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Uncorrected Author Proof

Journal of Sports Analytics xx (2021) x–xx

DOI 10.3233/JSA-200588

IOS Press

1

Investigating the efﬁciency of the Asian

handicap football betting market with

ratings and Bayesian networks

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3

Anthony C. Constantinoua,b,∗

4

aBayesian Artiﬁcial Intelligence Research lab, School of Electronic Engineering and Computer Science, Queen

Mary University of London (QMUL), London, E1 4NS, UK

5

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bThe Alan Turing Institute, UK7

Abstract. Despite the massive popularity of the Asian Handicap (AH) football (soccer) betting market, its efﬁciency has not

been adequately studied by the relevant literature. This paper combines rating systems with Bayesian networks and presents

the ﬁrst published model speciﬁcally developed for prediction and assessment of the efﬁciency of the AH betting market. The

results are based on 13 English Premier League seasons and are compared to the traditional market, where the bets are for

win, lose or draw. Different betting situations have been examined including a) both average and maximum (best available)

market odds, b) all possible betting decision thresholds between predicted and published odds, c) optimisations for both

return-on-investment and proﬁt, and d) simple stake adjustments to investigate how the variance of returns changes when

targeting equivalent proﬁt in both traditional and AH markets. While the AH market is found to share the inefﬁciencies of

the traditional market, the ﬁndings reveal both interesting differences as well as similarities between the two.

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Keywords: Directed acyclic graph, football prediction, graphical models, proﬁtability, rating system, return-on-investment,

soccer prediction

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1. Introduction19

Football prediction models have become immen-20

sely popular over the last couple of decades, and21

this is due to the increasing popularity of football

22

betting. In the academic literature, such models typ-

23

ically focus on predicting the outcome of a match

24

in terms of home win, draw, or away win; known as

25

the 1X2 distribution. Several types of models have

26

been published for this purpose and include rating27

systems (Leitner et al., 2008; Hvattum & Arntzen,28

2010; Constantinou & Fenton, 2013; Wunderlich29

& Memmert, 2018), statistical methods (Dixon &

30

Coles, 1997; Rue & Salvesen, 2000; Crowder et al.,31

2002; Goddard, 2005; Angelini & De Angelis, 2017),

32

machine learning techniques (Huang & Chang, 2010;

33

∗Corresponding author: Anthony C. Constantinou. E-mail: a.

constantinou@qmul.ac.uk.

Arabzad et al., 2014; Pena, 2014), knowledge-based 34

systems (Joseph et al., 2006), and hybrid methods 35

that combine any of the above (Constantinou & Fen- 36

ton, 2017; Constantinou, 2018; Hubacek et al., 2018). 37

In the recent special issue international competition 38

Machine Learning for Soccer, the models that topped 39

the performance table are hybrid and heavily rely on 40

rating systems (Constantinou, 2018; Hubacek et al., 41

2018). 42

In Asia, the most popular form of betting (also 43

common in Europe) is the so-called Asian Handicap 44

(AH). This form of betting introduces a hypotheti- 45

cal handicap (i.e., advantage) typically in favour of 46

the weaker team. Speciﬁcally, traditional AH intro- 47

duces a goal deﬁcit to the team more likely to win 48

before kick-off. The manipulation of the match out- 49

come creates interesting situations in which betting 50

is determined by hypothetical, rather than actual, 51

match outcome. Examples of the various types of AH 52

ISSN 2215-020X © 2021 – The authors. Published by IOS Press. This is an Open Access article distributed under the terms

of the Creative Commons Attribution-NonCommercial License (CC BY-NC 4.0).

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2A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

betting are provided in Section 2. This type of bet-53

ting has also become popular in the UK over the last

54

couple of decades. Football (soccer) syndicates are55

rumoured to bet millions per week, often on behalf56

of clients, on AH outcomes offered by bookmakers57

in Asia (Williams-Grut, 2016). This is because the

58

Asian markets attract higher volumes of bets and offer59

greater market liquidity. Estimates suggest that over60

70% of the betting turnover for football is recorded61

with Asian bookmakers (Kerr, 2018).

62

Whereas there is a large literature analysing63

traditional betting strategies, with researchers inves-64

tigating how to optimise their betting, there has been

65

only four published papers involving some analysis66

related to AH. Speciﬁcally, Vlastakis et al (2008)

67

used AH odds as one of their model variables to68

predict match scores and showed that they are a69

strong predictor of match outcomes. Grant et al

70

(2018) used AH odds, in conjunction with 1X2 odds,71

to analyse arbitrage opportunities and showed that72

these exist across a large number of ﬁxed-odds and73

exchange market odds. Hofer and Leitner (2017)74

described how to derive information from live AH75

and Under/Over odds in order to maximise expected76

returns. Finally, and in an effort to educate gamblers,

77

Hassanniakalager and Newall (n.d.) investigated the

78

product risk associated with different football odds79

and showed that the AH odds would generally gener-

80

ate lower losses compared to other popular types of81

bet such as the 1X2, Under/Over, and correct score.82

Remarkably, no previous published work involves a

83

model speciﬁcally designed for, and assessed with,84

AH bets.

85

The purpose of this paper is to investigate the

86

efﬁciency of the AH market in relation to the 1X287

market. The 1X2 market has been extensively studied88

and the literature provides mixed empirical evidence89

regarding its efﬁciency, with most evidence point-90

ing towards a weak-form efﬁcient market (Giovanni91

& De Angelis, 2019). In this paper, the efﬁciency of

92

both markets is measured in terms of the ability of the93

model in discovering proﬁtable betting opportunities

94

given both average and maximum market odds. The95

model is based on a modiﬁed version of the pi-rating96

system, which is a previously published football97

rating system (Constantinou & Fenton, 2013), that98

generates ratings that reﬂect team scoring abil-99

ity. The ratings are provided as input to a novel100

hybrid Bayesian Network (BN) model speciﬁcally

101

constructed to simulate the inﬂuential relationships

102

between possession, shots, and goals, to predict both

103

1X2 and AH outcomes.

104

A BN is a type of a probabilistic model intro- 105

duced by Pearl (Pearl, 1985) that consists of nodes 106

and arcs. Nodes represent variables and arcs repre- 107

sent conditional dependencies. A BN that consists 108

of both discrete and continuous variables, such as 109

the one constructed in this study, is called a hybrid 110

BN. Each variable has a corresponding Conditional 111

Probability Distribution (CPD) that captures the mag- 112

nitude as well as the shape of the relationship between 113

directly linked variables. If we assume that the arcs in 114

the BN represent inﬂuential relationships, then such a 115

model can be viewed as a causal graph and represents 116

a unique Directed Acyclic Graph (DAG) that can be 117

used for interventional analysis. Otherwise, the arcs 118

represent conditional dependencies that are not nec- 119

essarily causal relationships, and such a BN is not a 120

unique DAG but rather a Partial DAG that represents 121

an equivalence class of DAGs. For a quick introduc- 122

tion to BNs, with a focus on football examples, see 123

(Constantinou & Fenton, 2018). 124

Based on 13 English Premier League seasons 125

and betting simulations under different assumptions, 126

the ﬁndings reveal interesting differences as well as 127

similarities between the AH and 1X2 markets. Impor- 128

tantly, the AH market is found to share inefﬁciencies 129

with the traditional 1X2 market, and this provides 130

opportunities for ‘beating’ the market. The paper is 131

structured as follows: Section 2 describes the rules of 132

the AH betting market, Section 3 describes the model, 133

Section 4 covers the data and the process of model 134

ﬁtting, Section 5 presents the results, and Section 6 135

provides the concluding remarks. 136

2. Asian handicap betting rules 137

In what follows, the decimal odds system is used 138

(also known as European odds) for payoff in the event 139

of winning a bet. The decimal odds represent the total 140

return ratio of the stake; implying that the stake is 141

already included in the decimal number. For example, 142

a payoff of ‘3’ returns three times the stake; i.e., a bet 143

of £1 would return 1 ×3=£3(£2 proﬁt). Odds also 144

reﬂect probability that incorporates the bookmakers’ 145

proﬁt margin. An example from data is the Arsenal 146

versus Crystal Palace match played on 21/04/2019 147

with average 1X2 market odds 1.54, 4.44, 6.03, corre- 148

sponding to probabilities {64.94%, 22.52%, 16.58%}.149

Summing up these probabilities gives us 104.04%, 150

and the implied average proﬁt margin of 4.04%. 151

The AH betting market operates such that adver- 152

saries are handicapped according to their difference in 153

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A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 3

Table 1

The whole-goal AH outcome for different hypothetical scores between Arsenal and Crystal Palace

Arsenal Crystal Palace Score Handicap Settlement AH winner

goals goals difference score

1 0 1 –1 0 Void

1 1 0 –1 –1 Crystal Palace

3 1 2 –1 1 Arsenal

4 1 3 –1 2 Arsenal

0 0 0 –1 –1 Crystal Palace

0 1 –1 –1 –2 Crystal Palace

1 2 –1 –1 –2 Crystal Palace

strength. The term handicap means that one team is

154

assigned a hypothetical score (including fractional)155

advantage before the match is played. All types of156

AH betting offered by the bookmakers involve two157

possible outcomes. This means that the AH betting158

market reduces the possible match outcomes from

159

three (i.e., 1X2) to two. Each binary outcome corre-

160

sponds to each team winning, with the odds adjusted161

subject to the given handicap.

162

The standard AH betting involves assigning a163

hypothetical score advantage to the underdog. This164

represents the most common type of AH betting, and165

aims to make the contest equal. That is, the handicap

166

applied is the one1that optimises the odds, for both167

teams to win, towards 2 (or 50% chance of winning)168

and hence, it maximises the uniformity of the AH169

payoff distribution. While this represents the most170

popular type of bet, the AH market offers different171

types of score advantage, each of which we discuss172

below, including the possibility to assign a hypothet-173

ical score advantage to the favourite rather than the

174

underdog.175

The market offers three types of AH betting that176

need to be modelled explicitly into the model, as well177

as in the betting simulation. These are:

178

i. Whole goal handicap: A team is given a whole-179

goal handicap such as −1 or +1. In this case, the180

possibility of a draw is eliminated by removing

181

the draw outcome from the equation and nor-182

malising the probabilities of the residual two

183

outcomes to sum up to 1. If a handicap draw184

is observed, the bet is voided (refunded).

185

Let us revisit the example from data discussed186

at the beginning of this section, involving Arse-187

nal versus Crystal Palace with average 1X2188

market odds {1.54,4.44,6.03}. Arsenal was the189

1The other handicaps do not share the same market liquidity;

implying limited stakes and possibly also higher proﬁt margins,

for the bookmaker, due to lower competition.

strong favourite. The bookmakers introduced the 190

handicap of –1, which maximised the uniformity 191

of the AH distribution with odds {1.87,1.99}.192

The match ended 2-3; i.e., –1 for Arsenal. The 193

AH winner was Crystal Palace since it won the 194

match by one goal difference, which makes it 195

two goals difference given the handicap; i.e., this 196

made the settlement score, which is the match 197

result after the handicap is considered, equal to 198

2. Table 1 illustrates how the whole-goal AH 199

is determined based on other hypothetical score 200

lines between Arsenal and Crystal Palace. 201

ii. Half-goal handicap: A team is given a half-goal 202

handicap such as −1.5 or +1.5. Assuming a 203

match between Xand Yand a handicap of +1.5 204

(i.e., Xreceives a 1.5 goal advantage), and 205

that a bet is placed on X, the bet would win 206

as long Xdoes not lose by more than one goal 207

difference; otherwise the bet is lost. In this case, 208

the possibility of a draw is eliminated by the 209

handicap itself, since it is not possible for the 210

settlement score to be a draw. 211

An example from data is the Liverpool versus 212

Wolves match played on 12/05/2019 with 213

average 1X2 market odds {1.30, 5.62, 10.17}.214

Liverpool was the strong favourite. The book- 215

makers introduced the handicap of −1.5, which 216

maximised the uniformity of the AH distribution 217

with odds {1.91,1.95}. The match ended 2–0 218

(i.e., +2) in favour of Liverpool. The AH winner 219

was Liverpool since it won the match by two 220

goals difference; i.e., 0.5 goals more than the 221

handicap. This made the settlement score equal 222

to 0.5. Table 2 illustrates how the half-goal AH 223

is determined based on other hypothetical score 224

lines between Liverpool and Wolves. 225

iii. Quarter-goal handicap: A team is given a 226

quarter-goal handicap such as –0.25 or +0.25. 227

This type of handicap is, in fact, a com- 228

bined whole-goal and a half-goal handicap. 229

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4A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Table 2

The half-goal AH outcome for different hypothetical scores between Liverpool and Wolves

Liverpool Wolves Score Handicap Settlement AH winner

goals goals difference score

1 0 1 –1.5 –0.5 Wolves

1 1 0 –1.5 –1.5 Wolves

3 1 2 –1.5 0.5 Liverpool

4 1 3 –1.5 1.5 Liverpool

0 0 0 –1.5 –1.5 Wolves

0 1 –1 –1.5 –2.5 Wolves

1 2 –1 –1.5 –2.5 Wolves

Table 3

The quarter-goal AH outcome for different hypothetical scores between Fulham and Newcastle

Fulham Newcastle Score Handicap Settlement AH winner

Goals goals difference score

1 0 1 –0.25 (0 and –0.5) 1 and 0.5 Fulham

1 1 0 –0.25 (0 and –0.5) 0 and –0.5 Void and Newcastle

3 1 2 –0.25 (0 and –0.5) 2 and 1.5 Fulham

4 1 3 –0.25 (0 and –0.5) 3 and 2.5 Fulham

0 0 0 –0.25 (0 and –0.5) 0 and –0.5 Void and Newcastle

0 1 –1 –0.25 (0 and –0.5) –1 and –1.5 Newcastle

1 2 –1 –0.25 (0 and –0.5) –1 and –1.5 Newcastle

For example, if we bet £10 on the away team230

to win given AH –0.25 with odds 2 (i.e., 50%),

231

the stake would be divided between the nearest232

whole-goal and half-goal handicaps. That is, a

233

£5 bet will be placed on the away team to win234

given AH ±02with odds ∼2.5 (i.e., 40%) and235

another £5 bet on the away team to win given236

AH –0.5 with odds ∼1.66 (i.e., 60%). Note that237

the odds for the quarter-goal handicap reﬂect238

the average payoff, in terms of probability,

239

of the two nearest handicaps. Since this is a

240

combination of two bets, each bet is executed

241

independently. For example, a score of 0–0

242

would have resulted in voiding AH±0 (i.e.,243

£5 are returned) and winning AH –0.5 (i.e.,

244

£5×1.66 = £8.3 are returned).245

An example from data is the Fulham versus246

Newcastle match played on 12/05/2019 with

247

average 1X2 market odds {2.50,3.53,2.78}.248

Fulham was the weak favourite. The bookmak-249

ers introduced the handicap of –0.25, which250

maximised the uniformity of the AH distribu-

251

tion with odds {2.15,1.75}. The match ended252

0–4(i.e., –4) in favour of Newcastle. The AH

253

winner was Newcastle, since it won the match

254

by four goals difference; i.e., 4.25 goals more

255

than the handicap. This made the settlement256

2A zero-goal AH implies no handicap, but that there must be a

match winner; otherwise, the bet is voided.

score equal to –4.25. Table 3 illustrates how the 257

quarter-goal AH is determined based on other 258

hypothetical score lines between Fulham and 259

Newcastle. 260

3. The model 261

The overall model combines ratings with BNs. The 262

rating system captures the skill of teams over time, 263

and provides the ratings as an input into the BN model 264

which captures the magnitude of the relationships 265

between variables of interest. The two subsections 266

that follow describe the rating system and the BN 267

model respectively. 268

3.1. The rating system 269

The pi-rating is a football rating system that deter- 270

mines team ability based on the relative discrepancies 271

in scores between adversaries. It was ﬁrst introduced 272

in (Constantinou & Fenton, 2013) and thereafter used 273

in (Constantinou, 2018; Hubacek et al., 2018; 2019; 274

Van Cutsem, 2019; Wheatcroft, 2020). Modiﬁed ver- 275

sions of the pi-rating also formed part of the top two 276

performing models in the international competition 277

Machine Learning for Soccer (Constantinou, 2018; 278

Hubacek et al., 2018). This paper makes use of the 279

original pi-rating system (Constantinou & Fenton, 280

2013), with two modiﬁcations described below. 281

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 5

The pi-ratings assign a ‘home’ (H) and an ‘away’

(A) rating to each team, to account for team-speciﬁc

home advantage and away disadvantage. Therefore,

when a team Xplays against team Y, the match pre-

diction is determined by team’s Xrating Hversus

team’s Yrating A. The ratings are revised after each

match based on two learning rates: a) the learning

rate λwhich determines to what extent new match

results override previous match results in terms of the

impact in determining current team ratings, and b) the

learning rate γwhich determines to what extent per-

formances at home grounds inﬂuence a team’s away

rating and vice versa. Therefore, at the end of a match

between teams Xand Y, the new ratings at time tare

revised given the most recent ratings at time t−1as

follows:

X’s Hrating : Rt

XH =Rt−1

XH +eHλ

X’s Arating : Rt

XA =Rt−1

XA +γ(Rr

XH −Rt−1

XH )

Y’s Arating : Rt

YH =Rt−1

YH +eAλ

Y’s Hrating : Rt

YH =Rt−1

YH +γ(Rt

YA −Rt−1

YA )

where eis the error between the observed goal differ-

ence oand rating difference p which, for home

and away teams, is measured as follows:

eH=oH −pH and eA=oA −pA

respectively, where

oH =GoH −GoH and oA =GoA −GoH

pH =GpH −GpA and pA =GpA −GpH

where GoH and GoA are goals observed for home282

and away teams respectively, and similarly GpH and283

GpA are goals predicted for home and away teams.284

While the original pi-ratings represent a dimin-285

ished expectation of goal difference against the286

average opponent in the data, in this paper they287

represent the actual goal difference expectation.288

Speciﬁcally, the rating equation in this paper is289

simpliﬁed not to include the deterministic function

290

(e)=3×log10(1 +e) deﬁned in the original paper291

(Constantinou & Fenton, 2013), which is a function292

that diminishes the importance of each additional293

goal difference under the assumption that a win is

294

more important than increasing goal difference. The

295

justiﬁcation for this ﬁrst modiﬁcation is that, in AH,

296

we are only interested in goal differences and thus,297

the motivation here is to optimise for goal difference298

rather than the ability to win matches.299

The second modiﬁcation involves the learning rate

λ. In this paper, λis multiplied by kwhen a match

involves at least one team which had previously

played less than 38 matches, according to available

data. This modiﬁcation aims to increase the speed by

which team ratings converge for new teams during

their ﬁrst EPL season (each team plays 38 matches

in a season), and is expected to be especially impact-

ful during the very ﬁrst season in the data since, at

that point, all teams are considered ‘new’ by the rat-

ing. Therefore, the revised pi-ratings exclude ψ(e),

deﬁned above, and include k, as follows:

Rt

XH =Rt−1

XH +eHλk and Rt

YA =Rt−1

YA +eAλk

where k=3 for match instances in which both teams 300

had previously played less than 38 matches; other- 301

wise k=1. The parameter kwas optimised in terms 302

of minimising prediction error e. A limitation here 303

is that the kparameter was optimised given integer 304

inputs from 1 to 10. For future work, it is recom- 305

mended that the kparameter is optimised given real 306

numbers. 307

3.2. The BN model 308

The graph of a BN model can be automatically 309

discovered from data, determined by knowledge, 310

or a combination of the two. Learning the correct 311

graph of a BN from data remains a major challenge 312

in the ﬁelds of probabilistic machine learning and 313

causal discovery. While some structure learning algo- 314

rithms perform well with synthetic data, it is widely 315

acknowledged that this level of performance does not 316

extend to real-world data which typically incorporate 317

noise and latent confounders. 318

In disciplines like bioinformatics, applying struc- 319

ture learning algorithms can reveal new insights that 320

would otherwise remain unknown. However, these 321

algorithms are less effective in areas with access to 322

domain knowledge, such as in sports. As a result, the 323

BN model in this paper has had its graphical struc- 324

ture determined by the temporal fact that possession 325

inﬂuences the number of shots created, which in turn 326

inﬂuence the number of shots on target, and which 327

in turn inﬂuence the number of goals scored. Each 328

of these factors is also dependent on the level of rat- 329

ing difference between the two teams, as illustrated in 330

Fig. 1. This type of model can also be characterised as 331

a hierarchical Bayesian model of which the network 332

is the structural representation, and where the nodes 333

represent variables or hyperparameters of statistical 334

distributions. 335

Uncorrected Author Proof

6A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Fig. 1. The BN model. The Rating Difference (RD) is the only observable node in the network, determined by the pi-ratings, and represents

the difference between the home team’s prior home rating and the away team’s prior away rating Rt−1

XH −Rt−1

YA .

The temporal order of events in the BN graph336

naturally captures the importance of each event in pre-337

dicting goals scored. For example, the graph assumes338

that shots on target have a direct inﬂuence on goals339

scored, whereas possession has an indirect inﬂuence

340

and hence, while inﬂuential, it is assumed to be less

341

impactful than shots on target. While the temporal

342

order deﬁnes direct and indirect inﬂuences, note that

343

the magnitude of direct inﬂuences is still determined344

by data.

345

For each match, the prior ratings are retrieved and

the difference in team ratings is used as an input

into the BN model, which is a Hybrid BN model

consisting of both discrete and continuous variables,

designed in AgenaRisk (Agena, 2019). Speciﬁcally,

the actual input is the difference between prior home

and away ratings, and is passed to the BN model as

an observation to node Rating Difference (RD)inthe

form of

Rt−1

XH −Rt−1

YA

To ensure that the BN model is trained accurately 346

with respect to the rating data, the parameterisation 347

of the CPDs is also restricted to match instances in 348

which both teams had previously played at least 38 349

matches. All the residual variables in the BN model 350

are latent. Speciﬁcally, 351

i. The node RD, which represents the observable

rating difference between teams, is a mix-

ture of Gaussian probability density functions

∼N(μ, σ2); one for each state of node Rat-

ing Difference Level (RDL). Speciﬁcally, for

−∞ <RD<∞,

f(RD|,σ2, RDL)=

1

√2πσ2e−(RD−)2

2σ2

RDL

where parent RDL is a discrete distribution, μ352

is the average rating difference and σ2the vari- 353

ance of the rating differences. RDL consists 354

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 7

Table 4

Predetermined levels of rating difference

RDL 1 2 3 . . . 21 22 23

RD ≥2.095 ≥1.93 ≥1.765 intervals of ≥–1.205 ≥–1.37 < –1.37

& & 0.165 & &

< 2.095 < 1.93 rating < –1.04 < –1.205

Data points 55 68 107 . . . 93 58 55

of 23 states3, where each state corresponds to355

a pre-determined level of rating difference as

356

shown in Table 4. For example, the rating dif-357

ference level 3 is parameterised based on all358

historical match instances in which adversaries359

had rating difference RD =Rt−1

XH −Rt

YA rang-

360

ing from 1.765 to < 1.93. The granularity of

361

the 23 states has been chosen to ensure that for362

any combination of rating difference there is363

enough data points (more than 50) for a rea-364

sonably well informed prior.365

ii. The node P, which represents ball possession,

is a mixture of probability density functions

∼Beta(a, β); one for each state of RDL. Specif-

ically, for P∈[0,1],

f(P|a, β, RDL)=

Pa−1(1 −P)β−1

Beta(a, β)

RDL

where Beta(α, β)istheBeta function, αis the

366

ﬁrst shape parameter of the Beta function, also367

known as the alpha parameter, and represents

368

the number of minutes the home team is in pos-369

session of the ball, and βis the second shape

370

parameter of the Beta function, also known as371

the beta parameter, that represents the number372

of minutes the away team is in possession of

373

the ball. Thus, Preﬂects the possession rate

374

associated with the home team, over a Beta

375

distribution, whereas for the possession of the

376

away team the model assumes 1 −P.377

iii. The node p(SM), which represents the prob-378

ability to generate a shot per minute spent

379

in possession of the ball, is also a mixture380

of probability density functions ∼Beta(a, β)

381

given RDL, where αis the number of shots and

3The decision to discretise RDL represents a practical choice for

Hybrid BN modelling. In this case, discretising RD into RDL was

necessary to capture conditional Beta-Binomial relationships from

Possession to Goals scored, given the rating difference between

adversaries.

βis the number of minutes minus the number 382

of shots. 383

iv. The node S, which represents the expected

number of shots, is a Binomial probability mass

function ∼B(n, p),

f(k, n, p)−Pr(k|n, p)=

Pr(S=k)=n!

k!(n−k)! pk(1−p)n−k

where nrepresents the number of minutes in 384

possession of the ball deﬁned as4P×90, under 385

the assumption a match lasts 90 playable min- 386

utes, and pis p(SM); i.e., the probability to 387

generate a shot per minute spent in possession 388

of the ball, as deﬁned above. 389

v. The node p(ST ), which represents the proba- 390

bility for a shot to be on target, is also a mixture 391

of probability density functions ∼Beta(a, β)392

given RDL, where ais the number of shots on 393

target, and βis the number of shots off target; 394

i.e., total shots minus shots on target. 395

vi. The node ST, which represents the expected 396

number of shots on target, is also a Binomial 397

probability mass function ∼B(n, p), where n398

is the expected number of shots Sand pis the 399

probability for a shot to be on target p(ST ). 400

vii. The node p(G), which represents the prob- 401

ability to score a goal, is also a mixture 402

of probability density functions ∼Beta(a, β)403

given RDL, where αis the number of goals 404

scored, and βis the number of shots on target 405

successfully defended; i.e., total shots on target 406

minus goals scored. 407

viii. The node G, which represents the expected 408

number of goals scored, is also a Binomial 409

probability mass function ∼B(n, p), where n410

is the expected number of shots on target ST,411

and pis the probability to score a goal p(G). 412

ix. The node 1X2 is a discrete distribution with 413

states Home win,Draw, and Away win, deter- 414

mined by the distributions Gof both home (H)415

4For the away team (i.e., AT)itis(1−P×90.

Uncorrected Author Proof

8A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Table 5

The data variables used to train the rating system (R), the BN model (BN), and to simulate betting (B)

Variable Details Used in

Date The date of the match R

Home team The team playing at home grounds R

Away team The team playing at away grounds R

Match outcome (1X2) The outcome of the match in terms of home win, draw, or away win BN, B

Home possession rate The ball possession rate of the team playing at home BN

Away possession rate The ball possession rate of the team playing away BN

Home shots The number of shots created by the home team BN

Away shots The number of shots created by the away team BN

Home shots on target The number of shots on target created by the home team BN

Away shots on target The number of shots on target created by the away team BN

Home goals The number of goals scored by the home team R, BN, B

Away goals The number of goals scored by the away team R, BN, B

Team ratings The difference between home team and away team ratings BN

Handicap The AH on which the market odds are based B

1X2 odds The average and maximum (i.e., best available) bookmaker 1X2 odds B

AH odds The average and maximum (i.e., best available) bookmaker AH odds B

and away (A) teams; i.e., 1X2 = “Home win”

416

if GoH >G

oA,“Away win”ifGoH <G

oA,

417

“Draw” otherwise.

418

x. The node GD, which represents goal differ-419

ence, is simply GoH −GoA.

420

xi. The node AH represents a set of nodes corre-421

sponding to all the possible AH outcomes with422

state probabilities for home and away wins,

423

given GD, as deﬁned in Section 2.424

It should be clear by this point that for both home425

and away teams: a) nodes Pand p(SM) are hyperpa-426

rameters of Beta node S, b) nodes Sand p(ST ) are427

hyperparameters of Beta node ST, and c) nodes ST

428

and p(G) are hyperparameters of node G; effectively

429

creating a Beta-Binomial Hybrid BN process.

430

4. Data and model ﬁtting

431

4.1. Data432

The rating method, the BN model, and the bet-

433

ting simulation are based on data collected from

434

www.football-data.co.uk and manually recorded

435

from www.nowgoal.com. Table 5 speciﬁes which of

436

the data variables are used by the rating system, the437

BN model, and the betting simulation. For exam-438

ple, the rating system only requires information about439

goal data and hence, it only considers variables Date,440

Home team,Away team,Home goals, and Away goals.441

Since the ratings are used as an input into the BN442

model, they represent an additional BN variable and

443

at the same time make the BN model independent of

444

variables Date,Home team and Away team.

445

The data are based on the English Premier League 446

(EPL) seasons 1992/93 to 2018/19. However, AH 447

odds data were available only from season 2006/07 448

onwards, whereas ball possession data which is 449

needed by the BN model were available only from 450

season 2009/10 onwards. As a result, the rating sys- 451

tem is trained with up to 27 seasons of data, the BN is 452

parameterised with up to 10 seasons of data (since it 453

requires possession data), and the betting simulation 454

is performed over 13 seasons (since it requires AH 455

odds data). 456

4.2. Model ﬁtting 457

By deﬁnition, the ratings are developed in a tem- 458

poral manner. That is, for a match prediction at time 459

tthe model considers team ratings at time t−1. For 460

any match prediction, a team’s rating will always be 461

based on the most recent rating prior to the date of 462

the match under prediction, and a team’s rating will 463

always be based on past match results. 464

Conversely, the BN model functions as a machine 465

learning model independent of time and is validated 466

using leave-one-out cross validation (LOOCV). A 467

prediction between teams that have rating difference 468

Z, where Zis one of the 23 RDLs as deﬁned in Table 4, 469

is derived from all data matches with rating differ- 470

ence Z, excluding the match under prediction during 471

validation. 472

This combination of model parameterisation and 473

validation with a rating system and a BN model 474

is adopted by (Constantinou, 2018). The validation 475

approach is unconventional because the BN model 476

assumes no temporal relationships. When applied 477

to past matches, it generates predictions at time t478

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 9

based on the whole data set which may include future479

match results. The reason this approach works well,

480

without overestimating the future accuracy of the481

model, is because it does not matter whether the data482

comes from past or future. This is because the model483

assumes that the relationship between, for example,

484

shots on target and goals scored remains invariant485

over time for the average EPL team, and empirical486

results support this claim. These include:487

i. The results presented in Sections 5.2 and 5.3488

which show that predictive accuracy is consis-489

tent across all 13 seasons tested, including the490

three seasons 2006/07 to 2008/09 which did not491

form part of the BN’s training data;492

ii. The model in (Constantinou, 2018) which was

493

based on this approach and ranked 2nd in

494

the international Machine Learning for Soccer495

competition, with a prediction error consistent496

with the validation error.

497

The empirical support for the model extends to498

demonstrating that the model can produce good pre-499

dictions for matches between teams Xand Yeven500

when the prediction is derived from match data that

501

neither X nor Y participated in (Constantinou, 2018).

502

This claim is also supported by the results presented503

in this paper. Speciﬁcally, during Seasons 2006/07 to504

2008/09 the following teams have had their perfor-

505

mance determined by data that did not include any

506

of their matches: a) Shefﬁeld United, b) Charlton,

507

and c) Derby. The reason this occurred is because,508

as discussed above, the BN model was trained with

509

data from season 2009/10 onwards, which does not

510

include any match instances associated with these511

teams. Their performance in terms of possession,512

shots, shots on target and goals scored was derived513

by other similar match situations in terms of rating

514

difference between home and away teams.

515

This approach has advantages and disadvantages.516

The disadvantage is that, for those who are interested

517

in how such a model would have performed in the518

past, the results only approximate past performance

519

under the assumption the model would have been520

trained with at least the same amount of data as the521

test model. On the other hand, the advantage is that522

this approach allows us to preserve the sample size523

of the training data throughout validation, and this524

enables us to validate how the resulting model would525

have performed over multiple seasons without modi-

526

fying its parameterisation (excluding the removal of a

527

single sample; i.e. the match under assessment during

528

validation).

529

To understand why this is important, consider eval- 530

uating match instances ﬁve seasons in the past. A 531

temporal model would require the removal of the 532

ﬁve most recent seasons from the training data. This 533

would have led to limited samples for some of the 534

predetermined levels of rating difference shown in 535

Table 4. The limited data issue can only be over- 536

come by reducing the number of predetermined levels 537

of rating difference (i.e., the dimensionality of the 538

model); but doing so would produce a different model 539

than the one described. Instead, the approach adopted 540

by (Constantinou, 2018) enables us to address the 541

temporal aspect of the problem through the ratings 542

and to preserve the ﬁtting of the BN across all sea- 543

sons tested; effectively enabling us to test the current 544

parameterised model on multiple seasons indepen- 545

dent of time. 546

5. Results 547

The results are reported in terms of rating (i.e., goal 548

difference) error, predictive accuracy and proﬁtabil- 549

ity. Speciﬁcally, Section 5.1 assesses the accuracy of 550

the modiﬁed pi-ratings in terms of expected goal dif- 551

ference error, Section 5.2 assesses the accuracy of 552

the overall model in predicting both AH and 1X2 553

outcomes, and Section 5.3 assesses the capability of 554

the model in terms proﬁtability in both 1X2 and AH 555

markets. 556

5.1. Pi-ratings accuracy and overall model 557

ﬁtting 558

Figure 2 shows that the optimal λand ␥parameters, 559

that minimise the goal difference error as deﬁned in 560

Section 3, are λ= 0.018 and ␥= 0.7. Note that while 561

the results are based on training data from seasons 562

1992/93 to 2018/19, the optimisation is restricted to 563

match instances in which both teams had previously 564

played at least 38 matches; a total of 9,073 match 565

instances. This is to ensure that the model is optimised 566

on matches in which both teams have had their ratings 567

developed by at least one football season. 568

The optimal learning rates are fairly consistent 569

with those reported in (Constantinou & Fenton, 2013) 570

(i.e., λ= 0.035 and ␥= 0.7) on the basis of minimis- 571

ing goal difference error over ﬁve EPL seasons, with 572

those reported in (Van Cutsem, 2019) (i.e., where 573

λ= 0.06 and ␥= 0.6) on the basis of minimising mean 574

squared goal difference error over eight EPL sea- 575

sons, with those reported in (Constantinou, 2018), 576

Uncorrected Author Proof

10 A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Fig. 2. The optimal modiﬁed pi-rating learning rates and associated prediction error e,givenk= 3, are λ=0.018 and ␥= 0.7. The results are

based on training data from seasons 1992/93 to 2018/19. The optimisation is restricted to match instances in which both teams had previously

played at least 38 matches; a total of 9,073 match instances.

λ= 0.054 and ␥= 0.79, on the basis of minimising577

the Rank Probability Score (RPS) error metric (Con-578

stantinou & Fenton, 2012) over multiple leagues579

worldwide, and with those reported in (Hubacek et580

al., 2018), λ= 0.06 and ␥= 0.5, where pi-ratings had

581

been used in conjunction with Gradient boosted trees582

parameters to minimise RPS over multiple leagues583

worldwide.

584

However, note that the optimal learning rate λ585

is lower in this study, and this is likely due to the

586

modiﬁcation that performs more aggressive rating587

revisions to the ﬁrst 38 matches of each team, since

588

it is intended to improve the speed of rating conver-

589

gence. Interestingly, the overall mean goal difference

590

error shown in Fig. 2, e= 1.2283 (or e2= 1.509), is591

considerably lower than those reported in (Constanti-

592

nou & Fenton, 2013) and (Van Cutsem, 2019), where593

e2= 2.625 and e2= 2.66 respectively, and this sug-

594

gests that the modiﬁcations have had a positiveimpact

595

on the ratings.596

Figure 3 illustrates the expected goal difference597

for each of the 23 rating difference levels. Level 23598

represents the highest rating discrepancy in favour of

599

the away team, where the average expectation of the600

match is a score difference of –1.38 (or 1.38 goals in601

favour of the away team), and level 1 represents the

602

highest rating discrepancy in favour of the home team603

with an expected score difference of 2.18 (or 2.18

604

goals in favour of the home team). The graph reveals

605

a linear relationship between rating discrepancy and606

score discrepancy.

Fig. 3. Sensitivity analysis between the 23 states of RDL node and

the expected goal difference, with linear trendline superimposed

as a dashed red line.

As shown in Table 4, the granularity of the 23 607

intervals was selected to ensure that for any rating 608

difference state there are at least 50 data points for a 609

reasonably well-informed prior of observed goal dif- 610

ference. As with any discretised variable, different 611

splits produce slightly different results. In the case 612

of the RDL distribution, any changes in discretisation 613

will remain faithful to the linear relationship illus- 614

trated in Fig. 3; implying that we should expect minor 615

changes to the interval averages as long as the num- 616

ber of splits remains invariant and data points for each 617

interval are maintained above 50. 618

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 11

Table 6

The discrepancy in prediction error efor the different combinations of λand ␥, relative to the optimal inputs of λ=0.018 and ␥= 0.7. Darker

green cells represent lower discrepancy, whereas darker red cells represent higher discrepancy. Recommended inputs for λand ␥are the

ones that generate up to 0.01% discrepancy

Any minor model amendment is naturally expected619

to have minor impact on the predicted probabili-620

ties, and any minor impact is expected to have some621

inﬂuence on the results based on small discrepancies622

between predicted and published market odds (i.e.,623

small θvalues as deﬁned later in subsection 5.3, such

624

as θ=1or2). However, no changes are expected625

for larger discrepancies. Since the conclusions in this

626

paper are not driven by results that are based on such627

small differences between predicted and observed628

odds, any minor modiﬁcation is not expected to alter629

concluding remarks.

630

Table 6 presents the discrepancy in prediction631

error efor the different hyperparameter combinations632

of λand ␥, relative to the prediction error gener-

633

ated by the optimal inputs of λ= 0.018 and ␥= 0.7.634

Recommended values for λand ␥, that could be

635

considered as an alternative to the optimal values636

of λ= 0.018 and ␥= 0.7, are the ones that gener-637

ate up to 0.01% discrepancy in prediction error. A638

total of 31 different combinations, excluding the639

optimal combination, generate discrepancy within

640

the 0.01% threshold. By keeping the value for λ

641

static, the recommended hyperparameter combina-

642

tions are a) λ= 0.017 and ␥=0.65–0.73, b) λ= 0.018

643

and ␥= 0.65–0.74, c) λ= 0.019 and ␥= 0.65–0.74,644

and d) λ= 0.02 and ␥= 0.7–0.72.645

5.2.. Predictive accuracy

646

Predictive accuracy is measured for both 1X2 and

647

AH distributions. The Brier Score is used to mea-648

sure the accuracy of the binary AH outcome, and649

the RPS metric (Constantinou & Fenton, 2012) is650

used to measure the accuracy of the multinomial 1X2651

Table 7

Predictive accuracy across all seasons, based on the Rank Proba-

bility Score (RPS) for multinomial 1X2 predictions and the Brier

Score (BS) for binary AH predictions. Lower score indicates higher

predictive accuracy for both RPS and BS

Season RPS BS

(1X2 accuracy) (AH accuracy)

2006/07 0.197 0.252

2007/08 0.184 0.248

2008/09 0.192 0.229

2009/10 0.188 0.199

2010/11 0.202 0.248

2011/12 0.205 0.257

2012/13 0.191 0.258

2013/14 0.195 0.249

2014/15 0.199 0.254

2015/16 0.213 0.254

2016/17 0.191 0.267

2017/18 0.192 0.253

2018/19 0.191 0.260

Overall 0.195 0.248

distribution. The RPS can be viewed as a Brier Score 652

extended to multinomial ordinal distributions. 653

Table 7 shows that the RPS error for the 1X2 654

outcomes ranges from 0.184 to 0.213 with an aver- 655

age RPS of 0.195 across all 13 EPL seasons. This 656

result compares well relative to previous studies that 657

assumed pi-ratings. Speciﬁcally, in (Constantinou, 658

2018) the RPS ranged from 0.187 to 0.236 for 52 659

different leagues worldwide, with an average RPS 660

of 0.211 at validation, an average RPS of 0.203 for 661

EPL matches, and an average RPS of 0.208 in the 662

competition. Similarly, the average RPS was ∼0.2 in 663

(Hubacek et al., 2018), according to Fig. 3, and 0.206 664

in the competition. 665

These results are consistent with those reported 666

in Section 5.1, which show that the overall error e667

Uncorrected Author Proof

12 A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Fig. 4. Investigating the prediction shift over time, with reference to the four main variables of the BN model. The shift is measured in terms

of the expected value of the speciﬁed distribution, and by adding a football season’s worth of data, to the training data set, at a time.

optimises lower in this study; i.e., the ratings more

668

accurately predict score difference. The results from

669

proﬁtability presented in Section 5.3 are also consis-

670

tent with these ﬁndings.

671

5.2.1. Time-series analysis and sample size672

requirements673

The model is trained with data the covers 13 years674

of data, and not all the variables could be measured675

throughout this period. Still, the model seems to work676

well without evidence of bias. This subsection inves-677

tigates whether the variables used in the model show

678

any drift over time that the model might have ﬁl-679

tered out. Moreover, because the dimensionality of

680

the model has been adjusted relative to the available

681

sample size, this subsection also reports the sample

682

size required for the model priors to be well informed683

by data.684

Figure 4 presents the results from time-series anal-685

ysis in investigating potential shifts in the predictive686

outputs of the model over time. The analysis is per-

687

formed by increasing the training data set by a single688

football season’s worth of data at a time, and the shift689

is measured in terms of changes in the expected value

690

of the given distribution. As shown in Fig. 4, the anal-691

ysis focuses on the four main variables of the BN692

model; namely possession, shots, shots on target, and 693

goals scored. Moreover, the different levels of rating 694

difference (refer to Table 4) are categorised into four 695

groups, and are measured with reference to the 10 696

speciﬁed seasons. 697

The reason the ﬁrst three, out of the 13, seasons 698

are not considered here is because (as later shown in 699

Table 8) the BN model was not trained with data sam- 700

ples from the ﬁrst two seasons, and this also means 701

that any results obtained during the third season rely 702

on very low samples. As previously discussed in sub- 703

section 3.2, the reason the ﬁrst two seasons are not 704

considered by the BN model is because the BN is 705

trained with match instances in which both teams 706

had previously played at least 38 matches, to allow 707

for the pi-ratings to converge to reasonably accurate 708

estimates before considered for model training. 709

The results are discussed with reference to Table 8, 710

which presents the available samples for each of the 711

23 levels of rating difference, after each subsequent 712

league data set is added to the training data set that 713

was used to learn the BN model. The results show that 714

many of the shifts occur in the ﬁrst few seasons, and 715

this is reasonable since the ﬁrst seasons are the ones 716

which rely on fewer samples. Shifts are also observed 717

after adding the most recent leagues, but these shifts 718

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 13

Table 8

The number of data samples observed in each of the 23 intervals of Rating Difference Level (RDL) after each subsequent league data set

is added to the training data. Darker red cells represent sample size less than 19 (equivalent to less than half a season’s data), orange cells

represent sample size less than 38 (equivalent to more than half, and less than, a season’s data), and white cells represent sufﬁcient

sample size

are largely restricted to the outputs of possession and719

shots on target, and involve matches with a level of720

rating difference between 1 to 6. The most important721

output of the model, which is the goal difference,

722

remains essentially unchanged over time. Therefore,723

while it is reasonable to assume that shifts in playing724

style might occur naturally from season to season,

725

collectively these shifts appear to have no meaning-

726

ful impact on the prediction of goal difference, from727

which the 1X2 and AH distributions are formed.728

Lastly, the results in Table 8 suggest that the model

729

described in this paper should be trained with at least

730

ﬁve seasons of league data, to ensure that the model731

priors are well informed by data. Because the model

732

is trained with publicly available data, and which only733

increases over time, there is no motivation to use less

734

data than what is currently available and hence, this735

requirement should not be viewed as a limitation.736

5.3. Proﬁtability737

The assessment of proﬁtability is based on 13 EPL

738

seasons and considers both the average and the best739

available (maximum) market odds. The simulation

740

is evaluated both in terms of maximising proﬁt as

741

well as the Return On Investment (ROI). A stan-

742

dard betting strategy is used where ﬁxed singe-unit

743

bets (e.g., £1) are placed on 1X2 and AH outcomes744

with payoff that is higher than the model’s estimated745

unbiased payoff by at least θ, where θis the discrep-746

ancy between the predicted probability and the payoff747

probability. For example, if the model predicts 51%748

and the bookmakers’ payoff for that event is 50% (i.e.,749

odds of 2), then θ=1; i.e., the bookmakers pay 1%750

more than the model’s estimate. The betting simula- 751

tion is performed across all payoff decision thresholds 752

θ, for both 1X2 and AH outcomes. The results are ﬁrst 753

discussed in terms of 1X2 betting performance with 754

reference to Tables 9, 10 and 11; then in terms of AH 755

betting performance with reference to Tables 12, 13 756

and 14. 757

Table 9 presents the proﬁtability generated by 758

1X2 bets over all possible payoff decision thresh- 759

olds θ, assuming static θacross all 13 seasons, for 760

both average and maximum market odds. Unsurpris- 761

ingly, the results show that it is much easier for the 762

model to generate proﬁt when taking advantage of the 763

maximum market odds. Moreover, low thresholds θ764

(i.e., when the predictions are roughly in agreement 765

with market odds) are not proﬁtable. Interestingly, 766

ROI maximises at much higher thresholds θcom- 767

pared to proﬁt; i.e., proﬁt maximises at 8% and 9% 768

whereas ROI at 18% and 16%, for average and maxi- 769

mum market odds respectively. This is because lower 770

thresholds θgenerate a higher number of bets which 771

can generate higher proﬁt even if ROI is lower. 772

Tables 10 and 11 show how the proﬁtability 773

changes when we consider the threshold θthat max- 774

imises ROI (Table 10) or proﬁt (Table 11) per season, 775

rather than considering a static θacross all seasons 776

(Table 9), for both average and maximum market 777

odds. Proﬁt, once more, tends to maximise on lower 778

thresholds θcompared to ROI. As an example, Table 779

A1 provides the information used during the betting 780

simulation to assess proﬁtability for 1X2 outcomes, 781

based on average odds of season 2010/11 as shown 782

in Table 10. 783

The results show that the optimal threshold θvaries 784

dramatically between seasons, and there is much to 785

Uncorrected Author Proof

14 A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Table 9

Proﬁtability based on average (left) and maximum (right) market odds for 1X2 bets simulated over 13 EPL seasons; from 2006/09 to 2018/19

θAverage market odds Maximum market odds

Bets Bets Odds Win Returns Proﬁt ROI Bets Bets Odds Win Returns Proﬁt ROI

won Rate won Rate

0% 4334 1290 3.06 29.8% 3942.5 –391.5 –9.03% 4938 1514 3.22 30.7% 4879.0 –59.01 –1.20%

1% 3712 1105 3.06 29.8% 3375.9 –336.1 –9.05% 4794 1468 3.24 30.6% 4755.3 –38.67 –0.81%

2% 3109 937 3.11 30.1% 2909.9 –199.2 –6.41% 4305 1307 3.19 30.4% 4167.1 –137.91 –3.20%

3% 2538 775 3.09 30.5% 2397.8 –140.2 –5.52% 3658 1122 3.23 30.7% 3623.2 –34.81 –0.95%

4% 2072 643 3.09 31.0% 1987.2 –84.9 –4.10% 3067 950 3.26 31.0% 3096.4 29.38 0.96%

5% 1699 524 3.15 30.8% 1650.9 –48.1 –2.83% 2553 802 3.27 31.4% 2620.0 67.02 2.63%

6% 1339 415 3.19 31.0% 1325.3 –13.7 –1.02% 2076 656 3.29 31.6% 2157.8 81.77 3.94%

7% 1036 329 3.26 31.8% 1074.0 38.0 3.67% 1682 526 3.34 31.3% 1756.3 74.29 4.42%

8% 814 260 3.30 31.9% 858.7 44.7 5.49% 1345 422 3.42 31.4% 1444.6 99.56 7.40%

9% 612 198 3.17 32.4% 626.8 14.8 2.42% 1049 352 3.48 33.6% 1226.2 177.18 16.89%

10% 452 148 3.13 32.7% 462.6 10.6 2.34% 807 267 3.40 33.1% 906.6 99.63 12.35%

11% 320 101 3.12 31.6% 315.2 –4.8 –1.51% 608 199 3.28 32.7% 652.3 44.29 7.28%

12% 241 75 3.16 31.1% 236.9 –4.1 –1.68% 451 155 3.30 34.4% 511.8 60.78 13.48%

13% 191 65 3.27 34.0% 212.3 21.3 11.13% 324 108 3.27 33.3% 352.7 28.65 8.84%

14% 143 50 3.45 35.0% 172.5 29.5 20.59% 241 80 3.38 33.2% 270.4 29.37 12.19%

15% 103 37 3.10 35.9% 114.8 11.8 11.48% 184 62 3.48 33.7% 216.0 31.98 17.38%

16% 76 27 3.40 35.5% 91.8 15.8 20.84% 132 46 3.48 34.8% 160.0 27.99 21.20%

17% 51 17 3.64 33.3% 61.8 10.8 21.20% 101 37 3.31 36.6% 122.3 21.31 21.10%

18% 37 12 3.81 32.4% 45.7 8.7 23.59% 76 26 3.46 34.2% 89.9 13.91 18.30%

19% 24 7 3.10 29.2% 21.7 –2.3 –9.58% 52 18 3.82 34.6% 68.7 16.67 32.06%

20% 19 5 3.44 26.3% 17.2 –1.8 –9.37% 34 11 3.51 32.4% 38.6 4.61 13.56%

Table 10

The payoff discrepancies θthat maximise ROI per season (in yellow shading), based on 1X2 bets and for both average (left) and maximum

(right) market odds. The optimal θdiscrepancy is chosen over all θthat generate at least 30 bets in a single season

Season Average market odds Maximum market odds

θBets Bets Odds Win Returns Proﬁt ROI θBets Bets Odds Win Returns Proﬁt ROI

won Rate won Rate

2006/07 8% 34 13 3.4 38.2% 43.7 9.66 28.41% 10% 39 16 3.4 41.0% 54.6 15.6 40.00%

2007/08 7% 63 21 2.8 33.3% 58.4 –4.56 –7.24% 11% 41 15 3.1 36.6% 47.1 6.11 14.90%

2008/09 3% 181 72 3.1 39.8% 224.3 43.3 23.92% 5% 202 77 3.5 38.1% 268.0 66.04 32.69%

2009/10 2% 242 57 3.5 23.6% 199.9 –42.09 –17.39% 13% 30 9 3.7 30.0% 33.0 3.02 10.07%

2010/11 10% 33 17 2.8 51.5% 48.2 15.18 46.00% 10% 59 29 3.0 49.2% 86.6 27.56 46.71%

2011/12 0% 326 103 3.3 31.6% 338.7 12.68 3.89% 1% 370 119 3.6 32.2% 433.4 63.4 17.14%

2012/13 7% 65 22 3.5 33.8% 77.9 12.89 19.83% 9% 71 25 3.8 35.2% 94.5 23.45 33.03%

2013/14 10% 30 10 3.7 33.3% 36.6 6.59 21.97% 12% 33 13 3.8 39.4% 49.7 16.7 50.61%

2014/15 7% 73 33 3.1 45.2% 101.6 28.58 39.15% 9% 72 33 3.3 45.8% 109.5 37.51 52.10%

2015/16 10% 65 25 3.0 38.5% 74.7 9.65 14.85% 12% 57 21 3.2 36.8% 67.7 10.71 18.79%

2016/17 9% 58 14 4.8 24.1% 67.7 9.72 16.76% 10% 80 19 5.0 23.8% 94.7 14.73 18.41%

2017/18 8% 85 24 3.9 28.2% 93.3 8.28 9.74% 13% 31 11 3.5 35.5% 38.4 7.43 23.97%

2018/19 13% 32 11 3.5 34.4% 38.3 6.28 19.63% 14% 34 11 3.7 32.4% 40.4 6.36 18.71%

Overall 4.35% 1287 422 3.38 32.79% 1403.16 116.16 9.03% 8.01% 1119 398 3.59 35.57% 1417.62 298.62 26.69%

be gained by identifying the optimal θ. However, the786

high variance of θsuggests that is not reasonable to

787

expect that we will be able to successfully predict the788

optimal value of θbefore a season starts. Moreover,789

while the results are restricted to cases with 30 or

790

more bets in a single season, it is clear that in many

791

cases the sample size remains insufﬁcient for deriving

792

reliable and robust single-season conclusions. This 793

means that the maximised proﬁtability presented in 794

Tables 10 and 11 is not a realistic expectation of real- 795

world performance; only Table 9 is. These results are 796

important because they highlight the danger when 797

optimising models based on the results of a single 798

season (or few seasons), which is often the case in the 799

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 15

Table 11

The payoff discrepancies θthat maximise proﬁt per season (in yellow shading), based on 1X2 bets and for both average (left) and maximum

(right) market odds. The optimal θdiscrepancy is chosen over all θthat generate at least 30 bets in a single season

Season Average bookmaker odds Maximum bookmaker odds

θBets Bets Odds Win Returns Proﬁt ROI θBets Bets Odds Win Returns Proﬁt ROI

won Rate won Rate

2006/07 8% 34 13 3.4 38.2% 43.7 9.66 28.41% 5% 166 56 3.4 33.7% 188.4 22.39 13.49%

2007/08 8% 45 14 2.9 31.1% 40.8 –4.16 –9.24% 11% 41 15 3.1 36.6% 47.1 6.11 14.90%

2008/09 3% 181 72 3.1 39.8% 224.3 43.3 23.92% 5% 202 77 3.5 38.1% 268.0 66.04 32.69%

2009/10 10% 41 11 3.1 26.8% 33.7 –7.28 –17.76% 13% 30 9 3.7 30.0% 33.0 3.02 10.07%

2010/11 8% 59 27 2.8 45.8% 75.9 16.85 28.56% 6% 164 63 3.1 38.4% 197.2 33.19 20.24%

2011/12 0% 326 103 3.3 31.6% 338.7 12.68 3.89% 1% 370 119 3.6 32.2% 433.4 63.4 17.14%

2012/13 7% 65 22 3.5 33.8% 77.9 12.89 19.83% 9% 71 25 3.8 35.2% 94.5 23.45 33.03%

2013/14 5% 123 44 3.3 35.8% 143.2 20.21 16.43% 0% 380 127 3.4 33.4% 438.1 58.11 15.29%

2014/15 7% 73 33 3.1 45.2% 101.6 28.58 39.15% 8% 91 40 3.3 44.0% 133.2 42.22 46.40%

2015/16 8% 94 34 3.1 36.2% 104.7 10.66 11.34% 1% 371 130 3.1 35.0% 400.4 29.41 7.93%

2016/17 9% 58 14 4.8 24.1% 67.7 9.72 16.76% 10% 80 19 5.0 23.8% 94.7 14.73 18.41%

2017/18 8% 85 24 3.9 28.2% 93.3 8.28 9.74% 9% 93 26 4.2 28.0% 108.0 15.04 16.17%

2018/19 4% 188 67 3.1 35.6% 204.5 16.46 8.76% 5% 206 73 3.0 35.4% 222.3 16.25 7.89%

Overall 4.62% 1372 478 3.28 34.84% 1549.85 177.85 12.96% 4.61% 2265 779 3.52 34.39% 2658.36 393.36 17.37%

Table 12

Proﬁtability based on average (left) and maximum (right) market odds for AH bets simulated over 13 EPL seasons; from 2006/07 to 2018/19

θAverage market odds Maximum market odds

Bets Bets Odds Win Returns Proﬁt ROI Bets Bets Odds Win Returns Proﬁt ROI

won Rate won Rate

0% 3914 2329 1.62 59.5% 3762.8 –151.22 –3.86% 4703 2830 1.66 60.2% 4707.0 4.01 0.09%

1% 3471 2051 1.62 59.1% 3326.5 –144.52 –4.16% 4228 2527 1.66 59.8% 4203.2 –24.76 –0.59%

2% 3064 1817 1.61 59.3% 2929.4 –134.61 –4.39% 3788 2247 1.66 59.3% 3736.8 –51.25 –1.35%

3% 2665 1584 1.61 59.4% 2554.4 –110.65 –4.15% 3375 1998 1.67 59.2% 3340.7 –34.28 –1.02%

4% 2286 1357 1.62 59.4% 2201.3 –84.71 –3.71% 2974 1766 1.67 59.4% 2949.3 –24.73 –0.83%

5% 1932 1157 1.62 59.9% 1873.4 –58.56 –3.03% 2586 1526 1.67 59.0% 2545.8 –40.17 –1.55%

6% 1640 984 1.63 60.0% 1600.9 –39.09 –2.38% 2192 1304 1.67 59.5% 2171.3 –20.74 –0.95%

7% 1386 837 1.63 60.4% 1368.5 –17.52 –1.26% 1883 1124 1.67 59.7% 1877.9 –5.15 –0.27%

8% 1135 688 1.63 60.6% 1124.2 –10.78 –0.95% 1587 955 1.68 60.2% 1600.0 13.03 0.82%

9% 936 572 1.64 61.1% 938.5 2.45 0.26% 1307 790 1.68 60.4% 1325.1 18.06 1.38%

10% 765 460 1.65 60.1% 759.0 –5.98 –0.78% 1080 658 1.69 60.9% 1111.7 31.73 2.94%

11% 614 368 1.66 59.9% 612.1 –1.89 –0.31% 896 541 1.70 60.4% 920.6 24.61 2.75%

12% 488 292 1.65 59.8% 480.9 –7.12 –1.46% 728 427 1.71 58.7% 728.3 0.25 0.03%

13% 394 233 1.63 59.1% 379.7 –14.29 –3.63% 571 330 1.71 57.8% 564.5 –6.46 –1.13%

14% 312 191 1.65 61.2% 314.5 2.49 0.80% 463 279 1.71 60.3% 478.3 15.26 3.30%

15% 250 155 1.65 62.0% 255.1 5.07 2.03% 369 222 1.70 60.2% 376.8 7.76 2.10%

16% 201 126 1.67 62.7% 210.2 9.20 4.57% 300 183 1.68 61.0% 308.0 8.02 2.67%

17% 138 83 1.67 60.1% 138.8 0.80 0.58% 237 149 1.70 62.9% 253.7 16.75 7.07%

18% 113 70 1.68 61.9% 117.7 4.72 4.18% 180 108 1.69 60.0% 182.1 2.12 1.18%

19% 87 53 1.66 60.9% 87.9 0.86 0.99% 136 85 1.70 62.5% 144.5 8.48 6.24%

20% 62 40 1.74 64.5% 69.7 7.65 12.34% 108 69 1.72 63.9% 118.8 10.81 10.01%

21% 44 31 1.74 70.5% 54.0 10.01 22.75% 78 51 1.73 65.4% 88.3 10.34 13.26%

22% 30 20 1.87 66.7% 37.3 7.31 24.37% 60 40 1.77 66.7% 70.8 10.81 18.02%

23% 24 17 1.85 70.8% 31.4 7.37 30.71% 41 26 1.95 63.4% 50.7 9.70 23.66%

24% 16 11 1.84 68.8% 20.2 4.20 26.25% 33 21 2.05 63.6% 43.0 9.98 30.24%

25% 11 8 1.81 72.7% 14.5 3.45 31.36% 27 19 2.04 70.4% 38.7 11.73 43.44%

literature. This outcome is also discussed in Section800

6, point iv.

801

Interestingly, while maximising proﬁt per season

802

is guaranteed to also maximise proﬁt over all seasons

803

(Table 11), the same does not apply to ROI (see 804

Table 10 and compare it to Table 11). That is, opti- 805

mising the betting strategy for maximum ROI per 806

season does not necessarily imply that ROI will 807

Uncorrected Author Proof

16 A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Table 13

The payoff discrepancies θthat maximise ROI per season (in yellow shading), based on AH bets and for both average (left) and maximum

(right) market odds. The optimal θdiscrepancy is chosen over all θthat generate at least 30 bets in a single season

Season Average market odds Maximum market odds

θBets Bets Odds Win Returns Proﬁt ROI θBets Bets Odds Win Returns Proﬁt ROI

won Rate won Rate

s

2006/07 7% 66 46 1.6 69.7% 74.9 8.945 13.55% 8% 80 57 1.6 71.3% 91.4 11.385 14.23%

2007/08 8% 71 47 1.5 66.2% 72.2 1.165 1.64% 10% 70 47 1.5 67.1% 72.7 2.745 3.92%

2008/09 12% 30 22 1.8 73.3% 38.8 8.78 29.27% 14% 31 21 1.9 67.7% 40.9 9.88 31.87%

2009/10 11% 52 33 1.5 63.5% 49.6 –2.39 –4.60% 16% 31 22 1.4 71.0% 31.6 0.61 1.97%

2010/11 11% 43 28 1.8 65.1% 51.8 8.75 20.35% 14% 33 22 1.8 66.7% 39.0 6.01 18.21%

2011/12 10% 51 29 1.8 56.9% 51.7 0.74 1.45% 14% 34 20 1.8 58.8% 36.0 1.98 5.82%

2012/13 9% 55 40 1.6 72.7% 65.6 10.55 19.18% 13% 30 23 1.5 76.7% 35.4 5.415 18.05%

2013/14 10% 58 38 1.7 65.5% 64.0 5.96 10.28% 14% 33 22 1.7 66.7% 37.9 4.92 14.91%

2014/15 8% 64 39 1.8 60.9% 68.7 4.68 7.31% 9% 72 46 1.8 63.9% 83.4 11.41 15.85%

2015/16 8% 109 68 1.6 62.4% 107.2 –1.76 –1.61% 11% 87 55 1.7 63.2% 91.7 4.745 5.45%

2016/17 15% 33 21 1.6 63.6% 33.4 0.405 1.23% 17% 33 21 1.6 63.6% 34.2 1.18 3.58%

2017/18 14% 41 25 1.7 61.0% 43.3 2.29 5.59% 17% 31 20 1.8 64.5% 36.3 5.305 17.11%

2018/19 2% 267 163 1.7 61.0% 272.5 5.485 2.05% 4% 256 156 1.7 60.9% 266.6 10.59 4.14%

Overall 7.45% 940 599 1.66 63.7% 993.6 53.6 5.70% 10.04% 821 532 1.68 64.8% 897.2 76.175 9.28%

Table 14

The payoff discrepancies θthat maximise proﬁt per season (in yellow shading), based on AH bets and for both average (left) and maximum

Season Average market odds Maximum market odds

θBets Bets Odds Win Returns Proﬁt ROI θBets Bets Odds Win Returns Proﬁt ROI

won Rate won Rate

2006/07 1% 227 151 1.6 66.5% 244.2 17.22 7.59% 1% 308 204 1.7 66.2% 339.5 31.46 10.21%

2007/08 8% 71 47 1.5 66.2% 72.2 1.165 1.64% 8% 103 68 1.6 66.0% 106.6 3.64 3.53%

2008/09 9% 67 46 1.8 68.7% 80.9 13.89 20.73% 6% 168 102 1.8 60.7% 188.5 20.47 12.18%

2009/10 12% 39 25 1.5 64.1% 37.1 –1.89 –4.85% 16% 31 22 1.4 71.0% 31.6 0.61 1.97%

2010/11 11% 43 28 1.8 65.1% 51.8 8.75 20.35% 0% 356 220 1.7 61.8% 376.7 20.68 5.81%

2011/12 10% 51 29 1.8 56.9% 51.7 0.74 1.45% 14% 34 20 1.8 58.8% 36.0 1.98 5.82%

2012/13 9% 55 40 1.6 72.7% 65.6 10.55 19.18% 6% 154 104 1.6 67.5% 163.0 9.02 5.86%

2013/14 4% 172 111 1.7 64.5% 188.9 16.9 9.83% 6% 161 104 1.8 64.6% 183.0 22.035 13.69%

2014/15 6% 95 58 1.8 61.1% 101.6 6.64 6.99% 9% 72 46 1.8 63.9% 83.4 11.41 15.85%

2015/16 8% 109 68 1.6 62.4% 107.2 –1.76 –1.61% 11% 87 55 1.7 63.2% 91.7 4.745 5.45%

2016/17 15% 33 21 1.6 63.6% 33.4 0.405 1.23% 17% 33 21 1.6 63.6% 34.2 1.18 3.58%

2017/18 14% 41 25 1.7 61.0% 43.3 2.29 5.59% 0% 369 240 1.6 65.0% 382.6 13.565 3.68%

2018/19 2% 267 163 1.7 61.0% 272.5 5.485 2.05% 2% 318 197 1.7 61.9% 330.5 12.515 3.94%

Overall 5.57% 1270 812 1.66 63.9% 1350.4 80.385 6.33% 5.55% 2194 1403 1.69 63.9% 2347.3 153.31 6.99%

maximise across all seasons. Table 10 shows that808

even though ROI is maximised for each season

809

independently, the overall ROI across all 13 sea-810

sons is 9.03% in the case of average odds, which811

is notably lower compared to the respective overall812

ROI of 12.96% in Table 11. However, this observa-813

tion does not extend to the case of maximum market814

odds. This outcome is also discussed in Section 6,

815

point vi.816

Tables 12, 13, and 14 repeat the analysis of 817

Tables 9, 10, and 11, but for AH rather than 1X2 818

betting. Table A2 presents an example of the infor- 819

mation used during the betting simulation to assess 820

proﬁtability from AH bets, and it is based on aver- 821

age odds of season 2010/11 as shown in Table 13. 822

Overall, the AH bets appear to generate both lower 823

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 17

proﬁt as well as ROI compared to 1X2 bets. As with824

1X2 bets, optimising for maximum ROI per season

825

leads to a lower ROI across all seasons, compared to826

maximising proﬁt. Speciﬁcally, Table 13 shows that827

when maximising ROI per season leads to an overall828

ROI of 5.7% for average odds, which is lower than the

829

respective average ROI of 6.33% in Table 14 when830

maximising proﬁt. Once more, this observation only831

applies to the case of average market odds.832

An interesting observation is that AH betting gen-

833

erates a lower number of bets when θis low, compared834

to 1X2 betting, and a higher number of bets when θis835

high. This suggests that AH betting is less sensitive

836

to the betting decision threshold θcompared to 1X2837

betting, for both average and maximum market odds.

838

Furthermore, the optimal threshold θfor AH bets does839

not vary as much as it did for 1X2 bets. Despite the840

relatively low variance of θand the common occur-

841

rence of winning 60+ out of 100 AH bets, proﬁtability842

is still inconsistent between seasons. This is because843

match bets do not share the same payoff.844

5.3.1. Odds of bets simulated845

When it comes to the bets simulated, the 1X2 bets846

tend to average odds greater than 3 which suggests847

that the model tends to recommend bets on outsiders;848

a behaviour that is consistent with previous stud-

849

ies including the original pi-rating (Constantinou &

850

Fenton, 2013; Constantinou, 2018). Conversely, the

851

AH bets tend to be simulated on favourite outcomes852

with average season odds typically ranging between

853

1.6 and 1.8. However, it is important to note that an

854

issue with the AH odds retrieved from www.football-855

data.co.uk is that they do not always represent the856

odds associated with the handicap that maximises857

the uniformity of the AH distribution, as discussed

858

in Section 2. For example, the AH odds for seasons

859

2009/10 and 2010/11 appear to be predominantly860

based on ±0 AH; i.e., no handicap, with the outcome

861

of draw eliminated. Examples of this issue can also862

be viewed in Table A2; e.g., refer to the imbalanced

863

AH odds for dates 14/08, 11/09 and 27/11.864

According to Table 15, at least part of the AH865

odds of the ﬁrst ﬁve seasons do not reﬂect the stan-866

dard AH outcome, whereas the eight most recent867

seasons appear to be correctly based on the stan-868

dard AH outcome that aims to make the competition869

equal. Results from predictive accuracy and prof-

870

itability suggest that there is no meaningful difference

871

between the ﬁrst ﬁve and the last eight seasons. There-

872

fore, we have no reason to assume that this might have

873

Table 15

The mean average and mean maximum AH odds for each of the

13 seasons

Average Maximum

odds odds

Season HT AT HT AT

2006/07 1.89 1.97 1.95 2.05

2007/08 1.92 2.01 2.00 2.09

2008/09 1.85 2.30 1.94 2.50

2009/10 2.08 3.01 2.24 3.38

2010/11 1.87 2.24 1.94 2.38

2011/12 1.93 1.94 1.99 2.01

2012/13 1.93 1.95 1.99 2.01

2013/14 1.93 1.94 2.00 2.01

2014/15 1.92 1.95 1.98 2.01

2015/16 1.94 1.93 1.99 1.99

2016/17 1.95 1.93 2.01 1.99

2017/18 1.95 1.93 2.00 1.99

2018/19 1.96 1.94 2.03 2.00

inﬂuenced the overall conclusions. Finally, the pref- 874

erence of the model to bet on favourite AH outcomes 875

remains consistent across all 13 seasons. This out- 876

come is also discussed in Section 6, point ii.877

A possible limitation here is that, while the AH 878

market offers multiple handicaps for each match, this 879

study has only considered one handicap per match. 880

However, it is reasonable to assume that the results 881

presented in this paper approximate the overall AH 882

market. This is because when the model suggests a 883

bet on team Xfor a given handicap, then we should 884

expect the model to suggest a bet on team X regardless 885

the handicap, since any handicap must remain faithful 886

to the expected goal difference of the match, which 887

determines θ.888

5.3.2. Betting stake adjustments 889

Figure 5 presents the cumulative proﬁt generated 890

over eight different betting scenarios that represent 891

the combinations of the following betting options: a) 892

optimising for maximum ROI or proﬁt, b) optimising 893

θper season or across all seasons, and c) simulating 894

1X2 or AH bets. The results illustrate how the differ- 895

ence in proﬁt and ROI evolves across the 13 seasons 896

between 1X2 and AH bets. While AH bets generate 897

considerably lower proﬁt and ROI, the proﬁtability 898

is much less volatile than 1X2 bets and hence, it is 899

subject to a lower risk of loss which can often be 900

detrimental. For example, note the signiﬁcant losses 901

for the two best performing scenarios during matches 902

1900 to 2100, which are both based on 1X2 bets. 903

However, the lower risk of loss also limits proﬁts. 904

A fairer assessment of risk between 1X2 and AH 905

proﬁts would be to simply optimise stakes such that, 906

Uncorrected Author Proof

18 A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Fig. 5. Cumulative proﬁt when the betting procedure is optimised for either proﬁt or ROI, overall or per season, and based on either 1X2 or

AH maximum market odds. The results are based on 13 EPL seasons; from 2006/09 to 2018/19. Optimisations for overall proﬁt and ROI,

across all 13 seasons, are restricted to θdiscrepancies that generate at least 100 bets over those 13 seasons.

Fig. 6. Comparing the volatility of proﬁts when the stakes of AH bets is increased by as much required for the cumulative proﬁt to match

that of 1X2 bets.

at the end of the betting period, they both pro-907

duce the same proﬁt. Figure 6 provides these results908

by extending the scenarios of Fig. 5 to include an909

additional betting scenario in which AH stakes are 910

increased proportional to the difference in cumulative 911

proﬁt between 1X2 and AH bets. For example, in 912

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 19

Fig. 6a the new AH betting scenario assumes an913

increase of 5.59 times the stakes of AH bets, in order

914

for the cumulative proﬁt to become equal to that gen-915

erated by 1X2 bets.916

Overall, the graphs suggest that if we want AH bets917

to generate as much proﬁt as 1X2 bets do, then proﬁt

918

from AH bets will likely be subject to a similar risk of919

loss as with 1X2 bets. Therefore, while AH is often920

preferred due to the lower variance of returns, this921

advantage is rather eliminated when we need to bet

922

proportionally larger to match the expected proﬁt pro-923

duced by the corresponding 1X2 bets. This outcome924

is also discussed in Section 6, point iii.

925

6. Discussion and concluding remarks926

This paper presented a model speciﬁcally devel-927

oped for the prediction and assessment of the AH928

football betting market. The model is based on a

929

modiﬁed version of the pi-ratings system which mea-

930

sures the relative scoring ability between teams. The

931

modiﬁed pi-ratings are used as an input into a novel932

BN model that had its graphical structure deter-933

mined by the temporal assumption Possession →934

Shots →Shots on Target →Goals scored, which935

captures the natural causal chain of these events via

936

a Beta-Binomial Hybrid BN modelling process. One937

example of this assumption is that possession occurs

938

before shots (or shots on target) and hence, shots939

are assumed to be more impactful than possession940

in terms of determining goals scored.

941

Using goal scoring data over the last 27 EPL sea-942

sons, the modiﬁed pi-ratings discovered a strong943

linear relationship between team rating difference

944

and expected goal difference. However, the linear

945

relationship is oscillatory (refer to Fig. 3) and this946

suggests that goal data alone may be insufﬁcient947

in completely explaining team ability. Future work

948

will investigate whether factors beyond goals scored

949

could better explain this relationship. For example,

950

in (Constantinou & Fenton, 2017) it was shown that951

the three teams who were promoted to the EPL, from952

the English Championship, tend to perform signiﬁ-953

cantly better than the teams they replace. This is an954

important factor not taken into consideration by the955

pi-ratings; i.e., the teams are promoted with either an956

ignorant rating (if it is their ﬁrst time in the EPL) or957

with the rating they had when they were last relegated,958

which clearly underestimates their performance once

959

they return to the EPL.960

AH betting is assessed with reference to the tra- 961

ditional 1X2 betting. The assessment is based on 962

both average and maximum market odds and over all 963

possible betting decision thresholds in terms of dis- 964

crepancy between predicted and offered market odds. 965

Furthermore, the assessment differentiates between 966

betting strategies that are optimised for ROI and 967

betting strategies that are optimised for proﬁt. Key 968

observations include: 969

i. The previous literature has generally focused 970

on maximum market odds, and this is under- 971

standable since professional gamblers aim to 972

maximise payoff. Still, average odds are impor- 973

tant because they reveal the expected returns 974

for the average gambler. Moreover, maximum 975

odds are not attainable by everyone since many 976

countries do not allow access to many of 977

the online bookmakers, including exchange- 978

based websites which often offer the best odds 979

(excluding commission). This study shows that 980

the maximum available market odds increase 981

proﬁts by up to four times relative to aver- 982

age odds. Speciﬁcally, taking advantage of the 983

maximum market odds can lead to increased 984

proﬁts that range anywhere between 42% (refer 985

to overall proﬁts in Table 13) and 296% (refer 986

to maximised proﬁts in Table 9). 987

ii. The recommended AH bets tend to be on 988

favourite outcomes with odds that typically 989

average between 1.6 and 1.8 per season. Con- 990

versely, the recommended 1X2 bets tend to 991

be on outsider outcomes with odds averaging 992

above 3. The reduction of the problem from 993

a three-state multinomial to a binary distribu- 994

tion (i.e., from 1X2 to AH) explains why the 995

odds move from 1-in-3 to 1-in-2, but not why 996

the recommended bets switch from outsiders 997

to favourites. 998

iii. AH bets generate lower proﬁt as well as ROI 999

compared to 1X2 bets. Speciﬁcally, 1X2 bets 1000

are found to generate ∼2.5 to ∼5.5 times 1001

higher proﬁt and ∼2.5 to ∼4 times higher ROI 1002

compared to AH bets (refer to Fig. 6). For this 1003

reason, returns from AH bets tend to be consid- 1004

erably less volatile and subject to a lower risk of 1005

loss. While this outcome is in agreement with 1006

(Hassanniakalager & Newall, n.d.), this pre- 1007

sumed advantage of AH betting is ﬂawed. This 1008

is because, when the betting stakes of AH bets 1009

are increased proportional to the difference in 1010

cumulative proﬁt between 1X2 and AH bets, 1011

Uncorrected Author Proof

20 A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

the variance of proﬁt from AH bets increases1012

towards the variance of proﬁt from 1X2 bets.

1013

This implies that, when aiming for the same1014

proﬁt at the end of the same period of time,1015

AH bets are not necessarily less risky than 1X21016

bets.

1017

iv. Past studies often focus on a single foot-1018

ball season, and proﬁtability tends to be1019

reported based on the betting decision thresh-1020

old that maximises ROI under the assumption

1021

that the optimal betting decision threshold1022

remains invariant between seasons. However,1023

the results in this paper show that the optimal

1024

betting decision threshold varies dramatically1025

between seasons, despite predictive accuracy

1026

being consistent across the 13 seasons, and this1027

applies to both 1X2 and AH bets; albeit to a1028

lower degree for AH bets.

1029

This implies that the proﬁtability presented1030

in Tables 10, 11, 13, and 14 is not a realis-1031

tic expectation of real-world performance. This1032

is because the optimal betting decision thresh-1033

old is not consistent between seasons, and the1034

high variance suggest that it is unreasonable to1035

assume we will be able to predict the decision

1036

threshold that maximises proﬁt or ROI before

1037

a season starts. Therefore, the choice of eval-1038

uating football models based on the threshold

1039

that maximises proﬁtability in a single football1040

season, which is often the case in the literature,1041

should be discouraged. Moreover, the optimal

1042

betting decision threshold is also dependent on1043

whether we would like to maximise ROI or

1044

proﬁt. On the other hand, Tables 9 and 12 repre-

1045

sent a more realistic expectation of real-world1046

performance, even though it is unlikely that we1047

will follow a static betting decision threshold1048

across these many seasons.1049

v. Neither proﬁt nor ROI are consistent between1050

seasons, and this applies to both 1X2 and AH

1051

bets. While the overall performance of the1052

model is good enough to beat the market, it is

1053

still possible for the best possible betting deci-1054

sion threshold to be lossmaking for a whole1055

season (see Tables 10, 11, 13, 14). While this1056

is true for average market odds, the risk is1057

eliminated when we consider maximum odds;1058

though some seasons were barely proﬁtable.1059

vi. Finally, the results show that choosing to opti-

1060

mise for maximum ROI per season will likely

1061

produce undesired results in the long term, and

1062

this applies to both 1X2 and AH bets. On the

1063

other hand, choosing to optimise for maximum 1064

proﬁt (rather than ROI) per season, not only 1065

guarantees that the proﬁt is maximised across 1066

all seasons, but also often generates a higher 1067

overall ROI, across all seasons, compared to 1068

the overall ROI generated when optimising for 1069

maximum ROI for each season independently. 1070

This ﬁnding is important since most of the pre- 1071

vious studies focus on maximising ROI, often 1072

for individual seasons. 1073

Lastly, it is important to note that this paper has 1074

considered football data up to season 2018/19. The 1075

two subsequent seasons have been partly affected by 1076

the COVID-19 pandemic, where many matches were 1077

played with fewer or without fans. Relevant stud- 1078

ies have shown that this event had an insigniﬁcant 1079

or a signiﬁcant negative effect on home advantage 1080

(Wunderlich et al., 2021; McCarrick et al., 2021). 1081

The model described in this paper does not to con- 1082

sider this event. However, because the model relies 1083

on pi-ratings which involve a home and an away 1084

rating for each team, it can be easily adjusted to con- 1085

sider such previously unseen events. For example, if 1086

we assume that home advantage is not relevant for 1087

a particular match, we could consider assigning the 1088

‘away’ ratings to both teams for that match. Future 1089

research works could investigate whether such mod- 1090

elling modiﬁcations, that take into consideration the 1091

effect of playing in empty stadiums, improve predic- 1092

tive accuracy. 1093

Acknowledgments 1094

This research was supported by the ERSRC Fel- 1095

lowship project EP/S001646/1 on Bayesian Artiﬁcial 1096

Intelligence for Decision Making under Uncertainty, 1097

by The Alan Turing Institute in the UK, and by Agena 1098

Ltd. 1099

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Uncorrected Author Proof

22 A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting

Appendix A: Sample results from betting simulations

Table A1

Details of proﬁtability for case in Table 10: Season= 2010/11, Odds =Average, Bets =1X2, Optimisation = ROI, and θ=10%

Model Average Bookmakers’ Payoff Bets Returns

predictions bookmakers’ unnormalised discrepancy simulated from bets

odds prediction

Date HT AT 1X2 p(1) p(X) p(2) Odds(1) Odds(X) Odds(2) p(1) p(X) p(2) θ(1) θ(X) θ(2) Bet(1) Bet(X) Bet(2) Return(1) Return(X) Return(2) Proﬁt

14/08/2010 Aston Villa West Ham 1 0.62 0.22 0.16 1.96 3.30 4.03 0.51 0.30 0.25 0.11 –0.08 –0.09 1 0 0 1.96 0 0 0.96

14/08/2010 Wigan Blackpool 2 0.33 0.26 0.41 1.82 3.45 4.50 0.55 0.29 0.22 –0.22 –0.03 0.19 0 0 1 0 0 4.5 3.5

26/09/2010 Wolves Aston Villa 2 0.25 0.25 0.50 2.83 3.25 2.50 0.35 0.31 0.40 –0.10 –0.06 0.10 0 0 1 0 0 2.5 1.5

02/10/2010 Sunderland Man United X 0.13 0.20 0.67 4.93 3.45 1.75 0.20 0.29 0.57 –0.08 –0.09 0.10 0 0 1 0 0 0 –1

23/10/2010 Birmingham Blackpool 1 0.39 0.27 0.34 1.85 3.48 4.31 0.54 0.29 0.23 –0.15 –0.02 0.11 0 0 1 0 0 0 –1

24/10/2010 Liverpool Blackburn 1 0.73 0.17 0.10 1.66 3.64 5.43 0.60 0.27 0.18 0.12 –0.11 –0.08 1 0 0 1.66 0 0 0.66

01/11/2010 Blackpool West Brom 1 0.47 0.26 0.26 2.79 3.25 2.54 0.36 0.31 0.39 0.11 –0.04 –0.13 1 0 0 2.79 0 0 1.79

10/11/2010 Man City Man United X 0.25 0.25 0.50 2.57 3.22 2.75 0.39 0.31 0.36 –0.14 –0.06 0.14 0 0 1 0 0 0 –1

27/11/2010 Bolton Blackpool X 0.41 0.27 0.32 1.57 3.96 5.82 0.64 0.25 0.17 –0.22 0.01 0.15 0 0 1 0 0 0 –1

11/12/2010 Aston Villa West Brom 1 0.62 0.22 0.16 2.10 3.30 3.53 0.48 0.30 0.28 0.15 –0.08 –0.13 1 0 0 2.1 0 0 1.1

11/12/2010 Stoke Blackpool 2 0.41 0.27 0.32 1.62 3.86 5.41 0.62 0.26 0.18 –0.21 0.01 0.14 0 0 1 0 0 5.41 4.41

12/12/2010 Tottenham Chelsea X 0.25 0.25 0.51 2.84 3.28 2.49 0.35 0.30 0.40 –0.10 –0.06 0.10 0 0 1 0 0 0 –1

13/12/2010 Man United Arsenal 1 0.62 0.22 0.16 1.95 3.40 3.92 0.51 0.29 0.26 0.11 –0.07 –0.09 1 0 0 1.95 0 0 0.95

28/12/2010 Sunderland Blackpool 2 0.41 0.27 0.32 1.60 3.81 5.82 0.63 0.26 0.17 –0.21 0.00 0.15 0 0 1 0 0 5.82 4.82

28/12/2010 West Brom Blackburn 2 0.36 0.27 0.37 1.82 3.48 4.49 0.55 0.29 0.22 –0.19 –0.02 0.15 0 0 1 0 0 4.49 3.49

05/01/2011 Arsenal Man City X 0.63 0.21 0.15 1.94 3.49 3.86 0.52 0.29 0.26 0.12 –0.07 –0.11 1 0 0 0 0 0 –1

05/01/2011 Everton Tottenham 1 0.48 0.26 0.26 2.66 3.24 2.64 0.38 0.31 0.38 0.10 –0.05 –0.12 1 0 0 2.66 0 0 1.66

15/01/2011 West Brom Blackpool 1 0.33 0.26 0.41 1.78 3.68 4.48 0.56 0.27 0.22 –0.23 –0.01 0.19 0 0 1 0 0 0 –1

16/01/2011 Liverpool Everton X 0.63 0.22 0.16 2.20 3.22 3.43 0.45 0.31 0.29 0.17 –0.09 –0.14 1 0 0 0 0 0 –1

23/01/2011 Blackburn West Brom 1 0.56 0.24 0.21 2.22 3.27 3.24 0.45 0.31 0.31 0.11 –0.07 –0.10 1 0 0 2.22 0 0 1.22

01/02/2011 West Brom Wigan X 0.41 0.27 0.32 1.71 3.58 5.13 0.58 0.28 0.19 –0.17 –0.01 0.13 0 0 1 0 0 0 –1

12/02/2011 Man United Man City 1 0.67 0.20 0.13 1.77 3.57 4.61 0.56 0.28 0.22 0.10 –0.08 –0.08 1 0 0 1.77 0 0 0.77

12/02/2011 West Brom West Ham X 0.36 0.27 0.37 1.93 3.52 3.86 0.52 0.28 0.26 –0.16 –0.02 0.11 0 0 1 0 0 0 –1

20/02/2011 West Brom Wolves X 0.36 0.26 0.37 1.86 3.40 4.33 0.54 0.29 0.23 –0.18 –0.03 0.14 0 0 1 0 0 0 –1

26/02/2011 Wolves Blackpool 1 0.36 0.26 0.37 1.83 3.58 4.28 0.55 0.28 0.23 –0.18 –0.02 0.14 0 0 1 0 0 0 –1

19/03/2011 Man United Bolton 1 0.83 0.12 0.06 1.40 4.41 8.54 0.71 0.23 0.12 0.11 –0.11 –0.06 1 0 0 1.4 0 0 0.4

19/03/2011 West Brom Arsenal X 0.12 0.20 0.68 4.67 3.59 1.76 0.21 0.28 0.57 –0.09 –0.08 0.11 0 0 1 0 0 0 –1

09/04/2011 Wolves Everton 2 0.25 0.25 0.50 2.55 3.25 2.80 0.39 0.31 0.36 –0.14 –0.06 0.14 0 0 1 0 0 2.8 1.8

11/04/2011 Liverpool Man City 1 0.55 0.24 0.21 2.59 3.21 2.76 0.39 0.31 0.36 0.16 –0.07 –0.15 1 0 0 2.59 0 0 1.59

14/05/2011 West Brom Everton 1 0.26 0.24 0.50 2.63 3.28 2.69 0.38 0.30 0.37 –0.12 –0.06 0.13 0 0 1 0 0 0 –1

22/05/2011 Bolton Man City 2 0.33 0.26 0.41 4.92 3.72 1.69 0.20 0.27 0.59 0.13 –0.01 –0.19 1 0 0 0 0 0 –1

22/05/2011 Man United Blackpool 1 0.82 0.12 0.06 1.56 4.13 5.57 0.64 0.24 0.18 0.18 –0.12 –0.12 1 0 0 1.56 0 0 0.56

22/05/2011 Stoke Wigan 2 0.55 0.24 0.21 2.76 3.42 2.45 0.36 0.29 0.41 0.19 –0.05 –0.20 1 0 0 0 0 0 –1

TOTAL 15 0 18 22.66 0 25.52 15.18

Uncorrected Author Proof

A.C. Constantinou / Investigating the efﬁciency of the Asian handicap football betting 23

Table A2

Details of proﬁtability for case in Table 13: Season= 2010/11, Odds =Average, Bets =AH, Optimisation = ROI, and θ= 11%

Goals Goal Model Average Bookmakers’ Payoff Bets Returns

predictions bookmakers’ unnormalized discrepancy simulated from bets

Date HT AT HT AT difer AH p(1) p(2) Odds(1) Odds(2) p(1) p(2) θ(1) θ(2) Bet(1) Bet(2) Return(1) Return(2) Proﬁt

14/08/2010 Wigan Blackpool 0 4 –4 0 0.45 0.55 1.32 3.19 0.76 0.31 –0.31 0.24 0 1 0 3.19 2.19

21/08/2010 Arsenal Blackpool 6 0 6 –2 0.28 0.72 1.76 2.12 0.57 0.47 –0.28 0.24 0 1 0 0 –1

11/09/2010 Newcastle Blackpool 0 2 –2 0 0.65 0.35 1.20 4.23 0.83 0.24 –0.18 0.11 0 1 0 4.23 3.23

25/09/2010 West Ham Tottenham 1 0 1 0 0.51 0.49 2.54 1.47 0.39 0.68 0.11 –0.19 1 0 2.54 0 1.54

23/10/2010 Birmingham Blackpool 2 0 2 –0.5 0.39 0.61 1.85 2.01 0.54 0.50 –0.15 0.11 0 1 0 0 –1

24/10/2010 Liverpool Blackburn 2 1 1 –0.75 0.69 0.31 1.85 2.02 0.54 0.50 0.15 –0.19 1 0 1.425 0 0.425

30/10/2010 Man United Tottenham 2 0 2 –1 0.65 0.35 2.04 1.83 0.49 0.55 0.16 –0.19 1 0 2.04 0 1.04

01/11/2010 Blackpool West Brom 2 1 1 0 0.64 0.36 1.99 1.81 0.50 0.55 0.14 –0.20 1 0 1.99 0 0.99

10/11/2010 Man City Man United 0 0 0 0 0.33 0.67 1.82 1.98 0.55 0.51 –0.22 0.16 0 1 0 1 0

20/11/2010 Birmingham Chelsea 1 0 1 0.75 0.36 0.64 1.88 2.00 0.53 0.50 –0.17 0.14 0 1 0 0 –1

27/11/2010 Bolton Blackpool 2 2 0 0 0.56 0.44 1.20 4.17 0.83 0.24 –0.27 0.20 0 1 0 1 0

11/12/2010 Aston Villa West Brom 2 1 1 0 0.80 0.20 1.50 2.49 0.67 0.40 0.13 –0.20 1 0 1.5 0 0.5

11/12/2010 Stoke Blackpool 0 1 –1 –1 0.24 0.76 2.07 1.82 0.48 0.55 –0.24 0.21 0 1 0 1.82 0.82

26/12/2010 Aston Villa Tottenham 1 2 –1 0 0.57 0.43 2.22 1.64 0.45 0.61 0.12 –0.18 1 0 0 0 –1

28/12/2010 Sunderland Blackpool 0 2 –2 –1 0.25 0.75 2.06 1.81 0.49 0.55 –0.24 0.20 0 1 0 1.81 0.81

28/12/2010 West Brom Blackburn 1 3 –2 –0.5 0.36 0.64 1.82 2.05 0.55 0.49 –0.19 0.15 0 1 0 2.05 1.05

29/12/2010 Chelsea Bolton 1 0 1 –1.5 0.62 0.38 1.98 1.89 0.51 0.53 0.12 –0.15 1 0 0 0 –1

01/01/2011 Man City Blackpool 1 0 1 –1.5 0.37 0.63 1.83 2.03 0.55 0.49 –0.17 0.13 0 1 0 2.03 1.03

05/01/2011 Arsenal Man City 0 0 0 –0.5 0.63 0.37 1.94 1.93 0.52 0.52 0.12 –0.15 1 0 0 0 –1

05/01/2011 Everton Tottenham 2 1 1 0 0.65 0.35 1.90 1.92 0.53 0.52 0.12 –0.17 1 0 1.9 0 0.9

15/01/2011 West Brom Blackpool 3 2 1 –0.75 0.26 0.74 1.96 1.91 0.51 0.52 –0.25 0.22 0 1 0 0.5 –0.5

16/01/2011 Liverpool Everton 2 2 0 0 0.80 0.20 1.55 2.41 0.65 0.41 0.16 –0.22 1 0 1 0 0

01/02/2011 West Brom Wigan 2 2 0 –0.75 0.34 0.66 1.89 1.98 0.53 0.51 –0.19 0.16 0 1 0 1.98 0.98

12/02/2011 Man United Man City 2 1 1 –0.75 0.62 0.38 1.99 1.89 0.50 0.53 0.12 –0.15 1 0 1.495 0 0.495

12/02/2011 West Brom West Ham 3 3 0 –0.5 0.36 0.64 1.92 1.96 0.52 0.51 –0.16 0.13 0 1 0 1.96 0.96

20/02/2011 West Brom Wolves 1 1 0 –0.5 0.36 0.64 1.88 2.01 0.53 0.50 –0.17 0.14 0 1 0 2.01 1.01

26/02/2011 Wolves Blackpool 4 0 4 –0.5 0.36 0.64 1.83 2.05 0.55 0.49 –0.18 0.15 0 1 0 0 –1

05/03/2011 Arsenal Sunderland 0 0 0 –1 0.67 0.33 1.82 2.06 0.55 0.49 0.12 –0.16 1 0 0 0 –1

19/03/2011 Man United Bolton 1 0 1 –1.25 0.70 0.30 2.01 1.87 0.50 0.53 0.20 –0.23 1 0 0.5 0 –0.5

19/03/2011 West Brom Arsenal 2 2 0 0.75 0.37 0.63 1.84 2.03 0.54 0.49 –0.18 0.14 0 1 0 0 –1

03/04/2011 Fulham Blackpool 3 0 3 –1 0.31 0.69 1.92 1.94 0.52 0.52 –0.21 0.17 0 1 0 0 –1

09/04/2011 Man United Fulham 2 0 2 –1 0.70 0.30 1.86 2.01 0.54 0.50 0.16 –0.20 1 0 1.86 0 0.86

09/04/2011 Wolves Everton 0 3 –3 0 0.34 0.66 1.82 2.00 0.55 0.50 –0.21 0.16 0 1 0 2 1

10/04/2011 Blackpool Arsenal 1 3 –2 1.5 0.65 0.35 1.87 2.00 0.53 0.50 0.11 –0.15 1 0 0 0 –1

11/04/2011 Liverpool Man City 3 0 3 0 0.72 0.28 1.86 1.97 0.54 0.51 0.19 –0.23 1 0 1.86 0 0.86

16/04/2011 West Brom Chelsea 1 3 –2 0.75 0.36 0.64 1.95 1.94 0.51 0.52 –0.16 0.13 0 1 0 1.94 0.94

07/05/2011 Tottenham Blackpool 1 1 0 –1.5 0.40 0.60 1.81 2.05 0.55 0.49 –0.16 0.12 0 1 0 2.05 1.05

14/05/2011 Sunderland Wolves 1 3 –2 0 0.65 0.35 1.88 1.95 0.53 0.51 0.12 –0.17 1 0 0 0 –1

14/05/2011 West Brom Everton 1 0 1 0 0.34 0.66 1.89 1.94 0.53 0.52 –0.19 0.14 0 1 0 0 –1

15/05/2011 Arsenal Aston Villa 1 2 –1 –1.25 0.40 0.60 1.81 2.07 0.55 0.48 –0.15 0.11 0 1 0 2.07 1.07

22/05/2011 Bolton Man City 0 2 –2 0.75 0.67 0.33 2.00 1.88 0.50 0.53 0.17 –0.20 1 0 0 0 –1

22/05/2011 Man United Blackpool 4 2 2 –1 0.77 0.23 2.00 1.88 0.50 0.53 0.27 –0.31 1 0 2 0 1

22/05/2011 Stoke Wigan 0 1 –1 0 0.72 0.28 2.00 1.85 0.50 0.54 0.22 –0.26 1 0 0 0 –1

TOTAL 20 23 20.11 31.64 8.75