Corruption and Referee Bias in Football: the Case of Calciopoli

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Corruption and Referee Bias in Football:

the Case of

Calciopoli

∗

Walter Distaso

†

Leone Leonida

‡

Dario Maimone Ansaldo Patti

§

Pietro Navarra

¶

February 2012

Abstract

Based on theCalciopoli scandal, which uncovered widespread corruption in Italian football, this paper quantifies the effect of referee bias on the performance of football teams. The impartiality of referees is often distorted by external factors which exert some emotional pressure in order to influence their decisions. On the other hand, corrupt referees consciously and deliberately try to distort the results of the sport contest, in order to favor the corrupting teams. Building on the implications of a model where performance in a sport contest depends on both effort and bribing, our results highlight the different effects of these two forms of bias, and help to shed light on several aspects of the corruption scandal.

∗_{We are grateful to Valentina Corradi, Abhinay Muthoo, Stefan Szymanski and Tommaso Valletti}

for helpful suggestions. Usual disclaimer applies. Distaso gratefully acknowledges ﬁnancial support from the ESRC, grant RES-062-23-0311.

†_{Corresponding Author.} _{Imperial College Business School, Imperial College London,}

Exhibi-tion Road, SW7 2AZ, London, UK. Tel.: +44 207 594 3293. Fax: +44 207 594 9189. Email:

w.distaso@imperial.ac.uk.

‡_{Department of Economics, Queen Mary University of London.} §_{Department of Economics, University of Essex.}

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Keywords: referee bias, contest success function, economics of sport.

JEL: C73, C35, L83.

1 Introduction

In Summer 2006 the nationwide enthusiasm generated by the Italian football team’s World Cup victory in Germany has been overshadowed by a game-ﬁxing investiga-tion that uncovered widespread corrupinvestiga-tion in Italian football. The scandal, commonly

referred to as Calciopoli, revolved around twenty months of wiretapped telephone

con-versations involving key ﬁgures of Italian football. The magistrates of the Italian

Football Federation (FIGC) formally investigated a total of 41 people and examined

19 matches played in Serie A(the top Italian league) during the season 2004/05. The

prosecutors believed that a group of individuals had conspired to inﬂuence the results of the matches.

The FIGC prosecutors formulated allegations of corruption and fraud against team managers and referees. The President of FIGC and his deputy resigned soon after the scandal erupted. Two of the top members of the referees’ Association stepped down, allegedly thought of being part of the conspiring group. The football clubs involved

in the scandal were AC Milan, Fiorentina, Juventus and Lazio. All of them have

been penalized: the clubs were sentenced with points deductions, and Juventus was

relegated to play the 2006/07 season in a lower division (Serie B). Table 1 summarizes

the ﬁnal sanctions inﬂicted by the FIGC to the four teams.

[Table 1 about here]

Calciopoli has not been an isolated case of corruption in professional sport. For example, in the summer of 2007 the US has been shaken by “...the worst that could happen to a professional sport league”, according to the National Basketball Asso-ciation (NBA) commissioner David Stern. An NBA referee, Timothy Donaghy, was accused of two felonies by a district court in Brooklyn. He admitted to betting on

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several basketball matches, including the ones he refereed. He also confessed to trying to manipulate results and passing sensitive information to professional gamblers.

The academic literature has extensively examined the role and impartiality of ref-erees in sports contests, even when corruption and/or other illegal practices have not been discovered. Several papers have investigated to what extent external forces can induce referees to make biased and wrong decisions. The fact of playing at home, as well as social forces such as composition of the crowd, type of the stadium and crowd’s proximity to the ﬁeld have been found to have an impact on referees’ decisions, even if referees do not get any advantage for their biased behaviour. The literature has

mainly focused on the analysis of the so called home team bias, i.e. the extent to

which referees tend to be biased in favor of home teams. Several studies document the existence of home team bias and argue that pressure from home fans tends to inﬂuence referees’ decisions. This type of advantage has been measured in terms of added extra time, penalties awarded, yellow and red cards given out and free-kicks and oﬀ-sides awarded. Empirical papers examining the existence of the home team bias seem to be unequivocal and consistent across countries. Garicano et al. (2005) examined the

SpanishPrimera Division, Dohmen (2005) studied the home advantage in the German

Bundesliga and, ﬁnally, Goddard (2005) and Dawson et al. (2007) investigated the

English Premier League. All these papers support the home team bias hypothesis.

The typical explanation for this common phenomenon is that referees are emotion-ally influenced by external forces which affect their decisions. In other words, they are not intentionally biased and it is not in their interest to make disputable decisions, but their impartiality is affected by the emotional pressure exerted by external forces often associated to the preferences of the crowd.

In this respect,Calciopoli represents an unique experimental case to analyze

refer-ees’ behavior and complements the previous results of the literature. In fact, in the case

of theCalciopoli scandal there is no need to adopt any data analysis to gather indirect

evidence of corruption, because its presence has been documented and conﬁrmed by the outcome of the investigation and can be taken as a fact. Referees’ decisions can therefore be disputable or wrong for two reasons: the presence of external forces which

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put them under pressure and alter their impartiality, and/or the intention to take de-cisions in order to favor one of the two corrupting teams. We call these two forms distortion from impartiality as “unconscious” and “conscious” bias, respectively.

In this paper we quantify the eﬀect of the two forms of bias on the performance

of the four teams involved in the scandal during the 2004/05 Serie A season. Our

study’s objective is to answer the following questions: did referees’ bias aﬀect the performance of the teams involved in the scandal? Did the the four teams involved in the corruption scandal indeed obtain abnormal positive results when directed by the

referees convicted of fraud (henceforth called “biased referees”), even after we control

for the presence of “unconscous” bias? Is it possible to identify speciﬁc interaction eﬀects among individual teams and single biased referees?

To this end, we construct a model where the probability of victory in a football match is aﬀected by both team eﬀort and bribing activities, and then test its

implica-tions using data from the 2004/05Serie Aseason. As mentioned above, the Calciopoli

case allows us to examine the simultaneous eﬀect of the two forms (conscious and unconscious) of referee bias. Therefore, controlling for the unconscious referee bias, our empirical results highlight the full eﬀect of the conscious bias (due to corruption activities) on the performance of the teams.

Some interesting results emerge from our empirical investigation. In line with the literature on the economics of football, the home team bias exerted relevant eﬀects

in the 2004/05 Serie A season: referees’ decisions, on average, tended to favor the

team playing at home. In addition, the effect of financial resources also emerges as a determinant for the probability of winning a game. Our results indicate that biased referees did not have a generalized effect on the teams involved in the scandal. Their performance has not been affected by the type of referee directing their matches.

Dis-aggregating the analysis by team shows that in fact Juventus performed worse than

usual when directed by a biased referee, whereas for the three other teams there is no significant effect. Finally, disaggregating the analysis by referee, we are able to uncover different individual referee effects that affected the performance of some of the teams involved in the scandal.

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All in all, these results are quite surprising because they seem to imply that engaging in costly and risky illegal practices has produced no improvement in the performance of the corrupting teams, and in one case has produced a worse than average performance. A possible interpretation of these results suggests that managers, team owners and referees compete and interact to influence the outcome of the matches. Therefore, each team’s overall performance in the season is affected by the net effect of the single collusive interactions it entertains.

The paper is organized as follows. In Section 2 we discuss the theoretical and empirical literature on referee bias in sport contests, and motivate our study. In Section 3 we describe our model and its testable implications. In Section 4 we present the data and the empirical framework. In Section 5 we comment on the main results. In Section 6 we draw some concluding remarks.

2 Performance in sport contests and the role of

ref-erees

The economics of sport has received large attention from the literature since Rotten-berg (1956). Broadly speaking, research has focused on two main questions. The first revolves around the optimal design of sport contests. It deals with devising the most appropriate incentive scheme to ensure that all participants unfold their best effort to win the prize. Rosen (1986) models the incentive properties of prizes in sequential elim-ination tournaments in order to determine the best contestants and support the survival of the fittest. Ehrenberg and Bognanno (1990) empirically examine how the structure of prizes in the Professional Golfers’ Association influences players’ performance and elicits their efforts. The importance of the optimal design of sport competitions is also clarified by Taylor and Trogdon (2002). They test how changes in the rules for deter-mining the draft order in NBA recruitment can have a negative incentive for teams, which may voluntarily underperform during the season. An extensive review of this branch of literature can be found in Szymanski (2003).

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that can aﬀect the probability of success in a sport competition. Two variables emerge as the main determinants of the probability of success: ﬁrst, the strength of the par-ticipants/teams. In football, this variable has been typically measured in terms of the number of goals scored and conceded in a match. Reep et al. (1971) use a Poisson and a negative binomial distribution to model the distributions of the number of goals scored and conceded by the teams participating in a football tournament, respectively. More recent studies use the relative attack and defense capabilities of the two com-peting teams to determine “strength” (Maher, 1982; Dixon and Coles, 1997; Rue and Salvesen, 2000), while Kuypers (2000) constructs the cumulative points gained by the competing teams. The second determinant of the probability of success is the so-called

home bias, defined as “the consistent finding that home teams in sport competitions win over 50% of the games played under a balanced home and away schedule” (Coruneya and Carron, 1992; p. 97). This effect has been extensively investigated by the lit-erature with respect to both team and individual sport contests. Bray and Carron (1993), Holder and Nevill (1997), and Nevill et al. (1997) find that home advantage can have a significant impact on the athletes’ performance in alpine skiing, tennis, golf and skating, respectively. However, in the case of individual sport contests, some of the evidence is not robust (Koning, 2005). In those cases the estimation of home bias is difficult since the athletes generally play elimination tournaments, in which there is no guarantee of reaching the final stage of the competition. The home bias has been studied more effectively in team sport contests. Nevill and Holder (1999) document how home advantage significantly helped the French football team in the 1998 World Cup. Clarke and Norman (1995) and Carmichael and Thomas (2005) show that the home bias was statistically significant in English football matches played in the seasons going from 1981 to 1996. Bray (1999) reaches the same conclusion by examining the National Hockey League matches from 1974 to 1993.

According to the literature, home bias is due to five different factors: crowd, learn-ing/familiarity, travel, rule and referees. The crowd factor is defined as the spectators’ support for the home team which, normally, largely outweighs that for the visiting team. This unbalance is expected to affect the relative performance of the two

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com-peting teams and, in turn, the outcome of the game. The learning/familiarity factor denotes the advantage for the home team stemming from the knowledge of the envi-ronment in which the match is played. Again, this unbalance is expected to affect the outcome of the match. The travel factor is related to the resources and time that the visiting team has to spend to reach the hosting field. Finally, the rule factor acknowl-edges that, sometimes, the rules of the game may favor the home teams in certain sports. For example, in ice hockey the home team is given the last opportunity to make a line change. This implies that the home teams can observe the players chosen by their opponents and perhaps catch a glimpse of their strategy. The team playing away has no such opportunity and must change players without knowledge of what their opponent is doing. Another example of a rule benefiting the home team occurs in baseball, where the home team always bats last to end the game. If the home team is losing going into the last inning, they still have one last chance to win the game.

The literature has also highlighted how the surrounding environment, apart from influencing the outcome directly, may also have an indirect impact by affecting referees’ decisions. In other words, because of the pressure the environment is able to exert on referees, they are likely to treat more favorably the home team. Mohr and Larsen (1998) analyze the results of the Australian football championship and support the existence of the home referee bias, measured as the number of free kicks awarded to the home team compared to those given to the teams playing away. In an interesting paper, Nevill et al. (2002) conducted an experiment using two groups of qualified referees. Referees were asked to award free kicks while watching the same matches. A first group of referees watched the matches hearing the crowd noise, while the other evaluated the same episodes without sound. The results of the experiment strongly support the existence of a home referee bias: the first group of officials awarded fewer free kicks to home teams, whereas the number of free kicks awarded to the visiting team was unaffected.

Other more recent contributions oﬀer further support to the home referee bias. Sutter and Kocher (2004) ﬁnd that referees are more inclined to add extra time when the home team is not winning the match; Garicano et al. (2005) show that the referee

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bias depends on the number of spectators watching the match. Dawson et al. (2007) ﬁnd that the team playing away is more likely to face sanctions from the referees than

the home team.1 Finally, Dohmen (2005) and Scoppa (2008) highlight the eﬀect of

the type of stadium (e.g., with or without running track) and crowd’s proximity to the pitch on home team bias.

3 Eﬀort and bribing in sport: the model

When engaged in a game, two teams are contenders ﬁghting over a valuable prize (see, e.g., Szymanski, 2003). This section presents a simple theoretical model describing the optimal behavior of teams involved in a football match. The model tries to mimic the

situation highlighted by the Calciopoli investigation and hence explicitly introduces

bribing activities.

Several elements may influence the strategic behavior of the teams playing a football match. For example, not all the matches take place simultaneously. Therefore, the results of matches played earlier might influence the behavior of teams playing later in the day (or season). In addition, some teams are involved in both national and international (e.g., Champions League) football matches. Hence, they distribute their effort strategically between national and international games. Rather than modelling such a complex scenario, we assume that each team aims at winning all the matches. This implies that the competing teams play every match in isolation, without any concern for the results of other matches played and/or to be played.

We consider two competing teams whose aim is to maximise the economic value of winning a football match. The teams’ objective can be achieved by allocating their financial resources between the two inputs that affect the probability of success in the game: team effort and bribing the referee. For example, the investment in effort consists in resources spent to sign star players, as well as to build more effective infrastructure.

1_{This ﬁnding is in contrast to Ridder et al. (1994), who do not reject the hypothesis that the}

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In this framework, each team maximizes the following objective function:

Mi =πi(ei, ej, zi, zj, γi, γj)V,

s.t. peei+pzzi =Wi,

(1)

wherei̸=j andi, j ∈ {1,2}. Miis the expected monetary gain for teamifrom winning

the match, (Wi) denotes the ﬁnancial resources that team i allocates between eﬀort

(ei) and bribing (zi). γi denotes the prior probability that team i wins the match. V

is the monetary value of victory, and pe, pz are the unit prices of eﬀort and bribing,

respectively. Finally, πi(·) is the probability that team i wins the match. We model

this probability explicitly as a contest success function (CSF):

πi(ei, ej, zi, zj, γi, γj) = γi+eαi i z βi i γi+eαiiz βi i +γj+e αj j z βj j , (2)

where αi, βi ∈ (0,1) are the weights that teams attribute to the importance of eﬀort

and bribing in determining the probability of success. Note that if we let eαi

i z βi

i =Ki,

(2) simpliﬁes to the CSF proposed by Corchon (2000) and axiomatized by Rai and Sarin (2009).

As shown in Eq. (1), team i’s expected return from playing a match depends on

three factors: its own decision as well as teamj’s decision on how to distribute resources

between effort and bribing and on each team’s exogenously given prior probability. Prior probabilities are common knowledge and reflect, for example, whether the match is played at home or away. Differently to what happens in US professional sport, in soccer there is no upper limit to the amount of resources a team can spend on players’ wages, hence each team can sign better and better players, the higher the budget.

To define the optimal allocation of each team’s financial resources, maximize the following function with respect to effort and bribing:

Li = γi+eαiiz βi i γi+eαiiz βi i +γj +e αj j z βj j V −λ(peei+pzzi−Wi),

where, for simplicity,V,peandpz are team-invariant. Maximization yields the optimal

values of eﬀort and bribing as follows:

e∗_i = αi αi+βi Wi pe , (3) z_i∗ = βi αi+βi Wi pz . (4)

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Note that the equilibrium levels of eﬀort and bribing are inversely related to their respective prices and positively related to their respective weights and to the amount

of resources available to each team. Furthermore, e∗_i and z_i∗ do not depend on V and

on the prior probabilities, γi and γj. Substituting (3) and (4) for both teams into (2),

we derive the probability of winning the match for team i as follows:

π_i∗ = γi+ ( αi αi+βi Wi pe )αi( βi αi+βi Wi pz )βi γi+ ( αi αi+βi Wi pe )αi( βi αi+βi Wi pz )βi +γj + ( αj αj+βj Wj pe )αj( _β j αj+βj Wj pz )βj (5)

and plugging (5) into (1) yields the expected return:

M_i∗ = π_i∗V, M_j∗ = (1−π_i∗)V.

We use comparative statics to assess how changes in the parameters of the model affect the probability of winning a match. For the sake of simplicity, it is more conve-nient to deal with the ratio of expected returns M_i∗/M_j∗ =π_i∗/π_j∗. Three propositions are obtained. The first relates with the effect of a change in the prior probabilities of the two competing teams.

Proposition 1. The ratio of the teams’itoj probabilities of winning a match increases in γi and decreases in γj: ∂ ( π∗_i π∗_j ) ∂γi >0 and ∂ ( π_i∗ π_j∗ ) ∂γj <0.

The result of Proposition 1 is not surprising and reveals that, other things being equal,

π_i∗ are positively related to the prior probabilities (e.g., the likelihood of winning is

higher for the team hosting the match).

The next Proposition highlights the eﬀect of each team’s ﬁnancial resources.

Proposition 2. The ratio of the teams’ i to j probabilities of winning a match is increasing in Wi and decreasing in Wj:

∂ ( π∗_i π∗_j ) ∂Wi >0 and ∂ ( π_i∗ π_j∗ ) ∂Wj <0.

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A larger amount of available resources makes the team’s ﬁnancial constraint less binding and, consequently, more resources may be allocated to eﬀort and/or bribing. This, in turn, rises its probability of victory and reduces that of its opponent.

Our ﬁnal result refers to the relationship between the probabilities of victory and the weights that each team assigns to the importance of eﬀort and bribing.

Proposition 3. The ratio of the teams’ i to j probabilities of winning a match is increasing in αi and βi and decreasing in αj and βj:

∂ ( π∗_i π∗_j ) ∂αi >0, ∂ ( π∗_i π∗_j ) ∂βi >0 and ∂ ( π_i∗ π_j∗ ) ∂αj <0, ∂ ( π∗_i π∗_j ) ∂βj <0.

The higher the perceived impact of eﬀort (resp. bribing) on team i’s probability of

success, the larger its investment in eﬀort (resp. bribing) and the higher its probability of winning a match, other things being equal.

3.1 Testable hypotheses

The propositions presented in this section can be used to propose some testable hy-potheses about the determinants of a team’s probability of winning a football match.

The exogenously given prior probabilitiesγi in Proposition 1 depend, for example,

on whether the match is played at home or away. This leads us to state the following hypothesis:

H1 A team’s probability of winning is positively related to it hosting the game. H1 allows us to test for the existence of the home team bias. Proposition 2 refers to the marginal eﬀect of the size of the team’s budget on its probability of winning a game. This leads us to formulate the following hypothesis:

H2 A team’s probability of winning is positively related to its ﬁnancial resources and decreasing in that of its opponent.

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The result in Proposition 3 highlights the eﬀect of theβ’s, i.e. the relative perceived impact of bribing activities. Hence, one could be interested in analyzing the realized

eﬀect of β’s on the same team’s probability of winning a game and check whether the

two are positively correlated. Unfortunately this cannot be tested empirically, since of course teams’ preferences are not observable. Nor we can observe how the teams split their budget into bribing activities and effort to improve its quality. Generally, those managers who engage into illegal activities tend to not leave a trail after them. However, our data offer a very useful identification device, in that we can draw some

implications from the ﬁnal outcome of the Calciopoli investigation. More speciﬁcally,

we can distinguish between teams which distribute all their resources to eﬀort and teams which also engage into illegal activities. Likewise, we can distinguish between impartial and corrupt referees. Hence, we can say something about the eﬀectiveness of bribing activities undertaken by corrupting teams. This leads to the formulation of the following hypothesis:

H3 The probability of winning a match for teams involved in Calciopoli is

indepen-dent of whether the match is directed by a biased referee.

One-sided rejection of H3 might indicate that biased referees helped teams involved in the scandal to win matches. Conversely, not being able to reject H3 points towards the fact that corrupting teams did not perform any better when directed by biased referees. H3, therefore, allows us to test for the existence of conscious referee bias.

4 Empirical Framework

We base our analysis on the 2004/05 Serie A season, in which 20 teams competed for

the title. For each match, we collected data on the team playing at home, the result of the game and the name of the referee. Following Ferrall and Smith (1999) and Duggan and Levitt (2002) we consider each team in a match as observation units. Therefore, our data consists of 760 team-matches corresponding to the 380 games played during that season.

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Our theoretical model suggests that teami’s probability of winning a match depends on three factors: the home team bias, the team’s ﬁnancial resources, which determine the quality of the team’s past performance as compared to that of the challenger, and the amount of corrupting activities. We model this probability as follows:

Pr(W in)i = µ1+µ2Homei+µ3Strengthi+µ4T urnoveri+µ5Ref eree (s) i + +µ6T eam (j) i +µ7T eam (j) i ×Ref eree (s) i +εi, i= 1, . . . ,760, (6)

where the dependent variable Pr(W in) is the probability of winning a football match.

We use ordered logit models for estimating parameters to take into account all the outcomes of a football match (victory, draw and loss).

Homei is a dummy variable assuming value 1 if the team hosts the match and 0 otherwise. This variable captures, among other eﬀects, the home team bias, and it is a proxy of the unconscious referee bias. Therefore, according to H1, its associated

coeﬃcient (µ2) is expected to be positive.

T urnoverimeasures the diﬀerence in ﬁnancial resources between the two competing

teams, and it is calculated as (log) ratio between turnovers of the two teams. Strengthi

is a measure of the strength of the challenger. This variable is measured as the diﬀerence between the points accumulated in the league table by the opponent team up to when the match is played and the average number of points collected by all other teams. Therefore, it has a positive value if the challenger is stronger than average and a

negative value otherwise. We believe it is important to include the variable Strengthi,

because it reflects the effect of past financial resources available to the management in order to build a stronger team. According to H2, since (on average) the stronger the

challenger the lower the probability of winning, we expect the sign of the parameter µ3

(associated to the strength of the opponent) to be negative and that of µ4 (associated

to the relative ﬁnancial resources of the team) to be positive.

Ref eree(_is), for s= 1, . . . ,8 (the eight referees involved in Calciopoli), is a dummy

variable taking value 1 if the referees is involved in the scandal and 0 otherwise. This

variable captures the eﬀect exerted by each biased referee on the likelihood of winning a football match. T eam(_ij), for j = 1, . . . ,4 (the four teams involved in Calciopoli),

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is a dummy variable assuming value 1 if the team j is involved in the scandal and 0 otherwise. This variable captures the eﬀect of involved teams on the probability of winning a match.

In order to evaluate whether one (or more) of the eight biased referees favoured one (or more) of the four teams involved in the scandal, we constructed the interaction vari-able T eam(_ij)×Ref eree(_is). If the coeﬃcientµ7 is statistically signiﬁcant and positive,

then the likelihood of winning a match increases when a biased referee is associated

with a team involved in Calciopoli. This would imply rejection of H3 and provide

evi-dence of a positive eﬀect of a biased referee on the performance of a corrupting team. The variables Ref eree(_is), T eam(_ij), T eam(_ij)×Ref eree(_is), therefore, capture diﬀerent aspects of the conscious referee bias, after controlling for all other factors.

5 Empirical results

5.1 Preliminary analysis

In Table 2, we report the results of a preliminary analysis that builds on some

de-scriptive statistics. ColumnA reports the teams we focus on, and Column B the total

number of matches for each team, disaggregated by referees according to whether they

are (they are not) biased (B1 and B2, respectively). Column C displays the average

points per game for each set of teams, disaggregated by referee, again according to

whether they are (they are not) biased (C1 and C2, respectively). Finally, Column D

reports the t-statistics to test the null hypothesis that the diﬀerence between the two

means in C1 and C2 is zero.

Out of the whole sample of 760 observations, 274 matches have been directed by a biased referee (35%) and 486 by another referee (65%). The average points per game in

the two diﬀerent cases is similar (1.33vs 1.34), and in fact the associated test statistic

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Interestingly, this is also the case when only the subsample of the teams involved in Calciopoli is considered. The average points are still similar (1.76 vs 1.57), and the outcome of the test is the same. Therefore, this simple analysis seems to suggest that the biased referees did not, as a group, have a systematic eﬀect on the outcome of matches.

Results partially diﬀer when the analysis is conducted at team level. Fiorentina

was directed 13 times by a biased referee and 25 times by unbiased referees. In the

ﬁrst case, Fiorentina realized an average of 1 point per game and in the second case

1.16 points. However, the test does not reject the hypothesis that the two means are equal.

Juventus has been directed 20 times by a biased referee and 18 times by unbiased referees. This team had an average of 1.90 points in the ﬁrst case, and 2.67 in the second case. Moreover, the test strongly rejects the hypothesis that the diﬀerence between

the two means is zero. Hence, the evidence suggests that Juventus has performed

remarkably worse when refereed by biased oﬃcials.

Lazio has been refereed 13 times by a biased referee and 25 times by other referees. This team realized an average of 1.54 points per game in the ﬁrst case and 0.96 points in the second case. The associated test, however, does not reject the hypothesis that

the two means are equal. This is also the case for AC Milan. Its average points per

game when refereed by a biased oﬃcial is higher (2.35 vs 1.86), but the test does not

reject the null hypothesis of equality.

Summarizing, results from this preliminary analysis suggest thatFiorentinaachieved

a lower (but statistically not signiﬁcant) average of points per game when refereed by

bi-ased oﬃcials;Juventus achieved a lower (and statistically signiﬁcant) average of points

of per game when refereed by biased oﬃcials; Lazio and AC Milan had a higher than

normal (but statistically not signiﬁcant) average of points per game when refereed by biased oﬃcials.

These results, coupled with the evidence that the average number of points per

game realized by all the teams playing in Serie A is not inﬂuenced by the type

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have a generalized eﬀect. Rather surprisingly (e.g., Juventus), referee bias seems to inﬂuence the results of football matches in the opposite way, damaging the team it was supposed to favor.

5.2 Baseline results

The analysis in the previous section misses some important elements. Among these, the most important is the relative quality of the competing teams. If one really wants to determine the eﬀect of a referee on the outcome of a match, then the quality of

the opponent team should be properly controlled for. Take the case of Juventus, for

example. From the previous section, one concludes that its performance has been worse when refereed by biased oﬃcials. However, it could be the case that the matches refereed by biased referees have been played against very strong teams. So, the outcome of the previous test could be driven by the strength of the opponent, rather then by a biased referee. Therefore, it is important to disentangle these two eﬀects.

To this aim, we estimate model (6) using an ordered logit speciﬁcation. Results are reported in Table 3. We estimate six models, according to the level of aggregation of

the variable T eam.

In Column (A), T eam is a dummy variable assuming value 1 if the team belongs

to one of the four ones involved in the scandal, and 0 otherwise. Results suggest

that the variable Ref eree has a negative but not statistically signiﬁcant eﬀect on the

probability of winning the match. Therefore, the evidence suggests that the subset of biased referees did not change, on average, the likelihood of winning a match. Hence, H3 seems to be rejected by the data. As expected, the variable measuring the quality in the past performance of the opposite team is highly significant and displays the expected sign. This result seems to confirm the validity of H2. Moreover, in line with the empirical evidence emerging from the existing literature (Garicano et al., 2001; Goddard, 2005; Dohmen, 2005; Dawson et al., 2007), results confirm the existence

of home team bias. The variable Home is highly signiﬁcant and displays a positive

sign. On average, therefore, playing at home provides hosting teams with signiﬁcant

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average, therefore, richer team have higher probability of winning. These ﬁndings are in line with H1 and H2, and results are robust across all estimated models.

The variableT eamhas a positive sign and it is statistically signiﬁcant. This possibly

reﬂects the fact that two of the teams involved in the investigation (AC Milan and

Juventus) scored a very high number of wins during the season. Finally, the interaction

term, Ref eree×T eam, is not statistically diﬀerent from zero. This seems to suggest

that biased referees had no systematic effect on the outcome of football matches. Of course, these results may be due to some counterbalancing effects. If referees are biased in favor of a particular team and against another one, the effects of their actions on match results might turn out to be not statistically significant, if they are taken on aggregate. For this reason, we have repeated the analysis by creating a dummy variable for each of the four teams. Results of this exercise are reported in Columns (B) to (F).

Starting from Column (B), F iorentina has a negative (but not signiﬁcant) sign,

reﬂecting the low number of matches won during the season; the interaction term with

Ref ereeis also not signiﬁcant, suggesting no systematic inﬂuence of biased referees on the probability of winning a match.

The variable J uventus is statistically signiﬁcant and positive. This is not

surpris-ing, since the team won a high number of matches in the season. The interaction of

this variable with Ref eree is negative and statistically signiﬁcant. Therefore, biased

referees had a negative impact on the probability of Juventus winning games. This

result seems to reject H3, as far as J uventus is concerned. However, the direction of

rejection is actually the opposite of what we expected to see. Given all the evidence

about Juventus’ management pervasive corrupting activities, we would have expected

a better than average performance associated with biased referees. Instead, we ﬁne the opposite. The result is very surprising, given that prosecutors argued that the management of this team had established a network of connections able to illegally inﬂuence the outcome of matches and indeed of the entire season to its own advantage.

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Column (D) shows that Lazio’s poor performance in the season explains the

neg-ative and signiﬁcant sign of the variable T eam, while the interaction term is not

sig-niﬁcant, pointing towards non rejection of H3. Column (E) shows that the coeﬃcient

associated to AC M ilan is statistically signiﬁcant and positive, reﬂecting the good

performance of the team during the season. The interaction with the variableRef eree

is not statistically significant. This confirms the absence of any systematic effect of consciously biased referees on the probability of winning a match. H3 seems to be consistent with the data in this case as well. Finally, Column (F) reports results ob-tained including all the individual teams involved in the investigation. Qualitatively, it is possible to draw the same conclusions as those coming from models (B) to (E).

The evidence seems to suggest that all the teams penalized after the enquiry for trying to illegally inﬂuence the results of their matches did not in fact achieve any better performance (statistically) when refereed by biased oﬃcials. Therefore, H3 seems to be consistent with the data. Furthermore, quite surprisingly, our results, disaggregated

by team, suggest that Juventus, the team accused of masterminding the whole system

of corruption, has been the only one which consistently underperformed when directed by those referees.

5.3 From aggregate to referee-speciﬁc eﬀects

The results in Table 3 might be explained by possible individual eﬀects of biased referees compensating each other. Therefore, we check whether there is any systematic

relationship between each one of the four teams involved inCalciopoli and one or more

of the eight biased referees. Table 4 reports the results of a descriptive analysis linking

individual biased referees and the performance of the teams involved in Calciopoli.

Inspection of the Table reveals that Juventus played the highest number of matches

refereed by biased referees, compared to the other three teams. We also note that, in line with the results of difference in means tests shown in Table 2, there is no significant difference between the outcome of the matches refereed by biased and other referees for AC Milan, Fiorentina and Lazio. However, Juventus seems to have performed

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during the 2004/05 Serie A season and all of them were directed by biased referees.

In order to evaluate the eﬀect of individual biased referees, we estimate Model (6)

with the disaggregated variable Ref eree and report the results in Table 5.

We note that, for example, Ref 1, Ref 2 and Ref 4 are associated with a signiﬁcantly

worse than usual performance of Juventus. Ref 5 (resp. Ref 7) is associated with a

signiﬁcantly better (resp. worse) than usual performance of Fiorentina. Lazio’s and

AC Milan’s performances do not seem to be positively or negatively associated with any individual biased referee.

Table 6 summarizes the ﬁndings of Table 5: the performance of Juventus is

neg-atively associated to some of the biased referees, the performance of Fiorentina is

positively associated to one referee and negatively associated with a diﬀerent referee,

whereasLazio andAC Milan have obtained the same average results with both groups

of referees.

A potential concern about our results relates to the level of aggregation of the dependent variable. Indeed, our sample consists of all the matches played in the season. Hence, we have repeated the empirical analysis, focusing only on the matches played by the team(s) involved in the scandal.

Results are reported in Table 7, where we present six models. Column (A) is obtained by using observations from all the teams involved in the scandal. In line with our previous conclusions, results suggest that the stronger the opponent team, the lower the likelihood of winning a match. Moreover, the variable capturing the

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home team bias is statistically signiﬁcant and positive. Finally, the evidence rejects the hypothesis that biased referees, when considered as a group, have a diﬀerent impact on the probability of victory with respect to the other group of referees.

Column (B) reports the results obtained by using observations related to teams untouched by the scandal. Results are very similar to previous ones. Columns (C), (D), (E) and (F) report results obtained by estimating model (6) using observations onFiorentina, Juventus,Lazio and AC Milan are used, respectively. Results generally conﬁrm our analysis.

The variable Strength exerts a negative impact on the likelihood of winning a

match for Lazio and AC Milan, whereas in the case of Fiorentina and Juventus it

is not signiﬁcant. This ﬁnding can be explained by the fact that Juventus (resp.

Fiorentina) won (resp. lost) a large number of matches regardless of the strength of the opponent. Interestingly, the only team for which the home eﬀect is signiﬁcant is

Fiorentina. Juventus had, again, a signiﬁcantly worse than average performance when directed by biased referees. There seems to be no systematic eﬀect of biased referees

on the performance of Fiorentina. This indicates that the individual eﬀects of Ref 5

and Ref 7 possibly counterbalanced each other.

In Table 8 we report the results obtained with individual biased referees. Previous

results are conﬁrmed: Juventus has consistently under-performed when refereed by Ref

1, Ref 2 and Ref 4, while Fiorentina over (resp. under)-performed when refereed by

Ref 5 (resp. Ref 7).

5.4 Robustness Checks

In this Section, we run some robustness checks in order to further validate the results obtained and discussed in the previous Section. Our ﬁrst concern is to examine whether

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biased referees had an inﬂuence on the results of matches where corrupting teams faced strong challengers, competing for the same position in the tournament. To investigate this possibility we have replicated the analysis performed in Table 3, including a set of interaction terms to capture these eﬀects. The model we estimate is given by:

Pr(W in)i = µ1 +µ2Homei+µ3Strengthi+µ4T urnoveri+µ5Ref eree (s) i + +µ6T eam (j) i +µ7T eam (j) i ×Ref eree (s)

i +µ8Strengthi×Ref eree

(s)

i

+µ9Strengthi×Ref eree

(s)

i ×T eam

(j)

i +εi, i= 1, . . . ,760. (7)

The results of this analysis are reported in Table 9 where we show six diﬀerent models

according to the level of aggregation of the variable T eam. Reassuringly, the inclusion

of the interaction terms does not alter the effect and the significance of the regressors used in model (6). Furthermore, the results in Column (A) indicate that biased referees did not, as a group, have an effect on the probability of winning a match, when a team

faced a strong competitor (Ref eree×Strength). This result is further conﬁrmed by fact

that the eﬀect on the probability of success of the variableRef eree×Strength×T eam

is not statistically signiﬁcant.

It’s possible that these results are driven by some counterbalancing eﬀects of individ-ual referees eﬀects. Therefore, we replicate the analysis by including dummy variables for each team involved in the scandal. In Column (B) we observe that, although the

eﬀect of F iorentina×Strength is not statistically signiﬁcant, the triple interaction

termF iorentina×Strength×T eamis negative and signiﬁcant. This result does seem

in line with our ﬁndings in Table 5, where we noted that F iorentina’s probability of

winning games was favorably (resp. negatively) aﬀected by Ref 5 (resp. Ref 7).

Column (C) shows that J uventus performed better when it faced stronger teams

(J uventus×Strength). However, its probability of winning decreased when it played

against strong teams and the matches were directed by biased referees. (J uventus×

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in the previous Sections. Estimates in Columns (D) and (E) do not show any signiﬁcant statistical impact on the probability of winning emanating from the interaction eﬀects between the strength of the opponents and the biased referees in matches played by

Lazio and M ilan. Finally, in Column (F) we include all the teams involved in the scandal and note no diﬀerence with respect to speciﬁcations (A) to (E).

The second robustness check relates to the possible endogeneity of the variable

Strength. Corrupting teams likely get some advantage from their illegal activities, so

the variable Strength, which measures the quality of each team, compared to the rest

of the teams in the tournament could be inﬂuenced by these illegal practices. In order

to check for this, we use a suitably chosen instrument for Strength. In particular, for

each team we use the cumulative points obtained by the team ﬁnishing in the same ﬁnal

position at the end of the subsequent Serie A Championship.2 _{The goodness of the}

instrument is conﬁrmed by the ﬁrst stage regression R2 = 0.72. Results are reported

in Table 10 and they conﬁrm the ﬁndings of the previous Sections.

5.5 Discussion

How do we reconcile the empirical results with the body of evidence produced by the investigation, which has determined severe sanctions to the team involved? Our results show that, on aggregate, the performance of the corrupting teams in matches directed by corrupt referees was not statistically diﬀerent from the case when the match was

directed by an unbiased referee. In one speciﬁc case (Juventus) it was actually worse.

At first sight, these rather surprising results suggest that investing time and money in illegal activities did not in fact pay off. There is no direct evidence of a beneficial effect of biased referees on the performance of the corrupting teams.

A possible interpretation of these results is the existence of a sort of parallel cham-pionship where team managers and referees compete and interact to inﬂuence the outcome of the matches. In such a scenario, some teams undertake a collusive rela-tionship with a given referee or group of referees in order to manipulate match results.

2_{We use the subsequent Championship because, starting from 2004/2005,} _{Serie A} _{counted 20}

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Therefore, each team’s overall performance in the season is affected by the net effect of the single collusive interactions it entertains. For example, the fact that (on aggregate) the performance of two out of the four teams involved in the scandal is not statistically associated to biased referees, might be explained by possible effects coming out of in-dividual relationships between referees and teams compensating each other. Similarly,

the fact that the eﬀect of the conscious referee bias on the performance of Juventus is,

on aggregate, negative, could be a result of the net eﬀect of the individual relationships between each biased referee and each of the corrupting teams.

What are the implications of conscious referee bias on the fairness of the entireSerie

A championship? Referees are required to act impartially, and the team winning the

title fairly deserves it if all the matches of the championship are directed by unbiased referees. However, referees’ decisions can be unconsciously biased (for example, because of the home bias effect). Because referees favor the team playing hosting the game, the result of every single match is likely to be biased. However, because each team hosts the same number of matches within the season, the title should still be fairly assigned. In this case, the net effect of the home team bias on the overall outcome of the championship is likely to be zero. However, in the case of conscious referee bias, to the extent that biases do not exactly offset each other, the overall outcome of the championship should be considered unfair.

6 Concluding remarks

In the summer of 2006 a game-fixing scandal shook Italian football. Team managers were convicted for corrupting referees in order to manipulate match results. In this paper, we used a standard model borrowed from conflict theory to describe how the outcome of a football match may be affected by factors directly associated with the relative quality of the competing teams, as well as by resources used to corrupt referees.

We then used data on 2004/05 Serie A season to to quantify the eﬀects of the

diﬀerent determinants of the result of a match and to assess implications on teams’ performance. The ﬁndings of our paper are important to understand and evaluate

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Team Relegation Points Deduction Other Sanctions Juventus Serie B 9 points Stripped of the 2005/06 title; Out of the 2006/07 UEFA Champions League; 3 home games behind closed doors AC Milan None 8 points Deducted 30 points in 2005/06 season;

1 home game behind closed doors

Fiorentina None 15 points Out of the UEFA Champions League; 2 home games behind closed doors Lazio None 3 points Out of the UEFA Cup; 2 home games

behind closed doors

Table 1

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B B1 B2 C C1 C2 1 A ll the T eam s 760 274 486 1.34 1.33 1.34 -0.06 2 T ea ms In v o lv ed i n C a lc iopoli 152 63 89 1.65 1.76 1.57 0.89 3O th er T ea m s 608 211 397 1.25 1.20 1.28 -0.77 4 F iore ntina 38 13 25 1.11 1.00 1.16 -0.40 5J u v en tu s 38 20 18 2.26 1.90 2.67 -2.19** 6L a zi o 38 13 25 1.16 1.54 0.96 1.35 7 M ilan 38 17 21 2.08 2.35 1.86 1.28 Tab le 2 T e st fo r diffe re nc e in m e a n s C o lum n A re p o rts the subs et o f te a m s w e u se to te st the null h ypo the si s tha t the diffe re nc e in m ea ns is no t st atis tic a lly si gn if ic an t. Colu mn B re p orts th e total n u m b er o f o b se rvation s for ea ch su b se t of te ams , d is aggre gate d b y wh et h er th e ma tc h h a s b een re fe reed b y a b ia sed ref er ee (C o lu m n B1 ) o r o the rwis e (Colum n B2 ). Colum n C re p orts th e a ve rage p o in ts p er game for ea ch su b se t of te ams , d is aggre gate d b y wh et h er th e matc h h as b ee n re fe re ed b y a bias ed re fe re e (C o lum n C1 ) o r o the rwis e (Colum n C2 ). Fina lly, C o lum n D re po rts the t -s ta tis tic s to te st the null h y p o th es is th at th e d if fe re n ce a mon g th e two me an s is n ot st atis tic a lly d if fe re n t. * , * * a n d *** st an d for re je ct ion o f the null hypothe sis at 10% , 5% and 1% le ve l, re sp ec tive ly. N u mb er o f O b ser v a ti o n s A v er a g e P o in ts A D

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Variables (A) (B) (C) (D) (E) (F) Strength -0.020** -0.020** -0.022** -0.019** -0.022** -0.024** (0.010) (0.010) (0.009) (0.010) (0.010) (0.010) Home 0.971*** 0.964*** 0.975*** 0.971*** 0.982*** 1.002*** (0.142) (0.142) (0.142) (0.142) (0.144) (0.144) Referee -0.094 0.004 0.039 -0.047 -0.055 -0.093 (0.161) (0.149) (0.149) (0.147) (0.147) (0.161) Turnover 1.654*** 1.722*** 1.491*** 1.818*** 1.581*** 1.350*** (0.286) (0.270) (0.288) (0.272) (0.284) (0.317) Team -0.066 (0.235) Referee × Team 0.484 (0.368) Fiorentina -0.117 -0.126 (0.337) (0.344) Fiorentina × Referee 0.033 0.103 (0.584) (0.591) Juventus 1.839*** 1.870*** (0.690) (0.692) Juventus × Referee -1.571* -1.447* (0.822) (0.824) Lazio -1.062*** -0.926** (0.406) (0.402) Lazio × Referee 1.008 1.034 (0.775) (0.775) Milan 0.121 0.245 (0.485) (0.497) Milan × Referee 1.212 1.220 (0.764) (0.767) Table 3

Deteminants of the probability of winning a match

The dependent variable is the result of a match, with 0 indicating defeat, 1 draw and 2 victory, respectively. Number of observations: 760. Standard Errors are in parentheses. *, ** and *** stand for significance at 10%, 5% and 1% level, respectively.

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Referee W N L T W N L T W N L T W N L T Ref 1 0123 1203 0000 4206 Ref 2 1113 2125 0000 2013 Ref 3 1023 2002 1012 0000 Ref 4 0000 2114 2114 1001 Ref 5 1102 2013 1023 3003 Ref 6 0000 0000 1001 0000 Ref 7 0112 2103 0000 2204 Ref 8 0000 0000 2013 0000 Biased Referees 3 4 6 13 11 5 4 20 7 1 5 13 12 4 1 17 Other Referees 6 1 1 8 2 5 1 5 3 0 1 8 5 9 1 1 2 5 1 1 6 4 2 1 All Referees 9 1 5 1 4 3 8 2 6 8 4 3 8 1 2 1 0 1 6 3 8 2 3 1 0 5 3 8 Table 4

Teams and referees

The table reports the number and results of matches played by the teams involved in Calciopoli , disaggregated by referees. W

stands for victory, N stands for draw, and L stands for defeat, and T is the total.

Fiorentina

Juventus

Lazio

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Variable Variable

Strength -0.026*** Fiorentina ×Ref5 1.309*

(0.010) (0.785)

Home 1.011*** Fiorentina ×Ref7 -1.060*

(0.146) (0.584)

Turnover 1.359*** Juventus ×Ref1 -1.891*

(0.321) (1.007)

Fiorentina -0.097 Juventus ×Ref2 -2.487**

(0.346) (1.177)

Juventus 2.119*** Juventus ×Ref4 -2.556**

(0.700) (1.266)

Lazio -0.779* Juventus ×Ref5 -1.864

(0.406) (1.781)

Milan 0.661 Juventus ×Ref7 -0.840

(0.505) (0.932)

Ref1 -0.101 Lazio × Ref3 1.423

(0.290) (1.583)

Ref2 0.033 Lazio × Ref4 1.198

(0.400) (0.978)

Ref3 -0.131 Lazio × Ref5 -1.306

(0.517) (0.921)

Ref4 0.062 Lazio × Ref8 1.605

(0.408) (2.208)

Ref5 0.063 Milan ×Ref1 0.732

(0.263) (0.840)

Ref6 0.071 Milan ×Ref2 0.366

(0.825) (2.469)

Ref7 -0.004 Milan ×Ref7 -0.284

(0.279) (1.041) Ref8 -0.111 (0.661) Fiorentina × Ref1 -0.757 (1.123) Fiorentina × Ref2 0.181 (0.974) Fiorentina × Ref3 0.697 (2.006)

Determinants of the probability of winning a match. Interactions biased referees and teams

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Team Positive Impact Negative Impact Fiorentina Ref 5 Ref 7 Juventus

-Ref1; Ref 2; Ref 4

Lazio -Milan -Table 6

Impact of biased referees on the performance of the teams involved in

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Variable s (A) (B ) (C) (D) (E) (F) S tre n g th -0.028 -0.019* -0.017 -0.024 -0.130*** -0.035 (0.023) (0.011) (0.049) (0.053) (0.045) (0.045) Home 0.614* 1.062*** 1.320** 1.160 0.624 -0.060 (0.326) (0.158) (0.658) (0.755) (0.714) (0.754) Re fe re e 0 .360 -0.100 0.305 -1.571** 0.928 0.741 (0.333) (0.161) (0.777) (0.781) (0.790) (0.752) Tu rn ove r 2.263*** 1.397*** 4.618* 1.405 -0.997 0.565 (0.529) (0.343) (2.531) (1.415) (1.795) (1.325) De te m inants of the probability of w inning a m a tc h. Alte rnative s p e c ific a tions Tab le 7 T h e d ep ende nt va ria b le is the re sult o f a m a tc h, w ith 0 indic a ting a d ef ea t, 1 a dra w a n d 2 a v ict o ry resp ect iv el y . Co lu mn (A ) (r esp . (B) ) rep o rt s re su lt s o b ta in ed u si n g ma tc h es p la y ed b y te ams in v olve d (re sp . n ot in volve d ) in C alc iop oli. Colu mn s (C), (D), (E) an d (F) re p o rt re su lts o b ta in ed u si n g m a tch es p la y ed b y F io ren ti n a , J u v en tu s, L a zi o a n d A C M il a n , re sp ect iv el y . * , * * an d *** s tan d f o r s ign if ic an ce at 10%, 5% an d 1% le ve l, re sp ec tive ly.

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Variable s (A) (B ) (C) (D) (E) (F) S tre n g th -0.029 -0.019* -0.026 -0.015 -0.136*** -0.050 (0.023) (0.011) (0.055) (0.064) (0.045) (0.044) Home 0.651** 1.064*** 1.235 1.637** 0.720 -0.240 (0.325) (0.158) (0.795) (0.761) (0.687) (0.739) Re f1 0.290 -0.139 -0.993 -2.324* 0.376 (0.426) (0.292) (1.566) (1.218) (0.704) Re f2 0.070 0.004 1.079 -2.640** -0.300 (0.749) (0.395) (1.233) (1.163) (1.981) Re f3 1.255 -0.249 0.483 1.459 (0.985) (0.541) (2.299) (1.331) Re f4 0.093 -0.045 -3.031** 1.082 (0.741) (0.412) (1.217) (1.104) Re f5 0.327 -0.092 1.767* -2.007 -1.263 (0.774) (0.267) (0.991) (1.936) (1.328) Re f6 -0.475 _(0.712) Re f7 0.117 -0.046 -1.166 -0.710 -0.371 (0.497) (0.277) (0.832) (0.860) (0.973) Re f8 0.657 -0.127 1.569 (1.971) (0.655) (1.760) Tu rn ove r 2.345*** 1.397*** 5.144** 2.551 -0.878 0.135 (0.528) (0.343) (2.544) (1.715) (1.637) (1.461) Tab le 8 De te m inants of the probability of w inning a m a tc h. Alte rnative s p e c ific a tions dis a ggre gate d by re fe re e T h e d ep ende nt va ria b le is the re sult o f a m a tc h, w ith 0 indic a ting a d ef ea t, 1 a dra w a n d 2 a v ict o ry resp ect iv el y . Co lu mn (A ) (r esp . (B) ) rep o rt s re su lt s o b ta in ed u si n g ma tc h es p la y ed b y te ams in v olve d (re sp . n ot in volve d ) in C alc iop oli. Colu mn s (C), (D), (E) an d (F) re p o rt re su lts o b ta in ed u si n g m a tch es p la y ed b y F io ren ti n a , J u v en tu s, L a zi o a n d A C M il a n , re sp ect iv el y . * , * * an d *** s tan d f o r s ign if ic an ce at 10%, 5% an d 1% le ve l, re sp ec tive ly.

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Variables (A) (B) (C) (D) (E) (F) Strentgh -0.017 -0.019* -0.023** -0.016* -0.022** -0.020* (0.012) (0.010) (0.010) (0.010) (0.010) (0.011) Home 0.969*** 0.952*** 0.992*** 0.970*** 0.988*** 1.013*** (0.142) (0.142) (0.143) (0.142) (0.143) (0.145) Referee -0.100 0.002 0.040 -0.050 -0.055 -0.100 (0.160) (0.149) (0.149) (0.147) (0.148) (0.162) Turnover 1.624*** 1.721*** 1.485*** 1.820*** 1.592*** 1.349*** (0.288) (0.270) (0.289) (0.272) (0.285) (0.320) Team -0.059 (0.235) Referee × Team 0.482 (0.376) Referee × Strentgh 0.001 (0.019) Referee × Strentgh × Team -0.062

(0.037) Fiorentina -0.116 -0.128 (0.332) (0.340) Fiorentina × Referee -0.091 -0.009 (0.554) (0.560) Fiorentina × Strentgh 0.034 0.030 (0.027) (0.027)

Fiorentina × Strentgh × Referee -0.169* -0.170*

(0.094) (0.095) Juventus 1.943*** 1.972*** (0.611) (0.613) Juventus × Referee -1.678** -1.549** (0.768) (0.769) Juventus × Strentgh 0.092* 0.087 (0.055) (0.055)

Juventus × Strentgh × Referee -0.176** -0.179**

(0.079) (0.079) Lazio -1.154*** -1.018** (0.446) (0.440) Lazio × Referee 0.932 0.972 (0.820) (0.817) Lazio × Strentgh -0.067 -0.069 (0.048) (0.046)

Lazio × Strentgh × Referee -0.120 -0.119

(0.187) (0.185) Milan 0.174 0.305 (0.530) (0.541) Milan × Referee 1.140 1.158 (0.778) (0.784) Milan × Strentgh -0.026 -0.031 (0.056) (0.057)

Milan × Strentgh × Referee 0.063 0.061

(0.079) (0.080) Table 9

Deteminants of the probability of winning a match. Further interaction terms

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Variables (A) (B) (C) (D) (E) (F) Strength -0.024*** -0.024*** -0.026*** -0.023*** -0.025*** -0.027*** (0.006) (0.006) (0.006) (0.006) (0.006) (0.006) Home 0.568*** 0.565*** 0.573*** 0.568*** 0.571*** 0.582*** (0.083) (0.083) (0.084) (0.083) (0.083) (0.084) Referee -0.046 0.007 0.027 -0.024 -0.028 -0.046 (0.097) (0.088) (0.088) (0.088) (0.089) (0.097) Turnover 0.982*** 1.020*** 0.875*** 1.078*** 0.948*** 0.794*** (0.174) (0.158) (0.168) (0.160) (0.170) (0.191) Team -0.030 (0.143) Referee × Team 0.249 (0.215) Fiorentina -0.084 -0.088 (0.226) (0.226) Fiorentina × Referee -0.029 0.011 (0.405) (0.407) Juventus 1.123*** 1.141*** (0.371) (0.376) Juventus × Referee -0.945** -0.879* (0.453) (0.455) Lazio -0.592** -0.510** (0.236) (0.239) Lazio × Referee 0.561 0.572 (0.403) (0.404) Milan 0.047 0.132 (0.281) (0.288) Milan × Referee 0.625 0.622 (0.410) (0.411) Table 10

Deteminants of the probability of winning a match. Instrumenting for Strength