The Bradley-Terry Method: How Does It Work?

I discovered the Bradley-Terry method from the KRACH system used in college hockey. A more thorough explanation can be found at the prior link. To summarize, the ratings are calculated to meet two principles:

This system has a problem with teams who are either winless or undefeated; in order to match such a record, their ratings would have to be zero or infinite. To avoid this, each team is credited with three tie games (1/2 win and 1/2 loss each) against a hypothetical “average” team. (In KRACH, a rating of 100 is average, but this is arbitrary; scaling all the ratings by the same factor gives identical results. I use a rating of 1 as average instead.)

Prior to the 2011 football season, I used only one such tie game, but this tended to predict unrealistically high win percentages for unbeaten teams. The change to three fictional games tends to push down teams who have mediocre records against great schedules, as compressing the rating scale in this manner weakens more of their opponents' ratings than it improves. This mitigates one of the main criticisms of KRACH (which no longer uses the fictional game at all): a team can get to a fairly high rank purely by being in a tough conference, even if they do very poorly. This is exacerbated by the high percentage of games in conference in hockey (about 80%, compared to 65% for football or basketball).

Team A has played three games: a win over Team B (rating R_B = 2), a win over Team C (R_C = 3), and a loss to Team D (R_D = 4). Team A’s rating will affect the ratings of Teams B, C, and D, so all of these must be calculated iteratively, but for the moment we’ll assume R_B, R_C, and R_D are constants in calculating R_A.

Let’s start with a guess of R_A = 3. Team A’s expected wins:

This is too high, so R_A must be less than 3. Let’s try 2.5:

Still too high, but we’re pretty close.

Trial and error like the above would make calculation prohibitively difficult. Another way to interpret the rating is that a team’s rating is equal to their win ratio (wins divided by losses, including the fictional games) times their strength of schedule, defined as the rating of a team such that if their entire schedule (including the fictional games) were replaced by playing that team N times, their projected record would be exactly the same.

Strength of schedule can be calculated from the following formula:

`SOS = (sum R_o/(R_s+R_o))/(sum 1/(R_s+R_o))`

where R_s is your rating and R_o is your opponent’s, summed across all opponents. Because this is a nonlinear equation, an iterative process is used to calculate the ratings. Each team is assigned an initial guess of 1, then the strength of schedule and new rating for each team is caluclated using that guess. With the new ratings, each team’s strength of schedule is recomputed and a new set of ratings is generated. The process is repeated until the ratings converge.

The basic method has a significant flaw: A single-point 3OT squeaker and a 50-point blowout count exactly the same. There are methods which look at margin of victory, but many of these ignore record entirely and look only at pure points. An example of this is Ken Pomeroy’s basketball ratings, which adjust for tempo and strength of schedule to compute offensive and defensive ratings based on points scored and allowed per possession against an average team. While this is highly useful, it’s possible for the system to rate a team very highly even if that team has a terrible record, simply because that team loses a lot of close games and wins a few massive blowouts. (Illinois in 2008 is perhaps the most notable example: with a losing record, they ranked 40th out of 341 teams.)

One way to avoid overvaluing blowouts is to reduce the value of each additional point of scoring margin as you move away from zero. The way I do this is to assign a number of “victory points” (between 0 and 1) to each team based on the margin of the game:

`V_A = 1 / (1 + e^(-M_A/alpha))`

where M_A is Team A’s net scoring margin for the game (negative for a loss) and α is a factor that adjusts the balance between a pure points rating and a pure W-L rating. As α approaches 0, the rating system closely approximates the basic, pure W-L rating, with only extremely close games getting less than full credit for a win. As α grows, a team’s victory point “record” becomes nearly linearly related to total scoring margin. After calculating each team’s victory point total, the basic rating system as above is applied using victory points instead of the true number of wins.

Prediction of win probability and margin of victory is based on the following two equations:

The value of α is chosen to have a game within α points be considered “close” and a margin greater than 5α or 6α be a major blowout. For now, I am using α=5 for basketball and 6.5 for football. K_M and K_W are then approximated based on the available data.

The above methods completely ignore home field for computation. However, a team that managed the same results against the same teams while playing all of its games on the road would most likely be better than one that hosted all of its games. Home-field advantage is accounted for by adjusting each opponent’s rating while computing strength of schedule. If a game was played on the road, that opponent’s rating is multiplied by H for the purposes of computing strength of schedule; if the game was a home game, the opponent’s rating is divided by H instead. Most neutral site games receive no adjustment, but some are considered “semi-home” games (for instance, Georgia facing Boise State in Atlanta) and get a reduced adjustment factor of `sqrt(H)` instead.

For predicitve purposes, the home team’s rating is multiplied by H (or `sqrt(H)` for semi-home games) before calculating win probability and expected margin.

The value of H is determined empirically by computing the ratings with different H values and comparing the expected number of home wins in true home games against the true number. Early in the season, the value from the previous season is used; I generally switch over to calculating based on the current season around the midpoint of conference play.

Starting with the 2014 football season, I’ve included an adjustment in the early season that biases teams toward their rating from the previous year. This is to prevent cases where two strong teams play each other in week 1 and the loser ends up severely punished for it until the data set is large enough for the winner’s true strength to be recognized.

The adjustment is handled in much the same manner as the fictional games used for normalization: additional fake tie games are added, but this time against an opponent with rating equal to the team’s previous-year rating instead of an average team. However, these games are gradually removed from the ratings as real data comes in. Initially, the preseason adjustment is given a weight of five games. For football, that weight is decreased by 2/3 of a game for each game the team has played this season until it is completely removed after eight games. For basketball, the decay rate is 1/2 game instead of 2/3, so it does not disappear until after the team’s 10th game.

SOS values shown on the ratings page do not include the games put in for preseason adjustment.

Each team page has a list of their games played so far, with the following information:

Victory point totals are given including the normalization and preseason adjustment, then the SOS and victory point ratio which combine to produce the margin-aware rating. The page currently does not include the basic-method rating.