The team which is reduced in number suffers by scoring less goals than they would have done had they kept a full complement of players on the pitch and they also concede more goals than would have been so.
One of the most effective ways to model the expected progress of a soccer match is to calculate the average number of goals each team can expect to score at today's venue and against today's opponents.We can then use the Poisson distribution (with few tweaks) to calculate a multitude of predictions about the likely route the game will take.This is the basis for the game graphs on this site.However,if we want to carry this approach past the initial whistle we also need to know how the expected average number of goals each team will score today decays as the game progresses.Initial clues for a possible way forward lies in the distribution of goals scored in each half of play.In England approximately 45% of goals come before the interval and 55% afterwards.In fact if you plot the number of goals scored in various 10 minute segments over the course of a contest you'll find that the later segments see more goals than the earlier ones.In short the rate of goal scoring increases with time.
The best fit for goal expectancy throughout a game is;
Goal Expectancy during the Game = Initial Expectancy * (Proportion of game left)^0.85
where the initial expectancy is the average number of goals you expect the team to score in today's match up at the start of the game.If the initial expectancy is set to 1 and the proportion of the game remaining to 0.5,you'll get the answer 0.55,the observed fraction of second half goals.
This method gives us a route to determine the effect of a red card on the remaining goal expectancy of each of the two teams involved and thus the expected course of a match post that red card.As we have seen,red cards occur in about 15% of games.However they tend to occur later in matches,Kevin Pressman's red after 13 seconds and Vinnie Jones' yellow after three being exceptions.This makes collecting data for earlier game times problematical,often no data exists.What we can do is collect the average time of every red card and also record the average goal expectancy for the team receiving the red card and their opponents immediately prior to the incident.If we then record the average number of goals the red card side and their opponents actually went on to score,we will have the average change in goal difference between the sides pre and post the dismissal.
We find that over the last 6 seasons one average dismissal occurred just after the 60th minute and the average change in goal difference amounted to 0.61 of a goal.About 64% of the 0.61 difference occurred because the side with 11 players scored more than they would have expected to score prior to the card and the rest came about by the infringing side scoring less.
We can surmise that the change in goal differences bought about by the card decays at the same rate as ordinary goal expectancy.61 minutes means that about 36% of the game remains,so by plugging the figures into the equation we can find the amount by which two teams will see their goal expectancy difference changed by if the red card is shown at the very start.Usefully we can find out how much goal expectancy a team loses overall from a red card over an entire game.
The answer is about 1.45 goals.
So if a reasonably typical EPL team starts the game a man down within seconds they can be expected to score on average 0.5 of a goal less than they would have done with 11 players and concede 0.95 more goals resulting in a swing in overall goal expectancy of 1.45 of a goal.The figure of 1.45 goals can be re plugged into the decay equation to determine the respective figures at any time throughout the course of the game.For example a dismissal as early as the 33rd minute will cost a team a goals worth of expectancy,split between goals they don't score and extra ones they do and this number can be incorporated into the in game probabilities for that particular game.