Information and price efficiency in the absence of home crowd advantage

ABSTRACT

This paper evaluates the efficiency of betting market pricing for top league soccer matches played behind closed doors during the COVID-19 pandemic. The removal of the crowd component of home advantage results in substantial improvement in both the market predictions of goal difference and home team wins.

KEYWORDS:

• Market efficiency
• COVID-19
• home advantage
• soccer

JEL CLASSIFICATION:

• C53
• G14
• Z2
• Z21

I. Introduction

The study of how efficiently market prices incorporate all available information is a fundamental research area in economics. The efficient market hypothesis, in the weak form, postulates that asset prices contain all available information (Fama 1970).

Mainstream financial market environments are difficult to study, with consent flows of information and no defined end point for final settlement (an investor can hold a stock for many years). Betting markets are attractive as an alternative for investigating market efficiency, being similar to financial markets in many respects yet much simpler in structure. Betting markets have definite end points for settlement and provide researchers a natural laboratory in which agents make financial decisions under uncertainty (Thaler and Ziemba 1988).

The majority of research into efficiency of betting markets has concentrated on investigating biases which violate the efficient market hypothesis. Horse racing and soccer markets, due to high turnover and availability of data, are commonly subject to efficiency tests in the literature. Using prices from four high-street bookmakers, the seminal paper by Pope and Peel (1989) found the UK soccer betting market mostly efficient with no betting strategy profitable. More recently, Angelini and De Angelis (2019) uncovered a favourite – longshot bias, where systematic bets on favourites yield better results than bets on longshots, in certain European leagues. Using a large sample of games, Elaad (2020) highlights different degrees of efficiency around home field advantage across the higher and lower tier leagues of English professional soccer.

The home field advantage is a well-known phenomenon found in many different sports (Gómez, Pollard, and Luis-Pascual 2011). It is influenced by crowd noise (Balmer et al. 2007), event location (Pic and Castellano 2016, 2017), and increases with crowd density (Schwartz and Barsky 1977). Although the advantage is well-known, less is known about how efficiently it is incorporated into betting market prices. The COVID-19 pandemic of Spring 2020 provides an involuntary natural experiment to examine market pricing of the crowd component in home field advantage. As a response to the pandemic, the German professional soccer league introduced games behind closed doors in March 2020. Two parallel studies – Deutscher and Winkelmann (2020) – Fischer and Haucap (2020a) examine the efficiencies of the betting market odds for the 83-game sample played behind closed doors up to the end of the 2019/2020 season in Germany. Their findings suggest the presence of an away bias (i.e. profit from systematically betting away teams) and bookmaker mis-pricing of the games without crowds present for the top tier Bundesliga 1.

These interesting findings from the German top tier are observed in a small sample of games and could be due to randomness and not bookmaker mis-pricing of the home advantage. Randomness cannot be discounted due to the fact that in the initial 83 games behind closed doors, the percentage of home team victories dropped considerably (33%) compared with previous games in the same season (43%) in the top tier Bundesliga 1 (Fischer and Haucap 2020b).

This paper proposes a new technique to test market efficiency at the granular level of home win, away win and draw outcomes. The popular Asian Handicap soccer betting market is also utilized, for the first time in the literature, to provide a measure of how efficiently the market predicts the goal difference between home and away sides. Despite high volumes of turnover and popularity as a betting medium, the Asian Handicap market has not been given adequate attention by scholars, appearing only as a predictor for match outcomes in a handful of papers (Constantinou 2020).

Considering a large sample of games from the four main European leagues (Germany, Italy, England and Spain), evidence contrary to the preliminary – Deutscher and Winkelmann (2020) – Fischer and Haucap (2020a) findings is presented that the bookmaker market becomes more efficient with the removal of the crowd component in top tier soccer matches.

II. Methodology

Data

The outcomes and bookmakers’ odds for the matches played in the top divisions for each of the four major European soccer playing nations (Germany, Italy, England and Spain) were retrieved from www.football-data.co.uk, covering the most recent seasons 2019/2020 and 2020/2021.¹ In early 2020, due to the Covid-19 pandemic, decisions were made to restart the leagues without spectators present. Table 1 displays the dates each league moved to behind closed doors with column Games displaying the number of matches played without spectators for each selected league.² Across these 4 leagues, there were a total of 754 matches behind closed doors. An equally sized second set of games played immediately prior to the change are selected for comparison. For example, the 190 games prior to 17 June 2020 in the English Premier League are selected for comparison with the 190 games played behind closed doors after 17 June 2020.

Table 1. Number of matches per league for analysis

League	Behind closed doors from	Games	Comparison
Germany Bundesliga 1	11 March 2020	130	130
Italy Serie A	8 March 2020	220	220
England Premier League	17 June 2020	190	190
Spain La Liga Primera	10 March 2020	214	214
Total Games		754	754

The data-set contains both the opening and closing odds (for all possible outcomes: home, away, draw) offered by 13 large online bookmakers. In addition, the parallel Asian handicap market is included. The Asian handicap is a form of betting aimed at equalizing chances of the home and away sides. For example, an Asian handicap of −2.5 gives 2.5 goals to the away side at the start of the match, meaning that the home side must win by 3 goals or more for a bet on the home side to be successful.

Asian handicap efficiency

The Asian handicap $A_j$ for each game j represents the market estimate of the expected goal difference between home and away teams. The lowest increment for an Asian handicap is 0.25 of a goal.³ The following equation assumes a linear relationship between actual goal difference $D_j$ and $A_j$, and deliberately excludes any other possible factors which may effect the actual goal difference.

(1) $D_j={\alpha }_0+{\alpha }_1{A}_j+{\in }_j$(1)

The above model is estimated for both games played before and after the Covid change. The objective is to compare the predictive accuracy of the Asian handicap for games played behind closed doors to those played immediately prior. To evaluate predictive accuracy, the resulting goodness-of-fit measure $R^2$ from both models is compared following Sung, Johnson, and McDonald (2016).

(2) $R^2=1-\frac{S{S}_{res}}{S{S}_{tot}}$(2)

where $S{S}_{res}$ represents the sum of squares of residuals and $S{S}_{tot}$ are the total sum of squares. Distributions for the R-squared are estimated with a bootstrap method. For both sets (before and after), a random sampling of the matches with replacement is repeated and modelled 1,000 times. The null hypotheses tested is that the mean R-squared obtained from the bootstrap distributions does not differ for games played behind closed doors.

(3) $H_2:{\bar{x}}_1={\bar{x}}_2$(3)

where ${\bar{x}}_1$ and ${\bar{x}}_2$ are the mean R-squared from the distributions pre- and post-Covid, respectively. The pooled standard deviation s is calculated as follows:

(4) $s=\sqrt{\frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}}$(4)

where $s_1$ and $s_2$ represent the standard deviations from the two samples with sample sizes $n_1$ and $n_2$ (both 1,000). The standard error se of the difference between the two means is given by:

(5) $se\left(\overline{x_1}-\overline{x_2}\right)=s\times \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}$(5)

The significance of any difference between the two is highlighted using the t-test:

(6) $t=\frac{\overline{x_1}-\overline{x_2}}{\mathrm{s}\mathrm{e}\left(\overline{x_1}-\overline{x_2}\right)}$(6)

Market efficiency

A method adapted from the Sung, Johnson, and McDonald (2016) test of market efficiency for horse racing is employed to test the calibration between the objective probabilities of success revealed by the match outcomes and the implied probabilities from the bookmaker market (i.e. how well the market prices predict actual outcomes). The conditional logit model, in this case, is not suitable due to the independence of irrelevant alternatives property: The ratio of odds between any two of the three possibilities depends only on the characteristics of those possibilities and does not change with additional choices. The addition of a drawn match possibility to an odds ratio is not comparable to adding another participant as is the case in horse racing.

For each game, j = 1 … J, let i denote the outcome: a home win (i = 1), a draw (i = 2) or an away win (i = 3). The bookmakers publish odds in a decimal format $d_{ij}$, for example, decimal odds of 2.10 yield a return of €210 (profit of €110) for a successful €100 bet. The inverse of the decimal odds $p_{ij}=1/d_{ij}$ represent the implied probability prediction of the bookmaker market for each outcome i in game j. In the above example, the decimal odds of 2.10 represent a bookmaker probability estimate of 0.48 or a 48% chance of outcome i occurring in game j.

As a first step, the average bookmaker implied probabilities $p_{ij}$ are normalized using a standard normalization fitting coefficient ${\psi }_j$ derived from the bookmaker margin $M_j={\sum }_{i=1}^3{p}_{ij}-1$ for each game j:

(7) ${\psi }_j=\frac{1}{1+M_j}$(7)

(8) $\sum _{i=1}^3{q}_i=\sum _{i=1}^3{\psi }_j{p}_i=1$(8)

This normalization process yields three market implied probability estimates $q_i$ for each possible outcome i (home = h, away = a, draw = d) which sum to 1. To test market efficiency, three separate logistic regressions are estimated for each of the possible outcomes on both the before and after sets of games separately.

(9) $Y_{hj}={\gamma }_0+{\gamma }_1{q}_{hj}+{\in }_{hj}$(9)

(10) $Y_{aj}={\gamma }_0+{\gamma }_1{q}_{aj}+{\in }_{aj}$(10)

(11) $Y_{dj}={\gamma }_0+{\gamma }_1{q}_{dj}+{\in }_{dj}$(11)

where the independent variable is a binary indicator of success $(Pr(Succes{s}_{ij}=1))=Y_{ij}$ and is linked by the logit function. This analysis is consistent with previous tests of calibration between objective outcomes and bookmaker implied probabilities for soccer matches (Deutscher and Winkelmann 2020). However, rather than excluding the draws, three separate models are estimated to test the market’s ability to price all possible outcomes.

The models are fitted using maximum likelihood. Let $L_0$ be the value of the likelihood function for the extreme case of a model with no predictors, and let $L_m$ be the likelihood for each of the models estimated. The resulting McFadden pseudo-$R^2$ measures the predictive accuracy of each model.

(12) $R^2=1-\frac{L_m}{L_0}$(12)

Distributions for the McFadden pseudo-$R^2$s are estimated with a bootstrap method. For both sets of games, a random sampling with replacement is repeated and modelled 1,000 times. The null hypotheses tested is that the mean McFadden R-squared obtained from the bootstrap distribution does not differ for games played post Covid changes.

(13) $H_3:{\bar{\mu }}_1={\bar{\mu }}_2$(13)

where ${\bar{\mu }}_1$ and ${\bar{\mu }}_2$ are the mean McFadden R-squared from the distributions pre- and post-Covid respectively. The significance test follows the methodology outlined previously (Equations (4(4) $s=\sqrt{\frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}}$(4) )–(6))(6) $t=\frac{\overline{x_1}-\overline{x_2}}{\mathrm{s}\mathrm{e}\left(\overline{x_1}-\overline{x_2}\right)}$(6) .

III. Results

Asian handicap efficiency

Table 2 shows the sample statistics for the R-squared obtained from the bootstrapped before and after sets of matches. The mean R-squared is 4.6% higher for Asian Handicap predictions on the games played behind closed doors. This difference represents a 23 percentage point increase compared to the 754 games played prior and is statistically significant (Table 3).

Table 2. R-squared bootstrap for Asian handicap markets

	N	Mean	St. Dev	Min	Max
Before	1000	0.202	0.027	0.118	0.280
After	1000	0.248	0.029	0.162	0.340
Difference		0.046	0.002	0.044	0.061

Table 3. Difference between observed means in the two samples

Difference	0.046
Standard error	0.001
95% CI	[0.0439, 0.0487]
t-statistic	37.387
DF	1998
Significance level	$P<0.0001$

The results indicate the market better predicts the goal difference between home and away sides in top tier soccer matches played behind closed doors. Figure 1 is a density plot of the two R-squared distributions and illustrates the improvement in predictive accuracy with a substantial area of higher R-squared (for games played behind closed doors) without overlap.

Figure 1. Density plot of R-squared for Asian handicap markets

Market efficiency

To build upon the findings from the Asian Handicap market, a more granular approach is implemented in this section. Using the methodology outlined in section 2.3, the ability of the market to predict individual outcomes (home, away, draw) is investigated. A very strong improvement is evident in the market’s ability to predict home-team wins for games played behind closed doors. Tables 4 and 5 and Figure 2(a) illustrate a significant improvement in the mean McFadden R-squared of 5.6%, representing a 58 percentage point increase. No difference was found for the market efficiency related to the pricing of away wins and draw results show an increase with a high degree of overlap between the two distributions (see Table 4 and Figure 2(b,c) respectively).

Table 4. McFadden R-squared bootstrap for home, away and draw odds

		Home			Away			Draw
	Before	After	Difference	Before	After	Difference	Before	After	Difference
N	1000	1000		1000	1000		1000	1000
Mean	0.096	0.152	0.056	0.117	0.119	0.002	0.014	0.021	0.007
St. Dev	0.018	0.023	0.005	0.022	0.020	−0.002	0.008	0.009	0.002
Min	0.043	0.083	0.040	0.059	0.050	−0.009	0.000	0.000	0.000
Max	0.171	0.228	0.057	0.201	0.184	−0.017	0.046	0.062	0.017

Table 5. Difference between observed means in the two samples

	Home	Away	Draw
Difference	0.056	0.002	0.007
Standard error	0.001	0.001	0.000
95% CI	[0.0545, 0.0581]	[0.0001, 0.0038]	[0.0059, 0.0075]
t-statistic	61.71	2.027	17.448
DF	1998	1998	1998
Significance level	$P<0.0001$	$P=0.0428$	$P<0.0001$

Figure 2. Density plots of McFadden R-Squared distributions

The market’s overall ability to predict drawn matches is much lower than for either the home or away outcomes. This result is expected given the low variability of draw probabilities and is consistent with the findings in Pope and Peel (1989) & Angelini and De Angelis (2019) that draw odds contain little predictive content.

IV. Discussion

The analysis presented in this paper provides insights into the betting market’s adjustment to the removal of the crowd component of home advantage in the top leagues of European soccer. Contrary to recent preliminary research findings, results from longer time periods and multiple leagues are more efficient in estimating actual goal difference and the probability of home team wins for top tier matches played behind closed doors. The crowd component of home advantage is subjective and difficult for market makers to quantify. Its removal affords the market an opportunity to assess the teams on fundamental strengths, improving predictive accuracy.

Acknowledgments

I would like to thank Christopher Jepsen for his helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ Game data was collected up to and including the 2 December 2020, after this date England allowed spectators attend in certain stadia.

² At the start of the 2020/2021 season, 34 reduced capacity matches took place in Bundlesiga 1 and are excluded from the analysis.

³ For example, a handicap of −2.25 represents half the wager at −2.0 and half at −2.5.