and Stuart G. ColesReviewed work(s):Source: Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 46, No. 2(1997), pp. 265-280Published by: Wiley-Blackwell for the Royal Statistical Society

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Appl Statist. 1997)

46, No. 2, pp.

265-280

Modelling ssociation ootball coresand

Inefficienciesn

heFootball

ettingMarket

By MARK J. DIXONt and

STUART G. COLES

Lancaster

University,

K

[Received November

995. Revised eptember 996]

SUMMARY

Aparametric odel s developed nd fittedo Englisheague and cup football ata from 992 to1995. The

model s motivated y n aim to

exploitpotential nefficienciesn the

associationfootball ettingmarket,

and this s

examined

using bookmakers' dds from 995 to 1996. The

technique

s

based on a Poisson

regressionmodel

but s complicated y hedata structure

nd the dynamic ature fteams'

performances.

Maximum ikelihood

stimates re shown to

be

computationallybtainable, nd the model

is shown to

have a positive eturn

hen used as the basis of

betting trategy.

Keywords:Betting

trategy; xpectedreturn; ootball

soccer); Maximum

ikelihood; oissondistribution

1. Introduction

Bettingon the outcome of football (soccer) matches has a long tradition n theUK,

most

popularly

in

the

form of football

pools,

which

typically

nvolve the

selection of

matches that are

thought

to

be

those

most

likely

to be a draw. A

recently

ntroduced

type of

betting, ixed

odds

betting, s also

rapidly increasing

in

popularity.

Book-

makers offer

odds on the various

outcomes of a match. The

simplest

version of

this uses

just

the

outcome of the

match,

in

the sense

of it

being

a win

by

either the

team

playing at home or the team

playing away, or a draw. More

complicated

bets

can also be

placed

on

the

score or on the

half-time nd

full-time

esults.

In

making

bets the

challenge

then is to find

good

bets',

in

which the considered

probability

of

occurrence

is

higher

than the

corresponding probability

determined

by

the book-

makers' odds, so that there is a positive expected return. Unlike in other types

of

betting,

such

as

in

horse-racing,

the

odds are fixed around one week

before

the

matches are

played.

This

allows a detailed

comparison

of

the

bookmakers' odds

with estimated

probabilities

so that

any perceived

weaknesses

in

the

bookmakers'

specification

can

be

exploited.

Consequently,

a statistical model that

is

capable

of

accurately predicting

probabilities

of the

outcome of

football matches has the

potential

to form

the

basis of a

profitable

betting strategy.

This

paper

develops

a

model that meets this

requirement.

tAddress

or correspondence:epartment

f

Mathematics nd

Statistics,

ancaster

University, ancaster,

LA1

4YF,

UK.

E-mail: [email protected]

?

1997

Royal

tatistical

ociety

0035-9254/97/46265







266

DIXON AND COLES

Various roposals avebeen

madefor

modelling

he

utcome ffootball

matches;

these rereviewed

n

Section . For a bettingtrategy,owever,

robabilities ust e

estimated

n

a

team-specificasis, o that heprobabilities

f thevariousmatch

outcomes

etween

wo

specificeams n

a particularate can be calculated.

his

degree f resolutionalls utside he scope of mostof thepublishedmodels.An

exception o this s the model due to

Maher 1982), thatassumes

ndependent

Poisson istributionsor he

number f

goals

cored

yeachof

the

home nd

away

teams,

with

means hat respecificoeach

team's astperformance.hisforms

he

basisofour

modelling

pproach. owever,

n

attemptingo derive modelwhich

s

not

ust a reasonable

escriptionf thedata, but which

lso has thepotential o

provide etter stimatesf probabilities

han he subjective

stimatesscribed y

bookmakers,

e

havehad

to modify

nd

enhance his asic

model tructure.hese

modificationsccount

or

he

fluctuatingerformancef ndividualeams nd

also

enable he stimation

fmatch utcomes or

up competitions

n

which eams

rom

differenteagues layone another. neconsequencefthesemodificationss that

simple quations or hemaximum

ikelihoodstimatorsreno onger vailable,

ut

despite

he

high

dimensionality

f the modelwe

showthat

maximum

ikelihood

estimators

re still

vailable

umerically.rom

he

itted

odel,

he

probabilitiesf

the outcomes f each

match re calculated nd

compared

with he bookmakers'

odds;

his

nderlies

he

pecification

f

bettingtrategy

hich, sing

istorical

ata,

we show o have

positive

eturn.

Section reviews he

iterature

iscussing

heuse

of statistical

ethodology

n

summarizing

ata from

ootballmatches.

he

data available

o

us are describedn

Section

3.

Section

develops

statistical

odel,building

n

the

basic model

structuref Maher 1982).Theapplicationfthemodel o our assimilatedata s

described

ogether

ith ome

ample

esultsn

Section

.

The

utility

fthe

model s

thebasis for

bettingtrategy

s

outlined

n

Section .

Finally

ection

suggests

refinements

hich, e

believe,

ould ead

to

further

mprovements

n

return.

2. Literature

eview

Surprisingly

ew

apers

ave xaminedheuse

of tatistical

echniques

ormodel-

ling

ootball

ata.

American ational ootball

eague NFL) football as received

muchmore

ttention,

ut he ifferences

etweenhe

wo

ports

mean

hat

modelling

techniquesorNFL football o notnaturallyransfero association ootball.

Early

eferences

o statistical

odelling

ffootball ataconcentrate

ainly

n the

distributionf

the

number f

goals

scored

n

a

game. Moroney 1956)

briefly

examined his

problem

nd

suggestedhat, lthough

hePoissondistribution

ro-

vided n

adequate

it

o

scores, mprovements

ouldbe

obtained

y working

ith

the

negative

inomial

istribution.

eep

et al.

(1971)

similarly

xamined hefit

f

the

negative

inomial

istributiono scores rom

ootball

matches nd other

oal

scoring ames. hey

oncludedhat chance

ominates

he

game',

inding

o

way

of

predicting

utcomes ithinheir

lassofmodels

iven

he nherent

oise

n

observed

data.

In

contrast

ill

(1974) applied

simple omparisons

estforfinal

eague

placings ith xpert redictionsnddemonstratedsignificantorrelation.more

sophisticated

nalysis

f this

type

was

by

Fahrmeir

1994),

who

applied

newly

developed echniques

or

ime-dependent,

rdered, aired omparisons

o

German

football ata.







MODELLING

ASSOCIATION

FOOTBALL SCORES

267

These

ointsllustraten

apparent ichotomy:n the

ong un, t s not

difficulto

predict airly

ccurately

hicheams re ikely

o be successful,ut

hedevelopment

ofmodels

hathave

sufficientlyigh

esolutiono exploit his ong

runpredictive

capabilityor

ndividual atchess

substantiallyore

ifficult.o

ourknowledge,

theonly aper hat erives model or ootball cores na match etweenpecific

teams,

ccountingor

he

different

uality f the eams

nvolved,s that y

Maher

(1982).He obtained

maximum

ikelihood

stimatesor model n

which he cores f

the

home nd

away

eams n

anygame re

ndependentoisson

distributions,ith

meansmodelled

s

functionsf the

respective

eams'previous

erformances.his

approach

ormshe

basis of

ourmodel

n

Section .

With

omewhat ifferent

pplications

n

mind, everal

apershave ooked t

the

effect

f specific

ircumstancesn team

performances:arnett nd

Hilditch

1993)

applied tandard

onparametric

ests o see

whetherrtificial

itches,ubsequently

banned

n

the

English

eague, ave significant

dvantage

o thehome eam;

Ridder

etal.

(1994) nvestigatedhe ffectf he ending-offf player n the utcome f

footballmatch.

Other

papershave used

statistical odels

o describe

spects f

individual atches

hemselves:

hedzoy 1995)

nformallynvestigatedimes

when

goals

are

scored;Reep

and

Benjamin1968) modelled he number

nd

type

of

passing

moveswithin

game;

Clarke

nd Norman

1995) nvestigated

he

dvantage

of

playing

t home.

In

relationo

bettingtrategies,

apers n

the

fficiency

ndexploitationf

betting

marketsre

numerous

n

the conomics

iterature.

any

papers

ddress

orse-racing

and

NFL

football

etting,

nd a few

lso consider

etting

n football

matches,

though

ittle se

of statistical

ethodology

s made n

these. iscussions fvarious

betting arketsan be foundnGolec andTamarkin1991),Hausch

t al.

(1981)

and,

pecifically

n

the

ontext f football

etting,ope

and Peel

1989).

3.

Data

A wealth f

nformation

s

available

rom achfootball

match

layed.

bviously

scores re

recorded,

ut

lso the imes f he

oals,

he

oal scorers,

he eam's

eague

position

t the ime

f

playing

nd

so on.

An

individualeam's

erformance

n

any

particularame

ould

lso

be

affected

ymany

xternal

actors:

ewlyigned layers

or

the

acking

f

manager

or

xample. hough

his

nformations also

available,

t

is ess asily ormalizednd ts ualitativealue ubjective.onsequently,urmodel

exploits nly

ach

team's

istory

fmatch

cores,

hich

we have ssimilatedver

3-year

eriod,

hough

he

possibility

f

ncluding

ther

orms fdata

s

investigated

in

Section

.

The

available ata,which

omprise 629 full-time

eague nd

cup

match

esults

from he easons

992-93,

993-94 nd

1994-95,

ach

onsist f home core nd n

away

core. ata

from

995-96

re lso

available

ut

reused s a validation

ample

to

test he

utility

f

the

model

ubsequently

henused as

a basis for

betting

strategy.

he

datafrom 992

o 1995

rovide

ccurate

mpirical

stimatesfvarious

aggregated

eatures.able

1

gives

he

elative

requency,xpressed

s a

percentage,

of thescoresfrom -0 to

4-4.

Standard rrors n the basis of an underlying

multinomialodel re

hown

n

parentheses.

ggregating,

he atio f

frequencies

f

home

wins,

raws nd

away

wins sfound o be

46:27:27.

hus,

n

empirical

stimate

of

he

robability

f

randomly

elected atch

esulting

n

a

home

win,

or

xample,







268

DIXON AND COLES

TABLE 1

Empirical stimates f each scoreprobability

or oint and marginal robabilityunctionst

Home Estimatesf

core robabilities%) for he ollowingumbers

f way

oals:

goals

Away 0 1 2 3 4

33.4 (0.74) 36.4 (0.57) 19.5 0.49) 7.9

(0.42) 2.1 (0.16)

0

22.1

(0.36)

8.2

(0.32)

7.4

(0.28)

4.5 (0.23) 1.4 0.13) 0.4 (0.06)

1 33.0 (0.65) 10.3 0.38)

12.7 0.30) 6.4 (0.24) 2.7

(0.15) 0.6 (0.07)

2 24.5 (0.51)

8.2

(0.31)

9.1

(0.25)

4.8 (0.22) 1.9 0.14) 0.5 (0.09)

3 12.6 0.40)

4.2

(0.25)

4.5 (0.25)

2.3 (0.19) 1.2 0.11) 0.4 (0.06)

4 5.3 (0.31) 1.6 (0.14)

1.8 (0.13) 1.1 (0.13) 0.6

(0.07) 0.1 (0.04)

tStandard

errors re

given

n

parentheses.

is0.46.Because f he ize f he atabase,hesempiricalstimatesrovideccurate

estimatesf

randommatch

robabilities.

urobjective

n

ater

ectionss to obtain

estimatesn matches hich re

notrandomlyelected ut re

team pecific.

The assumption

hat hemarginal istribution

frandommatch cores s Poisson

can

be

examined

t this

tage.

itting Poisson istributiono

the ggregated

ome

and

away

cores

n

Table

1

reveals hat y ny riterionhe

oissonmodel s a near

perfectit o the ggregated

core ata.

This

gives

ome eassurancehat hePoisson

regression odeldeveloped

n

Section is at leastplausible

orour data,despite

concerns aised by

other esearchersbout

the

general ppropriateness

f the

Poisson ssumption.

further

ssumption

f

hebasic

model

n

Section

is

that

he

home ndaway cores re ndependent.o assess hevalidityf this ssumption,

Table

2

displays

f(i,]j)

fH

(i)fA(j)

for achhome

nd

away

core

i,j),

i

=

0,

..., 6

andj

=

0,.

..,

5,

where,ffH

ndfA

are

the

oint

nd

marginalmpiricalrobability

unctionsor ome nd

away

cores

respectively.ootstrap

tandard rrors

re

given

n

parentheses.

able

2

suggests

TABLE 2

Estimates

f

theratios

of

the observed

oint

probability

unction

nd the

empirical robabilityunction

obtained nder

he

ssumptionf ndependence

etween

hehome

nd

awayscorest

Home

Estimates

f

ratios

or

he

ollowing

umbers

f way oals:

goals

0 1

2

3 4 5

0

111.5

3.52)

92.0

(2.87)

103.4

4.18)

82.1

(7.67)

96.4

(15.31)

96.8

(28.12)

1 93.7

(2.43)

105.7

2.00)

99.3

(3.74)

103.7 (6.31) 86.9 (13.15)

108.3

19.99)

2 99.6

(2.91)

101.7

2.11)

99.2

(3.78)

97.4

(7.41)

95.9 (17.4)

106.7

23.77)

3 100.3

4.25)

98.5

(3.61)

91.8

(6.51)

116.6

11.03)

139.8 23.85)

75.4

(40.5)

4 91.0 (7.07) 93.8 (7.16) 108.6 10.74) 138.0 16.31) 111.7 32.86) 90.4 (55.33)

5 94.1

(13.24)

102.3

12.28)

114.3

20.6)

73.3

(31.01) 120.8 74.71)

130.4

129.7)

6

139.1

31.95)

49.1

(23.66)

146.4

41.33)

45.3

(57.84)

174.1

122.2)

tThe

numbers re

multiplied y

100 for

larity.

tandard rrors

re

given

n

parentheses.







MODELLING

ASSOCIATION

FOOTBALL SCORES

269

that he

assumptionf

ndependenceetween

cores s reasonable,

xcept

or he

scores

-0,1-0,

0-1

and

1-1.

Based on the stimates

nd errorslone,

he core -3

seems o be significantly

nderestimated

y the

ndependence

odel. However,

viewedwithin

he

ontext f all

theother esults,

e regard

his s due to sampling

error. modificationf the ndependencessumptionnthe ight f these bser-

vations

s considered

n

Section

.

4.

Model and

nference

4.1.

Model Specification

With

he

aim

of developing profitable

etting trategy,arious

eaturesre

required

f a statistical

odel or ootball

matches.or example:

(a) themodel hould ake nto ccount hedifferentbilitiesf both eamsn a

match;

(b)

there hould

e allowance or

he

fact

hat eams laying t

homegenerally

have ome dvantage

the o-called

home ffect';

(c)

the

most

reasonablemeasure f

a team's bility

s likely o be based

on a

summary

easure f

their ecent

erformance;

(d)

the

nature

f football

s such that a team's ability

s

likely

o

be best

summarized

n

separate

measures f heir bility o attack to

score oals)

nd

theirbilityo

defend

not

to concede oals);

(e)

in

summarizing

team's

performance

y

recent

esults,

ccount

hould

be

taken fthe bilityf the eams hat hey aveplayed gainst.

It

s

not

practical

o

obtain

mpirical

stimatesf

probabilities

fmatch utcomes

that account

for all

theseconstraints.nstead,

we

use

a

statisticalmodel

that

structurally

ncorporates

ach ofthese eatures.

ur basis

s the

model roposed y

Maher

1982),

with

modificationso enable

he nclusion fnon-complete

ata

sets,

and

data from

ifferentivisions imultaneously,

nd to allow

for

fluctuations

n

team erformance.

The basic ssumption

fMaher'smodel

s that henumber fgoals

cored y

the

home nd

away

teams

n

any

particular ame

re independent

oissonvariables,

whosemeans redeterminedy

the

respective

ttack nd defence

ualities

f

each

side.More xplicitly,na match etweeneamsndexedandj, et

Xijj

nd

Yi,j

ethe

number f

goals

scored

y

the

home

nd

away

ides espectively.

hen

Xij

-

Poisson(aj/37y),

(4.1)

Yi,j

-

Poisson(aj131),

where

ij

and

Yij

are

ndependent,

i, 1i

>

0, Vi,

he

i

measure

he attack' ate

f

the

eams,

he

/3i

measure he

defence' ates nd

>

0 is

a

parameter

hich

llows

for hehome

effect.

n

fact,

Maher

1982)

ncluded

more

general

model

peci-

fication

han

his,

llowing

or

eparate

ome

nd

away,

nd attack

nd

defence

parametersor veryeam.However,ikeMaher 1982),we havefoundmodel4.1)

to

be an

adequate

implification,

hough

here re

till

ssumptions

nthismodel

hat

would

not

be

supportedy

detailed

tudy

fmatch ata.

The essential

oint

s

that,

although

etails f

the model

may

be

inaccurate,

he

global

structure

hould

be







270

DIXON AND COLES

sufficientlyccurate o enable

hedevelopmentf a bettingtrategy

ith positive

expectedand realized) eturn.

Some aspects f themodel

re easily mproved

n, however. onsider irst he

assumptionf ndependence.

aher 1982) uggested

he

use ofa bivariate

oisson

familys anextensionfthebasicmodel, utthis amilys unable o representhe

departurerom

ndependenceor ow scoring ames

hatwe dentified

n Section

.

Instead,we propose

hefollowing odification

f model 4.1):

Pr(Xi,j

=

x,

Yij

=

Y)

=

-rA,y(X,

)

A

exp(-A) ,uy xp(-/i)

(4.2)

where

A

=

ai3j7,

and

(

-A,mp

if

x=y=O,

l+Ap

if

x=0,y=1,

7(X.y)<

l+,uLp

if

x=1,y=0,

i

-p

ifx=y=1,

1

1

otherwise.

In

thismodel, , where

max(-1/A,

l

/,)

<

p

<

min(I

/A,u,),

enters

s a dependencearameter:

=

0 correspondso ndependence,

ut therwise

the

ndependence

istributions perturbedor

ventswith

<s

1

and y

<s

1. It

is

easily hecked hat hecorresponding

arginal istributionsemain

oissonwith

means

A

and

,urespectively.

Another imitationf the

model s that t is staticthe

attack nd defence

parameters

f

each

team re regardeds constant hroughime.

his ssuewill

be

consideredn Section

.3.

4.2. Model Inference

It

follows

rom

model 4.2)

that with

n

teams there re attackparameters

{I

,

. .

*

C?n},

defence

arameters

,31,

. .,

nl1,

hedependencearameter and the

home ffect

arametery

o

be

estimated.

o

prevent

he

model

from eing ver-

parameterized,

e

mpose

he

onstraint

n

n-

i=

1.

i=1

For the

English

eague ystem,

hich

omprises

he remier

eague

nddivisions -

3 of theFootballLeague,n= 92,so themodelhas 185 dentifiablearameters.

Our

basic

tool of

inferences the ikelihood

unction.

ith

matches

ndexed

k

=

1,

.

.,

N,

and

corresponding

cores

Xk, Yk),

this takes

the

form, p

to

proportionality,







MODELLING ASSOCIATION FOOTBALL

SCORES

271

N

L(aj, p3i,,

7;

1,

.

.,

n)

=

17

Ak,Pk(xk,

yk)exp(-Ak)Axk

exp(-,uk),uyk

(43)

k=1

where

Ak

=

Ci(k)I3j(k)y,

,Uk

=

Oaj(k)I3i(k)4

and

(k)

and

(k)

denote espectively

he ndices

f

he

home nd

away eams laying

in

match

.

With

omplete ata,

n

the ense

feach eam

aving layed very ther

teamequally ften, nd

in

the simpler ase of independenceetween omeand

away

eam cores

p

=

0),

Maher

1982)

obtained

systemf inear quations hose

roots re

the

maximumikelihoodstimates.o achieve reater enerality,e are

restrictedo direct umerical aximizationf quation4.3).Thenear rthogonalityof

many arameterombinations

eans hat his s

straightforward,espite hehigh

dimensionality

fthemodel.

In

equation4.3),

eams rom ll four ivisionsre

ncludedn

the

ikelihood.

his

has two

onsequences:irstly,

he

arameters

or ach eam hould eflecthe

elative

quality

f the different

ivisions nd, secondly,

he

parameters ill

be

estimable

only

f there

s informationrommatches etween eamsof different

ivisions.

Fortunately,ecause

heres some

mobility

etweenhe eams

f

different

ivisions

at

the

tart f new eason

due

to

promotion

nd

relegation,

he ssue f

parameter

identifiability

s

resolved.

he

situation

s also

helped y

he

nclusionfresults rom

cup gameswhich nvolve eamsof differentivisions laying achother. hen,because heparametersre calibratedcross hedivisions,hemodel anvalidly e

used

to estimatehe

probabilities

f match utcomes

nvolving

eams f

different

divisions,

s

in

cupgames

or

xample.

hese

oints

re llustrated

y

Table 3,which

showsthe mean attack nd

defence

arameters

orteams

n

each division.As

expected,

he

verage

ttack nd

defence

ating

f

eams

ncreases ith

ighereague

status,

s measured

y increasing

nd

decreasing

mean values of

a

and

,3

res-

pectively.

4.3.

Model Enhancement

A structuralimitationf model 4.3) is that heparametersrestatic,.e. teams

are assumed o have constant

erformanceate,

s

determined

y

ai

and

3B,

ver

TABLE 3

Mean attack

nd

defencearametersor

teamswithinach

division

League

Mean

attack Mean

defence

parameter

parameter

3

Premier

1.38 0.68

Division

1

1.07 0.86

Division

2

0.83

1.14

Division 3 0.73

1.32







272

DIXONAND COLES

time. n reality,

team's erformanceends o be dynamic,

aryingrom ne time

period o another, nd

thisbehaviour

houldbe incorporatedn the model.

n

particular team'sperformances likely o be

moreclosely elated o their er-

formance

nrecent

atcheshan nearlier

matches.

n

principle,his ehaviour

ould

bemodelled yformalizingstochasticevelopmentf hemodel arameters;his s

considered

n

Section . In view f the

dimensionalityf the

model, owever,nd

sincewe shallalwaysneed to estimatehe

parameterst the fixed imepoint

f

making bet

rather

han, ay, orecastinghead,we take more

implistic

pproach

here.Thus we assume

hat

he

parametersre,

n

a loose

sense, ocally onstant

throughime ndthat istoricalnformations

of essvalue han ecent

nformation,

and

we determinearameter

stimates

or ach time

oint that

re

based

on the

history f

match

cores

up

to time .

Modifyingquation 4.3) we construct

'pseudolikelihood'

or ach

time

oint ,

Lt(i,

piA,

, 7; i

1,

. .,

n)

=

1

TAk,

Pk(Xk,

Yk)

exp(-Ak)Akxk

xp(-Pk)i4kkp)(t-tk)

kEA,

(4.5)

where

k

s the time

hat

matchk

was

played,

At

=

{k:

tk<

t},

Ak

and

Pk

are

as

in

equations4.4)

and

X

is

a non-increasing

unction

f

time. his

representsslight

abuse

of

notation

ince he

arameters

i,

,3i,

and are

themselvesime

ependent.

Maximizingquation

4.5)

at

time eads o

parameter

stimates

hich

re

based

on

games p

to time

only.

n

this

way,

hemodel as the

apacity

o

reflect

hanges

in

team erformance.

oreover,arying

he

hoice f

X

allows

historical

ata

to be

downweightednthe ikelihoodo a greaterr esser egree.

4.4.

Choice

of Weighting

unction

There re

various ossible hoices or

he

weighting

unction in

equation

4.5).

One

possibility

ouldbe

I() t

<,

to,

inwhichase, t time

,

allresults ithinhe astto ime nitswould egiven qual

weight

n

the nference.

nstead,

we workwith

he

model

0(t)

=

exp(-(t),

in

which

ll

previous

esults,ownweightedxponentially

ccording

o

a

parameter

(

>

0,

are includedn

the

nferencet time

.

The staticmodel

4.3)

arises

s

the

special

ase

=

0,

whereas

aking

ncreasinglyarge

alues

f

gives elatively

ore

weight

o themost ecent esults.

Optimizing

he

hoice f

is

problematic,

ince

quation

4.5)

defines

sequence

of

non-independent

likelihoods', hereas

we

require

such

that

the

overall

predictiveapabilityf themodel s maximized.n fact,nsubsequentections, e

restrict

ttentiono the

prediction

f

match

utcomes

ather

hanmatch

cores.

Therefore

t s

pragmatic

o choose to

optimize

he

prediction

f

outcomes. irst

note hat

he

probability

f a

homewin

n

match is estimated

s







MODELLING ASSOCIATION FOOTBALL SCORES

273

H

=

S

Pr(Xk

=

1,

Yk

=

m) (4.6)

1,mEBH

where

H

=

{(l, m):

1

> m},

and the score

probabilities

re

determinedrom he

maximizationfmodel4.5)att(k), he ime fmatch .Similarxpressionsoldfor

PA

and

PD,

the

probabilities

f

an

away

win nd a draw

espectively.

ow

define

N

SQ5)

=5E (6k log

pH

+

Sk

log

Pk

+

6k

log

PD)

(4.7)

k=1

where, or xample,

kH

1 ifmatch is a homewin nd

6kH

0

otherwise,

nd

k,

Pk

and

PD

are themaximum

ikelihoodstimatesrom

model

4.5),

with

weighting

parameter

et

t

. Consideringnly

he

utcomes,

nd

not

he

cores, quation4.7)

is the

nalogue

f

a

predictiverofileog-likelihood. plot

of

S(s) against ,

with

time nits aken obehalf-weeks,s givennFig. 1. The functionsmaximizedt

(

=

0.0065,

nd all

subsequent

esults re

given

with

espect

o

this

hoice

of

,

though

n fact heresults

re

robust

cross

range

f

-values.

4.5.

Parameter stimates nd

Results

The

complete

et

of

parameterstimates,

btained

y maximizingquation 4.5)

with

=

0.0065,

t each time

point t, gives

a

profile

f each team's

changing

performance

n

terms f

defence nd

attack

bilities. ata

from t least 60

half-

weeks

re

required

o estimate

arametersccurately,

o estimates

reobtained

or

rangingrom0to174.Forbrevitye show nly subset f he esultsfor he ull

set of

results or

the 1996

season,

ontact

M.

Dixon).

Tables

4

and 5

give

the

_3t.

LO

,

fr1 \ I

0

0.0

0.005

0.010

0.015 0.020

Valueof W

Fig.

1.

S(s)

versus

:

the

maximum

ccurs

t

=

0.0065







274 DIXON AND COLES

TABLE 4

Maximum ikelihood stimates nd standard rrors or the attack

and defence ateparameters, n August th,1995, or Premiership

teams in the 1995-96 season)

Team

&

se(&)

3

se(A)

Arsenal 1.235 0.151 0.527 0.078

Aston Villa 1.278 0.178 0.649 0.086

Blackburn 1.730 0.209 0.534 0.082

Bolton 1.183 0.141 0.760 0.100

Chelsea 1.238 0.169

0.658 0.089

Coventry 1.115 0.164 0.699 0.094

Everton 1.177 0.169 0.667 0.091

Leeds 1.510 0.186 0.583 0.088

Liverpool 1.448 0.180 0.561 0.082

Manchester

ity 1.232 0.170 0.728 0.091

ManchesterUnited 1.869 0.208 0.402 0.067

Middlesbrough 1.244 0.152 0.750 0.109

Newcastle 1.659 0.195

0.578

0.081

Nottingham orest 1.460 0.170 0.658 0.095

Queen's

Park

Rangers 1.497 0.195

0.717 0.095

Sheffield

ednesday

1.387 0.179 0.698

0.091

Southampton

1.446 0.183 0.772 0.098

Tottenham 1.622 0.201 0.775 0.100

West Ham 1.192 0.169 0.649 0.087

Wimbledon 1.281 0.174 0.732 0.094

TABLE

5

Maximum ikelihood stimates nd standard rrors or the attack

and

defence ate parameters n August5th, 1995, for

teams n

division (in

the

1995-96 season)

Team c se(&)

3

se(A)

Blackpool 0.858 0.110 1.357 0.163

Bournemouth 0.681 0.096

1.095 0.139

Bradford 0.832 0.107 1.175 0.149

Brentford

0.967

0.115

0.900 0.129

Brighton 0.820 0.107 1.032

0.137

Bristol

City

0.825 0.120 0.873

0.114

BristolRovers 0.917 0.113 0.965 0.138

Burnley

0.942 0.116 1.067

0.127

Carlisle

0.781 0.100 0.964

0.157

Chesterfield 0.764 0.099 1.024

0.160

Crewe

1.065

0.125

1.265

0.159

Hull 0.822 0.104 1.069 0.146

Notts

County

0.985 0.132 0.979 0.120

Oxford 0.956 0.119

0.951

0.119

Peterborough

0.829

0.111 1.161 0.133

Rotherham 0.852 0.106 1.136 0.143

Shrewsbury

0.764 0.104 1.060 0.145

Stockport

0.945 0.115 1.040 0.136

Swansea 0.798

0.106

0.899

0.122

Swindon 1.160 0.154 1.091 0.120

Walsall

0.911

0.114

1.116 0.162

Wrexham

0.957

0.116

1.257 0.150

Wycombe 0.813 0.105 0.984 0.137

|

York

0.916

0.113

0.926 0.131







MODELLING

ASSOCIATION

FOOTBALL

SCORES

275

maximum

ikelihood

stimates

f

the attack nd

defence arameters

n

August5th,

1995,

for

teams whichwere

n the Premier

ivision nd division

2

respectively

n

1995-96.

n

addition,Fig.

2

shows

thecorresponding

equenceof

estimates f a(t)

and ,3(t)

over time

forthree eams

the

non-uniformity

n these stimates

uggests

thatteams'performancesre genuinely ynamic.Also shownis the sequenceof

estimates

f the home

effect

arameter

y(t)

which,

s

might

be

expected,

emains

nearly

onstant ver

time.

The flatportion

rom

= 82

to

t

=

90 corresponds

o

the

summer

break in the football

season. Furthermore,

able

6 gives a sample

of

matches

nd the maximum

ikelihood

stimates f

the

outcome probabilities.

he

standard

rrors

f

the outcome stimates, articularly

hose of

thedraw

probability

estimates,

re

small

relative

to

the standard

errors of the attack

and

defence

parameter

stimates.

CD

......

.H .........

:.

.

E

E

-

...

.

..

Time

half

weeks)

(a)

Cu

.E X

0.

U

. .

. ..

...

E

IV,

Ior

60 80

100 120 140

160

Time half

weeks)

(b)

E

II:

(1)0

E'r

0c0 t

060

80 100

120 140 160

Time

half eeks)

(b)

Fig.

2.

(a),

(b)

Timeseries

f themaximum

ikelihood stimates

f attack nd

defence ate

parameters

forSheffield

nited

,Norwich

.....) and

Everton

---;(c)

variation

f thecommon

home

effect

arameter

with

ime







276

DIXON

AND COLES

TABLE

6

Maximum likelihood

estimates

for match outcome

probabilitiest

Match

Maximum ikelihood

stimates or the ollowing

utcomes:

Home win

Draw

Away win

Arsenal versus

Middlesbrough

0.535

0.069)

0.280 (0.030)

0.184 (0.046)

Aston Villa

versusManchesterUnited

0.214 0.054)

0.291 (0.029)

0.495 (0.072)

Blackburn ersus

ueen's Park

Rangers 0.615

(0.078) 0.221

(0.033) 0.164

0.049)

Chelsea

versus verton

0.457 (0.075)

0.298 (0.030)

0.245 0.057)

Liverpool

versus

heffield

ednesday 0.535

0.076) 0.262

(0.031) 0.203

(0.052)

Blackpool

versusWrexham

0.428 0.077)

0.240 (0.018)

0.332

(0.070)

Stockport

ersus

urnley 0.480

0.077) 0.259

(0.024) 0.261

(0.062)

Newcastleversus

toke

0.705 0.073)

0.198 (0.042)

0.097 (0.034)

tThe

matches re a

subset

f

thefixturesrom

August19th,

995,plus

one other

cross-division

atch.

Approximate

standard rrors re calculatedusing hedeltamethod. tandard rrors regiven nparentheses.

5.

Bettingtrategy

How usefuls themodel

erivedn

Section ,

when sed s the

basisfor

betting

strategygainst

dds

provided

ybookmakers?

detailed

nvestigation

ntobetting

strategiesor

ixed

dds

football

ettings

given

n

Popeand Peel

1989) nd Dixon

and

Pope

1996).

Herewe

address he

uestion

yreference

o a

new

etof

results,

corresponding

o

the

1995-96

eason, orwhichwe

have bothresults

nd

book-

makers' dds.We

first se

model

4.5),

with

=

0.0065, t each

new ime

oint

to

obtain urrentarameterstimates.hen, y comparingstimatedesult robabil-

itieswith

ookmakers'ddsfor

he

following

eek,

we

determine

hich

ames

re

most

dvantageous

o bet

on. We then

alculate

he

net

eturnrom uch

strategy.

A

typical

et f

bookmakers'

ddsfor

particular atch

might

e

8:13,

12:5, :1)

for

home

win,

raw

nd

away

win

espectively.

hus,

n

this

xample,

stake

f 13

units

n

a

home

win

would

yield

profitf 8

units

f

hat utcome

ccurred. dds

01:02

transformo

a

probabilityby

using

he

formula

p

=

02/(01

+

02).

The

above et

of

oddsthen

orresponds

o the

etof

probabilities

0.62,

.29,

0.20),

which as a sumof 1.11.Thisphenomenons standardnbetting arkets:fthe

bookmakersre accurate

n

their

robability

pecifications,hey

ave

an

in-built

'take',

orresponding

o their

xpected

rofit, hichnthe

bove xamples

11

%.

To

win

money

rom

ookmakers,

n

the

senseof

having

positive

xpected

eturn,

requires

determination

f

probabilities

hich

s

sufficiently

ore

ccurate han

those

btained rom heodds

n

order

o

overcome

he

bookmakers'

ake.We first

rescale

multiplicatively

he

bookmakers'dds

so that

hey

um

to

1. Denotethese

probabilitiesor

match

by

br,

bD

and

bA for home

win,draw

nd

away

win

respectively,

nd

similarly

et

Pk, Pk

and

Pk

be

the corresponding

aximum

likelihood

stimates

or

his

match nder

model

4.5).Comparisons

f he wo ets f

probabilitystimatesor achof theresult utcomes regivennFig. 3 for ach

match

n

our

database. verall

heres

reasonable

greement

etweenhe

robability

assessments,

ut he

variability

n

these

lots

ndicateshe

potential

or

ositive

ain

if

ourmodel

robabilities

re

accurate.







MODELLING

ASSOCIATION FOOTBALL

SCORES

277

+

+

++

t

+

cq ~~~~+

+

++

+

co

+

+

ci)

+ +

+

4

+

+

~

+

++

-o

2i~

W0

++

-a I

U; i X;+

C;

~~+ +++

+

O

+

6

0.2

0.3 0.4 0.5 0.6

0.7

Bookmakers' robabilities

(a)

ll,

+++

6a

U)

CI()

+%

o~~~~~~~~~~~~~~

0

0.20 0.25

0.30

Bookmakers'

robabilities

(b)

U)0

A)

~ (G =P

+b 51

02

If~~~~~~.

0.2 0.3 0.4

.5poaiiy hnteepctdi

ilb 0.60 r011i

h

b~~~~~~~~~~~ookmakers'taeiinlddInraiynihrPno kisheru

robabilitiesbu

Fig

3.tModel probaiivertyr

fustimates

ltedaais

ods foral

achestwhere ddscwrae tavalbe

(a)

hromeisb

druaws;nc.),

awa wnstrlbtigsrtg o n atclrgm st

e

saebton homewin, or xample,

s

3L6, ~ ()=~'b~ .

51

If b~' s he rue probabilitthnteepce anwllb .0 r-.1i

h

boomakrs ae i iclued.Inreaitynethe

Pkno

b~ isth tre pobbilty,bu

bookmakers'.~~~~~~~okmkesprbaiite

Fromeisb

quation

5.),

wawnatrlbtigsrtgso n atclrgm st

e

saebton

home

win,

or

xample,

f







278

DIXON AND

COLES

C\D

a~o

CD

I0-

1.0

1.1

1.2

1.3 1.4

Ratio

Fig

4.

Observed mean

return

plotted

against

the

ratio

of model

probabilities

o

bookmakers'

probabilities-

- -

-, return

f

-0.11,

the

expected

eturn

nder

random

betting,

which

s due to

the

bookmakers'

take of

11%

for each

match;

,

90%

bootstrap

ntervals):

he mean

return

s

calculated

only when there

re

more

than

10

sample values

where

k

denotes he

unscaled ookmakers'

robability

f a

home

win

n

match

,

for ome

predeterminedalue of

r

> 1,

with

corresponding

trategy

or

betson

away

wins

r draws.

ncreasing

eads

o

a

stricter

ettingegime,

ut

onsequently

fewer

ets.

The

effect

f a

range

f different

hoices

or can be seen

n

Fig. 3,

in

which

he

ine

p

=

rb s

plotted

orr

=

1.0, 1.1,

1.3. For a

particular

hoice

f

r,

points

alling

bove

that

ine

orrespond

o

matches

n

which

his

etting

trategy

would

have

ed

to a bet

being laced

n that

articular

utcome,

ith as

specified.

The success

f this

betting

trategy

an be

assessed

y

calculating

he

observed

return,

f

such

strategy

ad

been

dopted, iven

hematch

esults

hich

ctually

occurred. his splotteds a functionfr inFig.4,togetherith 0% confidence

intervalsbtained

sing

he

bootstrap.

here s

considerable

ariability

n

the

plot

which

makes

t

difficult

o draw

definitive

onclusions.

owever,

ith

=

1.2

our

betting

trategy

eads

to

a return

hich

s

borderline

ignificantly

ifferent

rom

-0.1

1,

the

xpected

eturn

nder

random

etting

trategy

ue

to the

bookmakers'

take,

nd

has

a

positive

xpected

bsolute eturn

or

ny

r

>

1.1. t

is

in

this

ense

that

we claim hat he

model

nd

nferencef

Section

meet

ur

tated

bjective

f

deriving

model

for

stimating

ootball

match

utcomeswhich s the basis

of a

bettingtrategy

ith

ositive

eturns.

6. Conclusions

Our aim has been

o derive

method or

stimating

he

probabilities

f

football

results ith

he

potential

o

achieve

positive

xpected

eturn

hen

sed s the

asis







MODELLING ASSOCIATION

FOOTBALL SCORES

279

of a betting

trategygainstbookmakers' dds.

Our basic model s simple

a

bivariate

oisson

distributionor henumbers

f goals scored y each team,

with

parameters

elated

o

pastperformancebut refinementsecessary

o improve

he

realism nd precision

f the model make

the associated

nference heavy

computationalurden.None-the-less,he computationsre manageable nd the

resulting odel s accurate

n many espects.

Our

bettingtrategy

s

equally imple:

e bet

on all outcomes orwhich

he

ratio

ofmodel o bookmakers'

robabilitiesxceeds

specifiedevel. or sufficiently

igh

levels,

we

have

shown

hatthis strategy ields positive

xpected

eturn,ven

allowing

or

he

n-built ias

n

the

bookmakers'

dds.

The simplicityf

our model and the associated

etting

trategys appealing.

However,o mprove

urthern theutility

f

our

pproach,

e perceive

hat

urther

modifications

ay

be desirable.

ne possibility

s to

refine

urther

he

Poisson

regression

odel. Stochastically

pdatedparameters

re a natural

dea in this

context,ut hedetailedmplementationaybe difficult.mith1981)considered

dynamicegression

rameworkor imple

oisson

models,

utthe

generalization

f

these deas

to the cale of model

4.5)

is not

mmediate.roadening

he

scope

of

our model

to

incorporate

dditional ovariate nformation

s a second

rea

for

development.

he

quantitative

alueof

suchdata s

not

always bvious,

o such

development

ight

eed a

Bayesian

tructureo

exploit

heir

ubjective

alue.

A

third

ossibility

s the

betting egime.

We

have restrictedttention

o

far

o fixed

oddsbets f

match

utcomes.

his

eads o

a

bettingtrategy

n

which

elatively

ew

bets re

actually laced.

As bookmakers

ffer dds

on specific

atch

cores,

he

probabilities

fwhich realso obtained

rom urmodel,

bettingtrategyasedon

match cores ould be developed.More radically,here re severalmarket-style'

indexbetting ptions

or

football

matches,

here oal margins

re bought

nd

sold

ikecommoditiese.g.

Jackson1994)

and

Dixon

and Robinson

1996));

the

implementation

f our model

formarket

trategies

n such n

option

s

a further

possibility.

The ure f

cientificmprovementf ur

model ndbettingtrategy,

ith

he

urely

incidental

y-productfwinning

oney

romhe

ookmakers,ncourages

s

to build

on the

pparent

uccess

f

the

resent

odel

n

the

arious

ways

iscussed

bove.

Acknowledgements

We thank

refereend

the ditor or ommentsn an

earlier

ersion

f he

paper

and Jonathan

awn for

ommentshat

helped

o

improve

he

handling

f

depen-

dence etween

cores.We are

grateful

o

many olleagues

t Lancaster

ho

helped

o

digitize

ootball

eague

ata

and bookmakers'

dds.

For

entry

fdata

and

odds,

we

thank

imon

Garside,

ara

Morris,

MikeRobinson nd

PeteSewart.

he

data were

obtained

rom

Williams

1992,1993, 994),

Hugman 1991)

nd

90-Minutes

weekly

magazine.

dds

were btained

romocal

branches f

major

UK bookmakers.

e

also

thank

Mike

Robinson

nd

Barry

owlingson

or

writing

ome f

the

oftware

used

n

the

handling

f data.

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