Estimating the Effect of the Red Card in Soccer: When to Commit an Offense in Exchange for Preventing a Goal Opportunity

Abstract

We study the effect of the red card in a soccer game. A red card is given by a referee to signify that a player has been sent off following serious misconduct. The player who has been sent off must leave the game immediately and cannot be replaced during the game. His team must continue the game with one player fewer. We estimate the effect of the red card from betting data on the FIFA World Cup 2006 and Euro 2008, showing that the scoring intensity of the penalized team drops significantly, while the scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an increased number of goals when a weaker team is punished. We also show when it is better to commit a red card offense in exchange for the prevention of a goal opportunity.

Full text

Estimating the Effect of the Red Card in Soccer
When to Commit an Offense
in Exchange for Preventing a Goal Opportunity
Jan Vecer, Frantisek Kopriva, Tomoyuki Ichiba,
Columbia University, Department of Statistics, New York, NY 10027, USA
October 10, 2008
Abstract
We study the effect of the red card in a soccer game. A red card is given by a referee to signify that
a player has been sent off following a serious misconduct. The player who has been sent off must leave
the game immediately and cannot be replaced during the game. His team must continue the game with
one player fewer. We estimate the effect of the red card from betting data on the FIFA World Cup 2006
and Euro 2008, showing that the scoring intensity of the penalized team drops significantly, while the
scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a
smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an
increased number of goals when a weaker team is punished. We also show when it is better to commit a
red card offense in exchange for the prevention of a goal opportunity.
1 Introduction
In this paper we study the effect of the red card in a soccer game. This problem was previously addressed by
Ridder et al. (1994) using data from 3 seasons of the Dutch professional league. Their study concluded that
the increase in the rate of scoring for the team that is not punished is statistically significant, but they did
not observe a statistically significant decrease in the rate of scoring of the penalized team. Their approach
used statistical methods to estimate the change of the rates of scoring following a red card.
A more recent and extensive study of the effect of the red card was done by Bar-Eli et al. (2006) using
data from 41 seasons of the German Bundesliga. They studied the red card in relation to psychological
effects. They showed that the red card reduces the scoring chances of the sanctioned team, while increasing
the scoring chances for the opposite team. However, they did not look directly at the changes in the intensity
of scoring which is the focus of our paper.
Previous research estimated the corresponding parameters such as probability of winning or losing the
game, or the expected time to score the next goal using the entire statistical sample; thus neglecting the
effects of individual situations in the games. Our approach uses a novel idea to look at the impact of a red
card in each individual game, where we estimate the rates of scoring with data obtained from in-play betting
markets. This enables us to incorporate more rare situations such as several red cards in a particular game.
In this case the estimate from a statistical sample becomes unreliable as the number of such games is small.
It has been argued that betting markets tend to be more efficient in estimating the parameters of the
models when compared to other methods. For instance, betting markets give better results when compared
to opinion polls as shown by Forsythe et al. (1992).
We show in our paper that the red card leads to a fairly substantial decline in scoring intensity of the
penalized team, which is in contrast with the previous study of Ridder et al. (1994), but is in accordance
with Bar-Eri et al. (2006). We also show that the red card leads to a slight increase in the scoring intensity
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of the opposite team. We use data on the top international teams that played in the last World Cup and
in the last Euro Cup, while the previous studies used only national league data where the dynamics of the
soccer game may differ slightly.
The combined effect of the total scoring intensity depends on the relative strength of the two teams. If a
stronger team is penalized, the total scoring intensity typically drops down. If a weaker team is penalized,
the total scoring intensity can increase or remain the same. The decline of the total rate of scoring is typically
larger when the stronger team is penalized when compared with the increase in the total scoring rate when
the weaker team is penalized. However a weaker team commits a red card offense more frequently, and so
the overall effect of the red card is neutral on the expected number of goals.
The last section of our paper determines when it is optimal to commit a red card offense in exchange for
prevention of a possible goal opportunity for the opposite team. We assume that the objective is not to lose
the game in the regulation time. This optimal time comes surprisingly early in the game. When the score
is tied, and the red card offense is not followed by a penalty kick, it is better to make the offense anytime
during the game when the opposing team has 57.5% or higher chances of scoring. If there is a penalty kick
on the top of the red card, the optimal time to stop a certain goal is anytime after the 51st minute. When
a team is leading by one goal, it is slightly less inclined to commit a red card offense. With no penalty kick,
the chances of scoring must be at least 73% in order to commit such an offense. With a penalty kick, a sure
goal should be stopped anytime after the 55st minute. When the team is trailing by a goal, it should become
more aggressive in preventing a goal opportunity when there is no penalty kick, stopping more than 53.2%
chances of scoring at any time during the game. However, the trailing team should become more conservative
when a penalty kick is involved. It is never optimal to commit such an offense.
2 Estimation of the Scoring Intensity
We use data obtained from betting markets on the FIFA World Cup 2006 and Euro 2008. Betting markets
on these two tournaments reached unprecedented efficiency and liquidity. It was also possible to buy and
sell futures type contracts on the outcome of the game or the number of goals scored, and trade them even
during the actual game. Prices of all traded events (win, draw or loss of a given team, and number of goals)
were immediately affected by a goal or a red card.
We analyze data from the betfair.com market to estimate the implied scoring rate of a given team as the
game progressed. The betting market on a given match traded the following most liquid contracts:
•Team 1 to win,
•Draw,
•Team 2 to win,
•Team 1 to score next,
•Team 2 to score next,
•No goal to be scored for the rest of the match,
•Three or more goals scored.
These contracts expired at the end of the regular time of a particular match (90 minutes + injury time).
They were traded before and during the actual game. The contracts are quoted in terms of odds 1 : x. One
can interpret such odds as the probability of this event. It is possible to buy the event (bet that this event
will happen, or “back” it), or sell the event (bet that this event will not happen, or “lay” it). The backer of
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the event bets $1, and receives $xback (creating a total profit of $(xback −1)) if the event happens, otherwise
he loses his $1 stake. The layer of the event bets $(xlay −1) and receives $xlay (creating a total profit of $1)
if the event does not happen, otherwise he loses his $(xlay −1) stake. The market is set up in such a way
that xback < xlay at all times. If the event happens, the backer wins xback −1, but the layer loses a larger
stake xlay −1. The odds xback and xlay are immediately available for backing or laying the event. One can
request better odds, in which case one’s order is queued in the market, and has to wait until it is matched
with a counter party. However, a matching bet may never come.
Before RC After RC
Contract Back Lay Back Lay
Italy 1.87 1.88 2.68 2.70
Australia 8.40 8.60 4.40 4.90
The Draw 2.78 2.80 2.42 2.44
Italy Next Goal 1.82 1.83 2.50 3.00
Australia Next Goal 6.40 7.00 3.55 4.20
No Goal 3.20 3.30 2.84 2.98
Three or More Goals 6.80 7.20 6.00 11.00
Table 1: Odds for different betting contacts for Italy - Australia game immediately before and after the red card
(Italy) in the 50th minute.
Table 1 illustrates this concept with odds taken directly before and after the moment of the red card
in the Italy - Australia game. Italy was sanctioned in the 50th minute of the game. The betting market
suspends all trading when there is an apparent goal or a penalty, so it is rather simple to identify the quotes
that immediately precede and follow such an event. If someone were to bet $1 on an Italian victory before
the red card he would receive $1.87 if Italy won, making a net profit of $0.87. After the red card, the same
event of an Italian victory became less likely with odds of 1:2.68. Note that the layer of the event (before the
red card) had odds of 1:1.88, betting $0.88 instead of $0.87 – a gain that collects the backer of the event. This
is due to the aforementioned market spread between buying and selling of an event. Note that the market
spread may be relatively large if the market exhibits a small liquidity as in the case of three or more goals
bet after the red card (1:6 for back and 1:11 for lay).
The market odds can serve as estimates of the likelihood of a given event. For example, the chances of
Italy winning were between 1:1.88 and 1:1.87 before the penalty, which corresponds to the interval (0.532,
0.535) for the probability of this event. After the penalty, this probability dropped to the interval (0.370,
0.373).
In order to analyze the change in probabilities as they evolved during the game, especially before and
after the red card event, we focus our attention on the scoring intensities of the two teams. Scoring intensities
of the two teams already determine probabilities of winning, drawing or losing the game, the probability that
a particular team scores next, as well as the probability of a certain number of goals. From this perspective,
scoring intensities are the fundamental parameters of our analysis.
In order to find the relationship between the scoring intensities and probabilities associated with the
game, we use the Poisson model for the scoring. We further assume that the scores of the two teams are
independent. This is a reasonable assumption, a typical correlation of the scores of the two teams is very
small. For example, the correlation of the scores in Euro 2008 tournament was only – 0.13. This model
was also assumed in previous research, such as in Wesson (2002). All bets in this case depend only on two
parameters: the scoring intensity of the first team λtand the scoring intensity of the second team µtat the
time t. Thus number of goals scored by each team follows the Poisson distribution. The market implies that
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λtand µtare the expected number of goals to be scored by Team 1 and Team 2 respectively in the rest of
the game (time between the current time tand the end of the match time T).
The theoretical probabilities that correspond to the betting contracts are given by the following formulas.
We assume the current score is Xt:Yt:
Win Team 1 = P(XT> YT) =
∞
X
k=0
P(XT=k−Xt, YT< k −Xt) =
∞
X
k=0 "e−λtλk
t
k!·
k+Xt−Yt−1
X
i=0
e−µtµi
t
i!#
Draw = P(XT=YT) =
∞
X
k=0 "e−(λt+µt)·λ(k+max(Xt+Yt)−Xt)
t
(k+ max(Xt+Yt)−Xt)! ·µ(k+max(Xt+Yt)−Yt)
t
(k+ max(Xt+Yt)−Yt)!#
Win Team 2 = P(YT> XT) =
∞
X
k=0
P(YT=k−Yt, XT< k −Yt) =
∞
X
k=0 "e−µtµk
t
k!·
k−Xt+Yt−1
X
i=0
e−λtλi
t
i!#
Team 1 Next Goal = λt
λt+µt
·h1−e−(λt+µt)i
Team 2 Next Goal = µt
λt+µt
·h1−e−(λt+µt)i
No Goal = e−(λt+µt)
Three+ = P(XT+YT≥3) =
∞
X
k=(3−Xt−Yt)+
e−(λt+µt)(λt+µt)k
k!
Note that we have 7 contracts that depend only on two parameters λtand µt, and so this problem is
over-parameterized. The market may even admit arbitrage (risk free profit) if the contract prices are not
properly related to each other. For instance, the probabilities of Win of Team 1, Draw, and Win of Team 2
should add up to 1. However, due to the inefficiencies of the market, such as the bid-ask spread, asynchronous
trading, and market order delay, it may be difficult to lock into these opportunities.
Let xback be the odds of backing a specific event, and let pbe the theoretical probability of this event.
Then the return of this bet is given by
xback ·p−1.
For instance if one was to bet on an outcome of a fair coin toss (p=1
2), the return would be zero if xback = 2.
If the odds were higher than 2, than the return would be positive, if the odds were smaller, than the return
would be negative. For example when xback = 2.02, the return is 1%, so one should expect to earn on average
$1 per $100 invested. If the odds were xback = 1.98, the return is -1%, so one should expect to lose on average
$1 per $100 invested.
The return for the laying of an event is calculated in a slightly different way because it requires a bet of
$(xlay −1) in order to win $1. The return is given by
xlay
xlay −1·(1 −p)−1.
Given a fair coin toss, the return is zero when xlay = 2. The return is positive when the odds are smaller,
and is negative when the odds are larger. For example, when xlay = 1.98, the corresponding return is 1.02%,
when xlay = 2.02, the corresponding return is -0.98%.
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In order to estimate λtand µtfrom the odds quoted by the betting market, we choose ˆ
λtand ˆµtthat
minimizes the best expected return of all contracts combined, namely finding a minimizer of the following
formula:
min
λt,µt
max
iÃxback
i·Pi(λt, µt)−1; xlay
i
xlay
i−1·(1 −Pi(λt, µt)) −1!,
where Pi(λt, µt) is the probability of i-th event given the intensities of scoring λtand µt. The properties of
the min-max estimator are discussed in Cox and Hinkley (1979).
Return
Contract 1/xlay 1/xback Pi(ˆ
λt,ˆµt) Back Lay
Italy 0.532 0.535 0.534 -0.002 -0.004
Australia 0.116 0.119 0.099 -0.171 0.020
The Draw 0.357 0.360 0.367 0.022 -0.016
Italy Next Goal 0.546 0.549 0.560 0.020 -0.031
Australia Next Goal 0.143 0.156 0.152 -0.030 -0.010
No Goal 0.303 0.313 0.288 -0.079 0.022
Three or More Goals 0.139 0.147 0.130 -0.113 0.010
Table 2: Probabilities implied by the betting odds for the Italy - Australia game right before the time of the red card
in the 50th minute (columns 1/xlay, 1/xback ), probabilities of these events using a Poisson model with the optimal
choice of parameters ˆ
λt= 0.980 and ˆµt= 0.265 (column Pi(ˆ
λt,ˆµt)), and returns that correspond to the buying or
selling these events. Positive returns are typed in bold.
Table 2 illustrates the choice of the optimal parameters for the Italy - Australia game right before the
moment of the red card in the 50th minute. It not only shows the probabilities implied by the betting odds,
but also the theoretical probability of any event that come with the optimal choice of the scoring parameters
ˆ
λt= 0.980 and ˆµt= 0.265. Note that some of the betting contracts yield a positive return for this choice
of parameters (such as backing The Draw, or Italy Next Goal, or laying Australia, No Goal, and Three or
More Goals), but the best expected return is 2.2%, which is well below the standard commission rate of 5%
charged by the market. The optimal choice of the scoring intensities keeps the best return at the lowest level,
because for any other choice of intensities this discrepancy is larger. Although we find some contracts with
positive return, it is still not clear whether they should be bought or sold. Their price looks favorable with
respect to the prices of other betting contracts, but if taken alone, they could still be quoted at a fair price.
After the red card, the scoring rate for Italy fell from 0.980 to 0.677, while the scoring rate for Australia
rose from 0.265 to 0.477.
It is possible to use different estimators of λtand µt. One example is the L2least square estimator
min
λt,µtX
i
µ1
xback
i
−Pi(λt, µt)¶2
+Ã1
xlay
i
−Pi(λt, µt)!2
,
another one is the L1estimator:
min
λt,µtX
i"¯¯¯¯
1
xback
i
−Pi(λt, µt)¯¯¯¯
+¯¯¯¯¯
1
xlay
i
−Pi(λt, µt)¯¯¯¯¯#.
For comparison, L2estimators for the Italy - Australia game prior to the red card are ˆ
λ2
t= 0.959 and
ˆµ2
t= 0.265, L1estimators are ˆ
λ1
t= 0.948 and ˆµ1
t= 0.263, which are close enough to the minmax estimator
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ˆ
λt= 0.980 and ˆµt= 0.265. The advantage of the minmax estimator is that it is more robust when there is
a significant liquidity gap, that is, when some of the discrepancies between 1
xiand Pi(λt, µt) may become so
large that they can affect an estimate that is based on L1or L2. Minmax estimators are not susceptible to
such effects.
Figures 1-3 show the market implied intensities using the minmax estimator for selected games from the
World Cup to illustrate this concept. We start our inference analysis at the beginning of the game (time
t= 0), and finish when the game ends in the regulation (90 minutes) plus injury time. We indicate the break
between the two halves, which lasts 15 minutes.
0
20
40
60
80
100
0.0
0.5
1.0
1.5
Australia v Italy
Italy
Australia
Figure 1: Implied scoring intensities in the Italy - Australia game. A red card was given to Italy in the 50th minute
of the game.
0
20
40
60
80
100
0.0
0.5
1.0
1.5
Italy v USA
USA
Italy
Figure 2: Implied scoring intensities in the Italy - USA game. Italy was sanctioned with a red card in the middle of
the first half (25’), followed by a red card given to the US team at the end of the first half (45’), and the beginning
of the second half (48’).
3 Estimating the Effect of the Red Card
Upon receiving a red card the sanctioned team must complete the rest of the match with one player less.
During the FIFA World Cup 2006 tournament, 28 players in total were sanctioned. During Euro 2008 tour-
nament 3 players received a red card. Our analysis includes a total of 27 cases of red cards; 2 happened
in overtime, and 2 at the very end of the game. We use the method described in the previous section and
estimate the scoring intensities of both teams before and after a red card. Multiple red cards were given in
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0
20
40
60
80
100
0.0
0.5
1.0
1.5
2.0
2.5
Sweden v Trinidad
Trinidad
Sweden
Figure 3: Implied scoring intensities in the Trinidad - Sweden game. Trinidad was sanctioned by a red card in the
early stages of the second half of the game (46’).
4 games; 2 red cards in the Cote d’Ivoire - Serbia game, 3 red cards in Italy - USA and Croatia - Australia
games and 4 red cards in the Portugal - Holland game.
The scoring rate of the sanctioned team is denoted by λ, with λold being the rate prior to the red card,
and λnew being the rate after the red card. Similarly, the rates of the opposing team are denoted by µ, with
µold being the rate prior to the red card and µnew being the rate after the red card. The complete list of
these rates of scoring is given in Table 3.
In order to predict the change in rates after the red card event we use the following regression model:
λnew =θ1·λold +²1
for a certain θ1for the team which was just penalized, and
µnew =θ2·µold +²2
for a certain θ2for the opposite team. The error terms in the model reflect individual situations within each
game such as the suspension of a key player.
Linear regression leads to the following estimates:
ˆ
θ1= 0.663 ≈2
3,
with R2= 0.972, and
ˆ
θ2= 1.237 ≈5
4,
with R2= 0.991. Thus the scoring intensity of the penalized team drops approximately to 2
3of the intensity
that precedes the red card, and the scoring intensity of the opposing team increases approximately to 5
4of
the original intensity before the red card. The graph that illustrates the change of intensity of the penalized
team is given in Figure 4 and the graph that illustrates the change of intensity of the opposing team is given
in Figure 5.
4 Red Card and Expected Number of Goals
This section discusses the question whether we are expected to see more or less goals after a red card. The
total scoring intensity of both teams combined before the red card is given by
λold +µold
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Game Red Card Time λold λnew µold µnew
Trinidad - Sweden Trinidad 46’ 0.224 0.048 1.501 1.741
Korea - Togo Togo 53’ 0.350 0.313 0.702 1.030
Spain - Ukraine Ukraine 47’ 0.430 0.244 0.728 0.858
Germany - Poland Poland 75’ 0.144 0.083 0.608 0.751
Argentina - Serbia Serbia 65’ 0.300 0.227 0.865 0.941
Mexico - Angola Angola 79’ 0.111 0.050 0.480 0.519
Italy - USA Italy 28’ 1.272 0.846 0.381 0.567
Italy - USA USA 45’ 0.563 0.325 0.769 1.037
Italy - USA USA 47’ 0.296 0.099 1.092 1.415
Czech - Ghana Czech 65’ 0.740 0.538 0.364 0.477
Cote d’Ivoire - Serbia Serbia 46’ 0.615 0.561 1.093 1.478
Cote d’Ivoire - Serbia Cote d’Ivoire 92’ – – – – – – – – – – – –
Mexico - Portugal Mexico 61’ 0.567 0.399 0.559 0.628
Italy - Czech Czech 47’ 0.715 0.468 0.968 1.303
Croatia - Australia Croatia 85’ 0.174 0.106 0.194 0.271
Croatia - Australia Australia 87’ 0.256 0.189 0.069 0.140
Croatia - Australia Croatia 93’ 0.088 0.110 0.115 0.065
Ukraine - Tunisia Tunisia 46’ 0.406 0.202 1.064 1.285
Germany - Sweden Sweden 35’ 0.547 0.493 1.054 1.306
Portugal - Holland Portugal 46’ 0.479 0.286 0.825 0.983
Portugal - Holland Holland 63’ 0.997 0.618 0.251 0.372
Portugal - Holland Portugal 78’ 0.245 0.153 0.427 0.523
Portugal - Holland Holland 95’ 0.048 0.046 0.038 0.036
Italy - Australia Italy 50’ 0.980 0.677 0.265 0.477
Brazil - Ghana Ghana 81’ 0.141 0.020 0.401 0.628
England - Portugal England 62’ 0.592 0.272 0.321 0.436
Croatia - Germany Germany 92’ – – – – – – – – – – – –
Turkey - Czech Turkey 92’ 0.079 0.051 0.064 0.101
France - Italy France 24’ 1.012 0.654 1.194 1.300
Table 3: Rates of scoring of the sanctioned and the opposing team immediately before and immediately after the red
card. The table contains 26 games from the World Cup 2006 (top), and 3 games from the Euro 2008 (bottom). In 2
cases the red card happened at the very end of the game.
and after the red card this expression becomes
λnew +µnew.
We want to find condition when
λnew +µnew ≥λold +µold.
Since λnew ≈2
3λold, and µnew ≈5
4µold, the above inequality is approximately valid when
µold ≥4
3λold.
Thus one should expect more goals if a weaker team is penalized, but fewer goals if a stronger or comparable
team is penalized. Figure 6 conveys this situation. It confirms that when a stronger or comparable team is
penalized, the decline of the total scoring rates is visible in most of such games. The stronger team that is
penalized loses more offensive power in comparison with its defensive capacities, and although the weaker
team improves its scoring rate, this increase is smaller. However, when a weaker team is penalized, the total
scoring rate sometimes increases, but sometimes it stays the same.
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Figure 4: Relationship between the scoring intensity prior to the red card (x-axis) and after the red card (y-axis) for
the sanctioned team. The line represents the best linear fit with slope 0.663.
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Figure 5: Relationship between the scoring intensity prior to the red card (x-axis) and after the red card (y-axis) for
the opposite team. The line represents the best linear fit with slope 1.237.
5 Red Card or Goal
Consider the situation where a player can prevent a goal with the help of misconduct and hence with the risk
of expulsion. Such situations occur quite often, such as when a goalie trips a player of the opposing team in
when he is in a favorable scoring position, or when a defender stops a ball flying into an empty goal with his
hands. When the game is tied and is approaching its end, it is better to commit such an offense, even in the
situation where a red card and a penalty kick are certain. Otherwise his team will lose the game anyway.
On the other hand, it may not be optimal to commit a red card offense in the early stages of the game, since
there would be enough time to score and come back into the game.
We consider the question: At what point in the game is better to commit a red card offense in exchange
for preventing a goal opportunity for the opposing team? This optimal time depends on the current score
and the objective of the team. Let us assume that the objective of the team is not to lose the game during
the regulation time and that the current score is either tied, or one of the teams leads by one goal. We look
at two situations: a red card is issued with a penalty kick, or a red card is issued without a penalty kick. We
further assume that the two teams are of comparable strength, with the average scoring rate at 1.1 goals per
game.
It has been observed in the previous research, such as in Garicano and Palacios-Huerta (2005), that soccer
teams often choose to prevent goal opportunities in exchange for illegal offenses. In our study we show that
such behavior, while unsportsmanlike, can be optimal to achieve a victory or a tie in a given situation. One
9

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0.2
0.4
0.6
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1.0
1.2
1.4
0.2
0.4
0.6
0.8
1.0
1.2
Figure 6: Change of the overall scoring rate given the relative strengths of the two teams: the x-axis represents the
scoring intensity of the opposing team prior to a red card (µold), the y-axis represents the scoring intensity of the
penalized team prior to a red card (λold ). The line λold =3
4µold should separate games with an increased scoring
intensity (below the line – when a weaker team is sanctioned) from games with a reduced scoring intensity (above the
line – when a stronger or a comparable team is sanctioned). Circle points represent games with an increased scoring
intensity that are located below the line, but they are mixed with square points that represent games with no change
of overall scoring intensity. Diamond points represent games with a reduced scoring intensity that are located above
the line.
of the reasons for this is that a soccer game has a relatively low number of goals, and thus a single goal often
makes a difference in the final outcome.
Taking into account the situation when the score is tied, Figure 7 shows optimal time when it is better to
commit a red card offense resulting in a penalty kick as a function of the probability of scoring by the opposite
team. We assume an 80% success rate of scoring from a penalty kick. Obviously, if the current chance of
scoring is less than 80%, it is better to let it play since a penalty kick would create a more favorable situation
for the opposing team. However, if the chance of scoring is higher, it is sometimes optimal to commit the
offense. Notably, when the chance of scoring is 100% (for example, when the ball is flying in an empty goal),
it is better to stop it in the 51st minute of the game.
0.2
0.4
0.6
0.8
1.0
20
40
60
80
Figure 7: Optimal time for a red card offense resulting in a penalty kick as a function of the probability of scoring a
goal in a given situation assuming the current score is tied.
When the violation does not lead to a penalty kick, but only a red card, optimal time to commit such an
offense comes even earlier in the game. When the chance of scoring is just at 57.5% or higher, it is better to
prevent such a scoring opportunity from the very beginning of the game as seen in Figure 8.
10

0.2
0.4
0.6
0.8
1.0
20
40
60
80
Figure 8: Optimal time for a red card offense that does not result in a penalty kick as a function of the probability
of scoring a goal in a given situation assuming the current score is tied.
Figure 9 captures the optimal time to commit a red card offense resulting in a penalty kick when the
sanctioned team is leading by a goal and when the objective is not to lose the game. The team should be
more conservative for such a violation since the one goal lead provides a certain cushion. If the chances of
scoring are below 86.6%, the team should let it play out. In the situation where the goal is certain if not
disrupted, the optimal time to commit an offense is anytime after the 55th minute, a little later than in the
case when the game is tied.
When the offense does not lead to a penalty kick, it is optimal to let the game go if the chance of scoring
is below 33.4%, but to commit such an offense anytime during the game when the chance is above 73% as
seen in Figure 10. Thus when the team is leading by a goal, it is less inclined to commit a red card offense
when compared to the tied score situation.
0.2
0.4
0.6
0.8
1.0
20
40
60
80
Figure 9: Optimal time for a red card offense resulting in a penalty kick as a function of the probability of scoring a
goal in a given situation assuming team committing the offense is leading by one goal.
An interesting situation occurs when the team who commits a red card offense is trailing by one goal.
That team should never risk an offense that results a penalty kick. It is optimal that the opposing team
scores with high probability, but leaves the trailing team with the full number of players. As seen in Figure
11, the optimal time to commit such an offense is at the end of the game. However, the trailing team has to
be more aggressive in stopping the scoring chance when there is no risk of a penalty kick as seen in Figure 12.
When the chance of scoring is just 53.2% or higher, it is optimal to stop such action from the very beginning
of the game when the penalized team is trailing by a goal. This should be compared to the committing an
11

0.2
0.4
0.6
0.8
1.0
20
40
60
80
Figure 10: Optimal time for a red card offense that does not result in a penalty kick as a function of the probability
of scoring a goal in a given situation assuming team committing the offense is leading by one goal.
offense anytime during the game with a 57.5% chance of scoring assuming the score is tied.
0.2
0.4
0.6
0.8
1.0
20
40
60
80
Figure 11: Optimal time for a red card offense resulting in a penalty kick as a function of the probability of scoring
a goal in a given situation assuming the team committing the offense is trailing by one goal. Note that the optimal
time is the end of the game in all situations.
Conclusion
We have introduced a new method for estimating the scoring intensity in a soccer game using data from in
play betting markets. We have shown that when one of the teams receives a red card, its scoring intensity
is reduced to about 2
3of the original intensity, whereas the intensity of the opposing team increases by a
factor of about 5
4. This observation has allowed us to study the effect of the combined scoring intensity, and
conclude that the expected number of goals decreases when a stronger or comparable team is penalized, while
the expected number of goals can increase or stay the same when a weaker team is penalized. We have also
shown when it is optimal to stop a scoring opportunity at the expense of a sanction (red card or a penalty
kick). The optimal time depends on the score. When the team is leading by a goal, it becomes less inclined
to commit a red card offense when compared to the tied score situation. When the team is trailing by a goal,
it becomes more cautious when it comes to a situation that involves a red card resulting in a penalty kick,
but more aggressive when only a red card sanction is involved.
12

0.2
0.4
0.6
0.8
1.0
20
40
60
80
Figure 12: Optimal time for a red card offense that does not result in a penalty kick as a function of the probability
of scoring a goal in a given situation assuming the team committing the offense is trailing by one goal.
References
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[4] Garicano, L., I. Palacios-Huerta, “Sabotage in Tournaments: Making the Beautiful Game a Bit
Less Beautiful.” Working Paper, 2005.
[5] Ridder, G., J. S. Cramer, P. Hopstaken, “Down to Ten: Estimating the Effect of a Red Card in
Soccer,” JASA, Vol. 89, No. 427, 1994.
[6] Wesson, J., The Science of Soccer, IoP, 2002.
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