A glossary of terms is in the pipeline,but for the meantime.....The win probability for each team is tracked on a minute by minute basis throughout each game.At any point in a game a team will have a probability of winning the game outright and an associated probability of drawing the game.By multiplying the probability of winning the game by 3 and the probability of drawing by 1 and adding the results together you will get an expected long term average for the number of points that team will achieve from that position or Expected Points (EP) for short.
It should come as no surprise to discover that in games where one red card is issued,then that card is more likely to be shown to the visiting side.The home side more often will have the better chance of winning the game,therefore the away side will be forced to do more defending,sometimes against superior players,which will lead to more fouls,more yellow cards and ultimately more chance of a red.In the 300+ EPL games spread over the last 7 seasons where a red card was shown over 60% of the time it was shown to the away side.To further test our idea that poorer sides are more likely to attract red cards we first need a way of quantifying pre game team quality.One very good way to represent the difference in quality between teams prior to the game starting is to compare each side's Pre Game Expected Points tallies.And once again we find that the underdog,irrespective of venue receives over 60% of dismissals in games where a single red card is shown.The average Pre Game Expected Points for the team who receives the red card is 1.29 compared to 1.45 for the team that doesn't see red.
There's nothing particularly novel about the previous finding,they are as you would expect and have probably been documented many times already.I therefore produced a regression line that described the relationship between pre game Expected Points and the likelihood that teams of varying EP would receive the game's only red card.I also split the games by venue and the plot can be seen below.
Each plot nicely illustrates the effect we are describing.The more out gunned a team initially appears to be the more likely it is to be the team to see red if one is shown.However,there's is a puzzling discrepancy.Very big underdogs whom have a pre game EP of 0.1 have just under a 70% chance of being the red carded side in games where one red is shown when they are at home,but when they are away this rises to almost 80%.This effect carries on through every pre game EP value.Massive pre game favourites with EP's of 2.8 are only likely to see red in 25% of these type of games at home,but the figure jumps to just under 40% on the road.
So the obvious question is why is there a difference.Venue is accounted for in the pre game team estimates,so we are looking a games which have identical pre game competitive balances.The only difference is that 80% of the crowd will be rooting for the pregame favourite when they are at home and 80% will be against them on the road.Obviously,home underdogs will also receive the bulk of the crowd support at home and will find that the majority of the crowd are against them on the road.
One possibility could be that for whatever reason teams with a similar pre game EP commit more fouls on their travels compared to when they have that level of EP at home.So to test this I plotted the regression line for Pre game EP and percentage of game fouls committed for both home and away teams.The graph is shown below.
Again,the graph illustrates the suspicion that poorer teams,in general concede the majority of the free kicks in a game.A team with an EP of 0.1 at the onset of the game will on average be responsible for around 60% of the transgressions made during that game,while the very best commit just over 40%.Two lines have been plotted to represent home and away teams and it is apparent that there is very little difference between the fouling rates of teams having the same pre game EP at different venues.On the road,teams commit very slightly more fouls compared to their EP dopplegangers at home,but the increase is only around 2% compared to the 20%+ increase in their likelihood of being the team to see the game's only red card.
Referees can only dismiss players with good reason and the very small increase in fouls committed by these teams on the road would seem insufficient reason for the disproportionately large increase in red cards they receive.It therefore looks highly likely that the influence of the home crowd and a subconscious desire to please the majority may be leading to a systematic and unfair increase in red cards shown to away sides in the EPL.
Red cards are relatively rare events and they also happen on average just after the hour mark,so although their influence can be large on individual games,their effect over a season is small and difficult to notice.Red cards redistribute about 25 points throughout a season and as teams can be both recipients and beneficiaries,the net effect per team can be small.However,if the effect is indeed valid and it does favour the home side disproportionately,it is one step to identifying a component that goes towards setting the size of a sport's home field advantage.
As a digression the explosion in red cards came about to promote more attacking play and more goals.Fans were thought to want to see more goals.However,by greatly punishing the weaker sides,this strategy has come at a cost of greatly reducing the competitiveness of the majority of games where a red card is shown.Fans can be reluctant to attend uncompetitive games,witness the reduced attendances and massive price cutting required when marketing unbalanced FA cup ties.So red cards are actually giving fans games they wouldn't have particularly attended if they'd known beforehand that a dismissal was going to occur.
Maybe it's time to adopt the NFL approach of expelling the player,but not reducing team numbers,except for a token ten minutes.