Adjusting for Players in the Elo Framework

When there are distinct players that you want to model, there are two different approaches that might be helpful.

If you know the adjustment you want to give a team based on the presence of one of its players, you should use the adjust() function.

If you want to model the player’s impact on the Elo rating, it can be helpful to use the players() function.

Consider American Football, in which the quarterback can exert a lot of influence on whether a team wins or not. If, halfway through a season, a good quarterback gets injured and no longer plays, a backup quarterback would fill in, but the team would not expect to keep its high Elo rating.

Consider this set of matchups:

games <- data.frame(
  team1 = c("A", "B", "A", "A"),
  team2 = c("B", "C", "C", "B"),
  quarterback1 = c("1", "2", "1", "4"),
  quarterback2 = c("2", "3", "3", "2"),
  won = c(1, 1, 1, 0)
)

We run elo.run() with the players() function. Behind the scenes, this is calculating an individual Elo score for the team and the quarterback. For updates, it sums the two Elos together for a composite Elo, compares this to the opponent’s composite Elo, and determines the Elo update. It divides this update according to the weights specified.

init <- c(A = 1500, B = 1500, C = 1500, "1" = 0, "2" = 0, "3" = 0, "4" = 0)
er <- elo.run(
  won ~ players(team1, quarterback1, weights = c(0.75, 0.25)) +
    players(team2, quarterback2, weights = c(0.75, 0.25)),
  data = games,
  k = 40,
  initial.elos = init
)
summary(er)

## 
## An object of class 'summary.elo.run', containing information on 7 teams and 4 matches.
## 
## Mean Square Error: 0.2536
## AUC: 0.3333
## Favored Teams vs. Actual Wins: 
##        Actual
## Favored 0 1
##   TRUE  1 1
##   (tie) 0 1
##   FALSE 0 1

final.elos(er)

##           1           2           3           4           A           B 
##    9.410557    5.676454   -9.698063   -5.388949 1512.064825 1517.029363 
##           C 
## 1470.905811

You can see that quarterback “1” adds 9 points to the effective Elo rating when he plays, and that quarterback “4” subtracts 5 points.

This also plays out in predictions on future games:

future.game <- data.frame(
  team1 = "A",
  team2 = "C",
  quarterback1 = c("1", "4"),
  quarterback2 = "3"
)

predict(er, newdata = future.game)

## [1] 0.5858725 0.5650629

Notice that the predicted win percentage is slightly lower with quarterback 4.

Another Application

This methodology has been used before to model an Ultimate Frisbee season. In this case, the players on the field at a given time are given a score (1 or 0) depending on whether they score. The first term is a players(player1, ..., player7), and the second term (because the opponent’s players weren’t kept track of) was simply the name of the opponent.