Estimating the Effect of the Red Card in Soccer

Transcription

1 Estmatng the Effect of the Red Card n Soccer When to Commt an Offense n Exchange for Preventng a Goal Opportunty Jan Vecer, Frantsek Koprva, Tomoyuk Ichba, Columba Unversty, Department of Statstcs, New York, NY 10027, USA October 10, 2008 Abstract We study the effect of the red card n a soccer game. A red card s gven by a referee to sgnfy that a player has been sent off followng a serous msconduct. The player who has been sent off must leave the game mmedately and cannot be replaced durng the game. Hs team must contnue the game wth one player fewer. We estmate the effect of the red card from bettng data on the FIFA World Cup 2006 and Euro 2008, showng that the scorng ntensty of the penalzed team drops sgnfcantly, whle the scorng ntensty of the opposng team ncreases slghtly. We show that a red card typcally leads to a smaller number of goals scored durng the game when a stronger team s penalzed, but t can lead to an ncreased number of goals when a weaker team s punshed. We also show when t s better to commt a red card offense n exchange for the preventon of a goal opportunty. 1 Introducton In ths paper we study the effect of the red card n a soccer game. Ths problem was prevously addressed by Rdder et al. (1994) usng data from 3 seasons of the Dutch professonal league. Ther study concluded that the ncrease n the rate of scorng for the team that s not punshed s statstcally sgnfcant, but they dd not observe a statstcally sgnfcant decrease n the rate of scorng of the penalzed team. Ther approach used statstcal methods to estmate the change of the rates of scorng followng a red card. A more recent and extensve study of the effect of the red card was done by Bar-El et al. (2006) usng data from 41 seasons of the German Bundeslga. They studed the red card n relaton to psychologcal effects. They showed that the red card reduces the scorng chances of the sanctoned team, whle ncreasng the scorng chances for the opposte team. However, they dd not look drectly at the changes n the ntensty of scorng whch s the focus of our paper. Prevous research estmated the correspondng parameters such as probablty of wnnng or losng the game, or the expected tme to score the next goal usng the entre statstcal sample; thus neglectng the effects of ndvdual stuatons n the games. Our approach uses a novel dea to look at the mpact of a red card n each ndvdual game, where we estmate the rates of scorng wth data obtaned from n-play bettng markets. Ths enables us to ncorporate more rare stuatons such as several red cards n a partcular game. In ths case the estmate from a statstcal sample becomes unrelable as the number of such games s small. It has been argued that bettng markets tend to be more effcent n estmatng the parameters of the models when compared to other methods. For nstance, bettng markets gve better results when compared to opnon polls as shown by Forsythe et al. (1992). We show n our paper that the red card leads to a farly substantal declne n scorng ntensty of the penalzed team, whch s n contrast wth the prevous study of Rdder et al. (1994), but s n accordance wth Bar-Er et al. (2006). We also show that the red card leads to a slght ncrease n the scorng ntensty 1

2 of the opposte team. We use data on the top nternatonal teams that played n the last World Cup and n the last Euro Cup, whle the prevous studes used only natonal league data where the dynamcs of the soccer game may dffer slghtly. The combned effect of the total scorng ntensty depends on the relatve strength of the two teams. If a stronger team s penalzed, the total scorng ntensty typcally drops down. If a weaker team s penalzed, the total scorng ntensty can ncrease or reman the same. The declne of the total rate of scorng s typcally larger when the stronger team s penalzed when compared wth the ncrease n the total scorng rate when the weaker team s penalzed. However a weaker team commts a red card offense more frequently, and so the overall effect of the red card s neutral on the expected number of goals. The last secton of our paper determnes when t s optmal to commt a red card offense n exchange for preventon of a possble goal opportunty for the opposte team. We assume that the objectve s not to lose the game n the regulaton tme. Ths optmal tme comes surprsngly early n the game. When the score s ted, and the red card offense s not followed by a penalty kck, t s better to make the offense anytme durng the game when the opposng team has 57.5% or hgher chances of scorng. If there s a penalty kck on the top of the red card, the optmal tme to stop a certan goal s anytme after the 51st mnute. When a team s leadng by one goal, t s slghtly less nclned to commt a red card offense. Wth no penalty kck, the chances of scorng must be at least 73% n order to commt such an offense. Wth a penalty kck, a sure goal should be stopped anytme after the 55st mnute. When the team s tralng by a goal, t should become more aggressve n preventng a goal opportunty when there s no penalty kck, stoppng more than 53.2% chances of scorng at any tme durng the game. However, the tralng team should become more conservatve when a penalty kck s nvolved. It s never optmal to commt such an offense. 2 Estmaton of the Scorng Intensty We use data obtaned from bettng markets on the FIFA World Cup 2006 and Euro Bettng markets on these two tournaments reached unprecedented effcency and lqudty. It was also possble to buy and sell futures type contracts on the outcome of the game or the number of goals scored, and trade them even durng the actual game. Prces of all traded events (wn, draw or loss of a gven team, and number of goals) were mmedately affected by a goal or a red card. We analyze data from the betfar.com market to estmate the mpled scorng rate of a gven team as the game progressed. The bettng market on a gven match traded the followng most lqud contracts: Team 1 to wn, Draw, Team 2 to wn, 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 expred at the end of the regular tme of a partcular match (90 mnutes + njury tme). They were traded before and durng the actual game. The contracts are quoted n terms of odds 1 : x. One can nterpret such odds as the probablty of ths event. It s possble to buy the event (bet that ths event wll happen, or back t), or sell the event (bet that ths event wll not happen, or lay t). The backer of 2

3 the event bets $1, and receves $x back (creatng a total proft of $(x back 1)) f the event happens, otherwse he loses hs $1 stake. The layer of the event bets $(x lay 1) and receves $x lay (creatng a total proft of $1) f the event does not happen, otherwse he loses hs $(x lay 1) stake. The market s set up n such a way that x back < x lay at all tmes. If the event happens, the backer wns x back 1, but the layer loses a larger stake x lay 1. The odds x back and x lay are mmedately avalable for backng or layng the event. One can request better odds, n whch case one s order s queued n the market, and has to wat untl t s matched wth a counter party. However, a matchng bet may never come. Before RC After RC Contract Back Lay Back Lay Italy Australa The Draw Italy Next Goal Australa Next Goal No Goal Three or More Goals Table 1: Odds for dfferent bettng contacts for Italy - Australa game mmedately before and after the red card (Italy) n the 50th mnute. Table 1 llustrates ths concept wth odds taken drectly before and after the moment of the red card n the Italy - Australa game. Italy was sanctoned n the 50th mnute of the game. The bettng market suspends all tradng when there s an apparent goal or a penalty, so t s rather smple to dentfy the quotes that mmedately precede and follow such an event. If someone were to bet $1 on an Italan vctory before the red card he would receve $1.87 f Italy won, makng a net proft of $0.87. After the red card, the same event of an Italan vctory became less lkely wth odds of 1:2.68. Note that the layer of the event (before the red card) had odds of 1:1.88, bettng $0.88 nstead of $0.87 a gan that collects the backer of the event. Ths s due to the aforementoned market spread between buyng and sellng of an event. Note that the market spread may be relatvely large f the market exhbts a small lqudty as n 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 estmates of the lkelhood of a gven event. For example, the chances of Italy wnnng were between 1:1.88 and 1:1.87 before the penalty, whch corresponds to the nterval (0.532, 0.535) for the probablty of ths event. After the penalty, ths probablty dropped to the nterval (0.370, 0.373). In order to analyze the change n probabltes as they evolved durng the game, especally before and after the red card event, we focus our attenton on the scorng ntenstes of the two teams. Scorng ntenstes of the two teams already determne probabltes of wnnng, drawng or losng the game, the probablty that a partcular team scores next, as well as the probablty of a certan number of goals. From ths perspectve, scorng ntenstes are the fundamental parameters of our analyss. In order to fnd the relatonshp between the scorng ntenstes and probabltes assocated wth the game, we use the Posson model for the scorng. We further assume that the scores of the two teams are ndependent. Ths s a reasonable assumpton, a typcal correlaton of the scores of the two teams s very small. For example, the correlaton of the scores n Euro 2008 tournament was only Ths model was also assumed n prevous research, such as n Wesson (2002). All bets n ths case depend only on two parameters: the scorng ntensty of the frst team λ t and the scorng ntensty of the second team µ t at the tme t. Thus number of goals scored by each team follows the Posson dstrbuton. The market mples that 3

4 λ t and µ t are the expected number of goals to be scored by Team 1 and Team 2 respectvely n the rest of the game (tme between the current tme t and the end of the match tme T ). The theoretcal probabltes that correspond to the bettng contracts are gven by the followng formulas. We assume the current score s X t : Y t : Wn Team 1 = P(X T > Y T ) = Draw = P(X T = Y T ) = Wn Team 2 = P(Y T > X T ) = [ P(X T = k X t, Y T < k X t ) = k=0 [ e (λ t+µ t) k=0 k=0 k=0 λ (k+max(xt+yt) Xt) t (k + max(x t + Y t ) X t )! [ P(Y T = k Y t, X T < k Y t ) = k=0 e λt λk t k! k+x t Y t 1 =0 e µt µ t! µ (k+max(xt+yt) Yt) t (k + max(x t + Y t ) Y t )! ] e µ t µk t k! k X t +Y t 1 =0 e λ t λ t! ] ] Team 1 Next Goal = Team 2 Next Goal = λ t [1 ] e (λ t+µ t ) λ t + µ t µ t [1 ] e (λ t+µ t ) λ t + µ t No Goal = e (λ t+µ t ) Three+ = P(X T + Y T 3) = e (λt+µt) (λ t + µ t )k k! k=(3 X t Y t) + Note that we have 7 contracts that depend only on two parameters λ t and µ t, and so ths problem s over-parameterzed. The market may even admt arbtrage (rsk free proft) f the contract prces are not properly related to each other. For nstance, the probabltes of Wn of Team 1, Draw, and Wn of Team 2 should add up to 1. However, due to the neffcences of the market, such as the bd-ask spread, asynchronous tradng, and market order delay, t may be dffcult to lock nto these opportuntes. Let x back be the odds of backng a specfc event, and let p be the theoretcal probablty of ths event. Then the return of ths bet s gven by x back p 1. For nstance f one was to bet on an outcome of a far con toss (p = 1 2 ), the return would be zero f xback = 2. If the odds were hgher than 2, than the return would be postve, f the odds were smaller, than the return would be negatve. For example when x back = 2.02, the return s 1%, so one should expect to earn on average $1 per $100 nvested. If the odds were x back = 1.98, the return s -1%, so one should expect to lose on average $1 per $100 nvested. The return for the layng of an event s calculated n a slghtly dfferent way because t requres a bet of $(x lay 1) n order to wn $1. The return s gven by x lay x lay (1 p) 1. 1 Gven a far con toss, the return s zero when x lay = 2. The return s postve when the odds are smaller, and s negatve when the odds are larger. For example, when x lay = 1.98, the correspondng return s 1.02%, when x lay = 2.02, the correspondng return s -0.98%. 4

5 In order to estmate λ t and µ t from the odds quoted by the bettng market, we choose ˆλ t and ˆµ t that mnmzes the best expected return of all contracts combned, namely fndng a mnmzer of the followng formula: ( ) mn max x back x lay P (λ t, µ t ) 1; λ t,µ t x lay 1 (1 P (λ t, µ t )) 1, where P (λ t, µ t ) s the probablty of -th event gven the ntenstes of scorng λ t and µ t. The propertes of the mn-max estmator are dscussed n Cox and Hnkley (1979). Return Contract 1/x lay 1/x back P (ˆλ t, ˆµ t ) Back Lay Italy Australa The Draw Italy Next Goal Australa Next Goal No Goal Three or More Goals Table 2: Probabltes mpled by the bettng odds for the Italy - Australa game rght before the tme of the red card n the 50th mnute (columns 1/x lay, 1/x back ), probabltes of these events usng a Posson model wth the optmal choce of parameters ˆλ t = and ˆµ t = (column P (ˆλ t, ˆµ t)), and returns that correspond to the buyng or sellng these events. Postve returns are typed n bold. Table 2 llustrates the choce of the optmal parameters for the Italy - Australa game rght before the moment of the red card n the 50th mnute. It not only shows the probabltes mpled by the bettng odds, but also the theoretcal probablty of any event that come wth the optmal choce of the scorng parameters ˆλ t = and ˆµ t = Note that some of the bettng contracts yeld a postve return for ths choce of parameters (such as backng The Draw, or Italy Next Goal, or layng Australa, No Goal, and Three or More Goals), but the best expected return s 2.2%, whch s well below the standard commsson rate of 5% charged by the market. The optmal choce of the scorng ntenstes keeps the best return at the lowest level, because for any other choce of ntenstes ths dscrepancy s larger. Although we fnd some contracts wth postve return, t s stll not clear whether they should be bought or sold. Ther prce looks favorable wth respect to the prces of other bettng contracts, but f taken alone, they could stll be quoted at a far prce. After the red card, the scorng rate for Italy fell from to 0.677, whle the scorng rate for Australa rose from to It s possble to use dfferent estmators of λ t and µ t. One example s the L 2 least square estmator ( ( ) 2 2 mn 1 1 P (λ t, µ t )) + P (λ t, µ t ), λ t,µ t x back x lay another one s the L 1 estmator: mn λ t,µ t [ 1 x back P (λ t, µ t ) + 1 x lay ] P (λ t, µ t ). For comparson, L 2 estmators for the Italy - Australa game pror to the red card are ˆλ 2 t = and ˆµ 2 t = 0.265, L 1 estmators are ˆλ 1 t = and ˆµ 1 t = 0.263, whch are close enough to the mnmax estmator 5

6 ˆλ t = and ˆµ t = The advantage of the mnmax estmator s that t s more robust when there s a sgnfcant lqudty gap, that s, when some of the dscrepances between 1 x and P (λ t, µ t ) may become so large that they can affect an estmate that s based on L 1 or L 2. Mnmax estmators are not susceptble to such effects. Fgures 1-3 show the market mpled ntenstes usng the mnmax estmator for selected games from the World Cup to llustrate ths concept. We start our nference analyss at the begnnng of the game (tme t = 0), and fnsh when the game ends n the regulaton (90 mnutes) plus njury tme. We ndcate the break between the two halves, whch lasts 15 mnutes. Australa v Italy Australa 1.5 Italy Fgure 1: Impled scorng ntenstes n the Italy - Australa game. A red card was gven to Italy n the 50th mnute of the game. Italy v USA Italy 1.5 USA Fgure 2: Impled scorng ntenstes n the Italy - USA game. Italy was sanctoned wth a red card n the mddle of the frst half (25 ), followed by a red card gven to the US team at the end of the frst half (45 ), and the begnnng of the second half (48 ). 3 Estmatng the Effect of the Red Card Upon recevng a red card the sanctoned team must complete the rest of the match wth one player less. Durng the FIFA World Cup 2006 tournament, 28 players n total were sanctoned. Durng Euro 2008 tournament 3 players receved a red card. Our analyss ncludes a total of 27 cases of red cards; 2 happened n overtme, and 2 at the very end of the game. We use the method descrbed n the prevous secton and estmate the scorng ntenstes of both teams before and after a red card. Multple red cards were gven n 6

7 Sweden v Trndad Sweden 2.5 Trndad Fgure 3: Impled scorng ntenstes n the Trndad - Sweden game. Trndad was sanctoned by a red card n the early stages of the second half of the game (46 ). 4 games; 2 red cards n the Cote d Ivore - Serba game, 3 red cards n Italy - USA and Croata - Australa games and 4 red cards n the Portugal - Holland game. The scorng rate of the sanctoned team s denoted by λ, wth λ old beng the rate pror to the red card, and λ new beng the rate after the red card. Smlarly, the rates of the opposng team are denoted by µ, wth µ old beng the rate pror to the red card and µ new beng the rate after the red card. The complete lst of these rates of scorng s gven n Table 3. In order to predct the change n rates after the red card event we use the followng regresson model: λ new = θ 1 λ old + ɛ 1 for a certan θ 1 for the team whch was just penalzed, and µ new = θ 2 µ old + ɛ 2 for a certan θ 2 for the opposte team. The error terms n the model reflect ndvdual stuatons wthn each game such as the suspenson of a key player. Lnear regresson leads to the followng estmates: ˆθ 1 = , wth R 2 = 0.972, and ˆθ 2 = , wth R 2 = Thus the scorng ntensty of the penalzed team drops approxmately to 2 3 of the ntensty that precedes the red card, and the scorng ntensty of the opposng team ncreases approxmately to 5 4 of the orgnal ntensty before the red card. The graph that llustrates the change of ntensty of the penalzed team s gven n Fgure 4 and the graph that llustrates the change of ntensty of the opposng team s gven n Fgure 5. 4 Red Card and Expected Number of Goals Ths secton dscusses the queston whether we are expected to see more or less goals after a red card. The total scorng ntensty of both teams combned before the red card s gven by λ old + µ old 7

8 Game Red Card Tme λ old λ new µ old µ new Trndad - Sweden Trndad Korea - Togo Togo Span - Ukrane Ukrane Germany - Poland Poland Argentna - Serba Serba Mexco - Angola Angola Italy - USA Italy Italy - USA USA Italy - USA USA Czech - Ghana Czech Cote d Ivore - Serba Serba Cote d Ivore - Serba Cote d Ivore 92 Mexco - Portugal Mexco Italy - Czech Czech Croata - Australa Croata Croata - Australa Australa Croata - Australa Croata Ukrane - Tunsa Tunsa Germany - Sweden Sweden Portugal - Holland Portugal Portugal - Holland Holland Portugal - Holland Portugal Portugal - Holland Holland Italy - Australa Italy Brazl - Ghana Ghana England - Portugal England Croata - Germany Germany 92 Turkey - Czech Turkey France - Italy France Table 3: Rates of scorng of the sanctoned and the opposng team mmedately before and mmedately after the red card. The table contans 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 ths expresson becomes We want to fnd condton when λ new + µ new. λ new + µ new λ old + µ old. Snce λ new 2 3 λold, and µ new 5 4 µold, the above nequalty s approxmately vald when µ old 4 3 λold. Thus one should expect more goals f a weaker team s penalzed, but fewer goals f a stronger or comparable team s penalzed. Fgure 6 conveys ths stuaton. It confrms that when a stronger or comparable team s penalzed, the declne of the total scorng rates s vsble n most of such games. The stronger team that s penalzed loses more offensve power n comparson wth ts defensve capactes, and although the weaker team mproves ts scorng rate, ths ncrease s smaller. However, when a weaker team s penalzed, the total scorng rate sometmes ncreases, but sometmes t stays the same. 8

9 Fgure 4: Relatonshp between the scorng ntensty pror to the red card (x-axs) and after the red card (y-axs) for the sanctoned team. The lne represents the best lnear ft wth slope Fgure 5: Relatonshp between the scorng ntensty pror to the red card (x-axs) and after the red card (y-axs) for the opposte team. The lne represents the best lnear ft wth slope Red Card or Goal Consder the stuaton where a player can prevent a goal wth the help of msconduct and hence wth the rsk of expulson. Such stuatons occur qute often, such as when a goale trps a player of the opposng team n when he s n a favorable scorng poston, or when a defender stops a ball flyng nto an empty goal wth hs hands. When the game s ted and s approachng ts end, t s better to commt such an offense, even n the stuaton where a red card and a penalty kck are certan. Otherwse hs team wll lose the game anyway. On the other hand, t may not be optmal to commt a red card offense n the early stages of the game, snce there would be enough tme to score and come back nto the game. We consder the queston: At what pont n the game s better to commt a red card offense n exchange for preventng a goal opportunty for the opposng team? Ths optmal tme depends on the current score and the objectve of the team. Let us assume that the objectve of the team s not to lose the game durng the regulaton tme and that the current score s ether ted, or one of the teams leads by one goal. We look at two stuatons: a red card s ssued wth a penalty kck, or a red card s ssued wthout a penalty kck. We further assume that the two teams are of comparable strength, wth the average scorng rate at 1.1 goals per game. It has been observed n the prevous research, such as n Garcano and Palacos-Huerta (2005), that soccer teams often choose to prevent goal opportuntes n exchange for llegal offenses. In our study we show that such behavor, whle unsportsmanlke, can be optmal to acheve a vctory or a te n a gven stuaton. One 9

10 Fgure 6: Change of the overall scorng rate gven the relatve strengths of the two teams: the x-axs represents the scorng ntensty of the opposng team pror to a red card (µ old ), the y-axs represents the scorng ntensty of the penalzed team pror to a red card (λ old ). The lne λ old = 3 4 µold should separate games wth an ncreased scorng ntensty (below the lne when a weaker team s sanctoned) from games wth a reduced scorng ntensty (above the lne when a stronger or a comparable team s sanctoned). Crcle ponts represent games wth an ncreased scorng ntensty that are located below the lne, but they are mxed wth square ponts that represent games wth no change of overall scorng ntensty. Damond ponts represent games wth a reduced scorng ntensty that are located above the lne. of the reasons for ths s that a soccer game has a relatvely low number of goals, and thus a sngle goal often makes a dfference n the fnal outcome. Takng nto account the stuaton when the score s ted, Fgure 7 shows optmal tme when t s better to commt a red card offense resultng n a penalty kck as a functon of the probablty of scorng by the opposte team. We assume an 80% success rate of scorng from a penalty kck. Obvously, f the current chance of scorng s less than 80%, t s better to let t play snce a penalty kck would create a more favorable stuaton for the opposng team. However, f the chance of scorng s hgher, t s sometmes optmal to commt the offense. Notably, when the chance of scorng s 100% (for example, when the ball s flyng n an empty goal), t s better to stop t n the 51st mnute of the game Fgure 7: Optmal tme for a red card offense resultng n a penalty kck as a functon of the probablty of scorng a goal n a gven stuaton assumng the current score s ted. When the volaton does not lead to a penalty kck, but only a red card, optmal tme to commt such an offense comes even earler n the game. When the chance of scorng s just at 57.5% or hgher, t s better to prevent such a scorng opportunty from the very begnnng of the game as seen n Fgure 8. 10

11 Fgure 8: Optmal tme for a red card offense that does not result n a penalty kck as a functon of the probablty of scorng a goal n a gven stuaton assumng the current score s ted. Fgure 9 captures the optmal tme to commt a red card offense resultng n a penalty kck when the sanctoned team s leadng by a goal and when the objectve s not to lose the game. The team should be more conservatve for such a volaton snce the one goal lead provdes a certan cushon. If the chances of scorng are below 86.6%, the team should let t play out. In the stuaton where the goal s certan f not dsrupted, the optmal tme to commt an offense s anytme after the 55th mnute, a lttle later than n the case when the game s ted. When the offense does not lead to a penalty kck, t s optmal to let the game go f the chance of scorng s below 33.4%, but to commt such an offense anytme durng the game when the chance s above 73% as seen n Fgure 10. Thus when the team s leadng by a goal, t s less nclned to commt a red card offense when compared to the ted score stuaton Fgure 9: Optmal tme for a red card offense resultng n a penalty kck as a functon of the probablty of scorng a goal n a gven stuaton assumng team commttng the offense s leadng by one goal. An nterestng stuaton occurs when the team who commts a red card offense s tralng by one goal. That team should never rsk an offense that results a penalty kck. It s optmal that the opposng team scores wth hgh probablty, but leaves the tralng team wth the full number of players. As seen n Fgure 11, the optmal tme to commt such an offense s at the end of the game. However, the tralng team has to be more aggressve n stoppng the scorng chance when there s no rsk of a penalty kck as seen n Fgure 12. When the chance of scorng s just 53.2% or hgher, t s optmal to stop such acton from the very begnnng of the game when the penalzed team s tralng by a goal. Ths should be compared to the commttng an 11

12 Fgure 10: Optmal tme for a red card offense that does not result n a penalty kck as a functon of the probablty of scorng a goal n a gven stuaton assumng team commttng the offense s leadng by one goal. offense anytme durng the game wth a 57.5% chance of scorng assumng the score s ted Fgure 11: Optmal tme for a red card offense resultng n a penalty kck as a functon of the probablty of scorng a goal n a gven stuaton assumng the team commttng the offense s tralng by one goal. Note that the optmal tme s the end of the game n all stuatons. Concluson We have ntroduced a new method for estmatng the scorng ntensty n a soccer game usng data from n play bettng markets. We have shown that when one of the teams receves a red card, ts scorng ntensty s reduced to about 2 3 of the orgnal ntensty, whereas the ntensty of the opposng team ncreases by a factor of about 5 4. Ths observaton has allowed us to study the effect of the combned scorng ntensty, and conclude that the expected number of goals decreases when a stronger or comparable team s penalzed, whle the expected number of goals can ncrease or stay the same when a weaker team s penalzed. We have also shown when t s optmal to stop a scorng opportunty at the expense of a sancton (red card or a penalty kck). The optmal tme depends on the score. When the team s leadng by a goal, t becomes less nclned to commt a red card offense when compared to the ted score stuaton. When the team s tralng by a goal, t becomes more cautous when t comes to a stuaton that nvolves a red card resultng n a penalty kck, but more aggressve when only a red card sancton s nvolved. 12

13 Fgure 12: Optmal tme for a red card offense that does not result n a penalty kck as a functon of the probablty of scorng a goal n a gven stuaton assumng the team commttng the offense s tralng by one goal. References [1] Bar-El, M., G. Tenenbaum, S. Gester, Consequences of Players Dsmssal n Professonal Soccer: A Crss Related Analyss of Group Sze Effect, Journal of Sport Scences, , [2] Cox, D., D. Hkley, Theoretcal Statstcs, Chapman - Hall, [3] Forsythe, R., F. Nelson, G. R. Neumann, J. Wrght, Anatomy of an Expermental Poltcal Stock Market, The Amercan Economc Revew, Vol. 82, No. 5, , [4] Garcano, L., I. Palacos-Huerta, Sabotage n Tournaments: Makng the Beautful Game a Bt Less Beautful. Workng Paper, [5] Rdder, G., J. S. Cramer, P. Hopstaken, Down to Ten: Estmatng the Effect of a Red Card n Soccer, JASA, Vol. 89, No. 427, [6] Wesson, J., The Scence of Soccer, IoP,