As the game progresses a team's success probability will ebb on flow dependent upon such things a current score,relative strengths of both team and whether a red card has been shown.By adding a team's chances of winning the match to half it's chances of drawing the game we can chart a teams expected success probability at various stages of the match.
At full time a team will have either won,gaining a success probability of 1.0 or drawn the game and shared the win equally with it's opponent gaining 0.5 wins each or of course lost.
By looking at the changes in success probability from the start of the game to full time,we can start to assign how much success probability should be attributed to the 11 or more participating players.
There are generally four fundamental stages to consider.
1) Change in SP due to time elapsed.
Each team will start a game with a likely success probability (SP) based on their relative strengths compared to today's opponents.This can be calculated by taking a weighted view of both team's recent results.How far you go back in time is a matter for individual taste,but I would suggest the last 25 games as a bare minimum requirement.
Since goals,both scored and conceded determine football games,it's sensible to expess the relative team merits in terms of how many goals each team will on average score in today's game,given the relative merits of both attack and defences.We can therefore use the pre game goal expectancies for both teams to calculate the expected success probability at the start of the game.This is the starting point and it's also team specific for today's game.
As the game develops without any goals occuring,the pregame goal expectancies for each team will start to decay.Thus we can re calculate each team's SP up to the point when a goal is scored.What generally happens is that a much superior team will see it's SP trend downwards towards 0.5 (as the teams are still level) and an inferior oppenent will see it's SP trend upwards.We can assign this change in SP respectively to each team along with any changes that occur in the lulls between significant events such as goals,penalty awards or red cards.
2) Goals scored.
To state the obvious,goals change games.If you score you always increase your success probability.The real bone of contention is how to distribute the credit for the goal between the scorer and the creator,similar to the situation in American Football where the QB throws a TD to his WR.I currently split the credit 80/20 in favour of the scorer.Some goals will be relative tap ins,although as Torres demonstrated recently,there maybe no such thing as a tap in,while other goals may result from a relatively simple pass from a team mate.
3) Goals conceded.
Allowing goals always decreases your SP and again we need to apportion the "blame".Harsh though it may seem,I've given the keepers 20% "credit" for the goal.The remaining 80% I've distributed between the 10 outfielders.55% of remaining fall in SP is divided amongst the defenders.I've chosen this figure because if we use goals scored as a proxy for "attacking play" we find that forwards score 55% of the goals,midfielders 35% and defenders 10%.Therefore it seems reasonable to flip these percentages when trying to apportion the less transparent skill of defending.Blaming forwards partly for goals conceded may seem strange,but firstly the proportion of fault is relatively small, they do actively defend at corners and goals often develop from failure to maintain possession higher up the field.
4) Red Cards.
It's not harder to play against ten men that eleven,it's easier.Reduced in numbers,a team finds it harder to score and harder to prevent goals being scored against them.A red card reduces that team's SP and the entire amount by which the SP is reduced counts against the transgressing player.
These are the (very) bare bones of a system to apportion blame and credit for team wins or draws amongst the individual players.So lets use a real example.I've chosen the Fulham Man City game from last week to illustrate the methodology and I've started with a pared down Success probability graph,illustrating the six major incidents from the game.Namely,kick off,two goals apiece and full time.
Significant Events from the Fulham Man City Draw.
0-1,Aguero ,18 mins,assist,Silva.
Kasami on for Duff,61 mins.
Ruiz on for Dembele,84 mins.
Zabaleta on for Silva,69 mins.
Johnson on for Nasri,81 mins.
Tevez on for Aguero,83 mins.
A deeper look at some of the SP changing events from the game.
Man City are the away side,but unsurprisingly given their vast investment are still considered more likely winners.If this game were to be repeated a large number of times I'd expect Man City to score on average 1.9 goals,concede 0.8 goals,win just over 60% of the time and draw just over 20% of the time.
So at kick off Man City have a SP of 0.741 compared to Fulham's 0.259.Just prior to the first Man City goal after 18 minutes,City's SP has declined to 0.722 and that fall is shared amongst the 11 starting players.Fulham's has risen to 0.278 from 0.259 and that is also shared amongst their starters.
Aguero's first goal).
Aguero scores from a Silva assist and Man City's SP goes from 0.722 to 0.902.80% of the credit goes to Aguero and 20% to Silva.By contrast Schwartzer is debited 20% of the 0.18 his team's SP has fallen,another 44% is shared by each member of the back four,28% by the four midfielders and 8% by the the two strikers.
His goal raises Fulham's SP from 0.07 to 0.39 of which 80% is attributed to him and 20% to Zamora,who set up the chance.
Last 20 minutes).
Fulham's SP rises from 0.39 at the time of the Murphy goal to 0.5 at the final whistle,this is shared amongst the players on the field at the time of the goal.Man City's players lose the same amount as their SP falls to 0.5 from 0.61 as they were still the most likely winners with the game tied at 2-2.
A summary of each players win shares from the game.
|Schwartzer.||-0.018||Conceded 2 goals,but benefitted from Fulham being big underdogs and claiming a point.|
|Kelly.||0.005||Unless they score,assist or are sent off,defenders score identically.Tackles won for example would re distribute the SP.|
|Duff.||0.006||He was substituted while Fulham were still trailing,so didn't reap the |
completed comeback rewards.
|Murphy.||0.271||A late equaliser is always going to be a big SP mover.|
|Dempsey.||0.030||Assist for Zamora goal.|
|Zamora.||0.145||His goal could have just been a consolation.Assist for Murphy's leveller.|
|Kasami.||0.010||Limited game time,but gained from coming on before the equaliser.|
|Silva.||0.097||Supplied an assist and had left the field before the equaliser.|
|Nasri.||0.056||He wasn't on the pitch at the very end when City's SP was slipping away quickly and,as a forward he wasn't heavily penalised for Fulham's goals.|
|Zabaleta.||-0.039||Appeared when City still led,but couldn't help them protect the lead.|
|Tevez.||-0.007||Appeared with the game stalemated,but couldn't grab a winner.|
This initial system rewards or penalises players for being involved in events that are integral to the score and the final result.Consequently,players who aren't involved in game changing incidents tend to score similar shares.As in American Football,it' quite easy to assign a number to depict how influential a quarterback has been to the game,but much more difficult when dealing with a lineman.An improvement to players being scored as a unit would be to re distribute the unit's total by using stats such as tackles made or passes completed.The problem here is that these kind of stats,even if they correlate well with talent,aren't readily available in an easy to use format.So at the moment it's a work in progress and the more obvious contributors to the fore.