Bradley–Terry Model

Basic Concepts

The objective of the Bradley-Terry model is to determine the ranking of n variables, representing competitors, based on pairwise competitions.

Let 1, 2, …., n represent the competitors. Let w_ij = the number of matches in which i wins against j. We assume there are no ties. We also define w_i = the number of wins by i.

Our objective is to obtain a ranking r_i such that r_i > r_j implies that i  is a stronger competitor than j. In fact, we seek r such that

The likelihood function is

Thus, the log-likelihood function is

It turns out that we can obtain estimates for the r_i that maximize LL via iteration. If the r_i are the current estimates, then revised estimates are obtained via

We then normalize these estimates via

and use these values as r_i to obtain the next estimate.

Example

Example 1: Figure 1 displays the wins of 4 video game players in head-to-head competitions. E.g. players B and C play 11 games, of which B wins 6 times (w₂₃ = 6) and C wins 5 times (w₃₂ = 5). Player A won 5 games (w₁ = SUM(B2:E2) = 5) in total and lost 0+3+2+4 = 9 times.

Determine the ranking r_i of the 4 players.

Figure 1 – Win table

Based on the sample in Figure 1, the probability of player i wins against player j is shown in Figure 2.

Figure 2 – Pairwise (sample) probabilities

E.g. since A beat B 1 time and lost 3 times, the probability that A beats B is 1/(1+3) = .25 (cell AG8). From Figure 1, we see that B and D never played each other and so the values in cells AI9 and AG11 are undefined.

Iterative Rankings

We need to estimate the rankings of the 4 players. We will do this so that the sum of the rankings is one. As described above, we create these ranking estimates iteratively, as shown in Figure 3 (using 17 iterations, stopping when the value in row 16 is equal to one.

We initialize the process by setting the ranks of each player to 1/n (1/4 = .25 for Example 1, as shown in M2:M5 of Figure 3).

Figure 3 – Iterations

To create the iteration in Figure 3, we use the symmetric table of w_ij + w_ji values shown in Figure 4.

Figure 4 – Total pairwise matches

E.g. the value in cell J3 (or I4) is calculated by the formula =D3+C4.

We now show how to calculate the values in the range M7:M10, namely

E.g. we place the formula

=$H2/(M2+M$2)+$I2/(M2+M$3)+$J2/(M2+M$4)+$K2/(M2+M$5)

in cell M7, highlight M7:M10, and press Ctrl-D.

The values in range M12:M15 correspond to the r_i′ = w_i/p_i. This is done by inserting the formula =$F2/M7 in cell M12, highlighting M12:M15, and then pressing Ctrl-D. We also calculate the sum of the r_i′ in cell M16 via the formula =SUM(M12:M15), Finally, we normalize the r_i′ as the next iteration of the ranks as shown in N2:N5. This is done by inserting the formula =M12/M$16 in cell N2, highlighting range N2:N5, and pressing Ctrl-D.

We now complete the rest of the calculations for iteration #2 by highlighting the range M7:N15 and pressing Ctrl-R. Finally, we fill in the rest of Figure 3 (iterations 3-17) by highlighting range N2:AC16 and pressing Ctrl-R. We highlight enough columns so that the value in row 16 becomes 1.

We see from range AC2:AC5 that the rankings for the 4 players are (.153, .300, .219, .328). Thus D is viewed as the strongest player, while A is the weakest. Notice that the order is not the same as the values in column F.

Pairwise Rankings

The pairwise ranking of the 4 players is shown in Figure 5.

Figure 5 – Pairwise (population) probabilities

For example, the value in cell AO9 is calculated via the formula =AC3/(AC3+AC5). This means that the probability that B beats D in any match is 47.8% (and the probability that D wins is 1-.478 = 52.2% as shown in cell AM11). Note that we are able to obtain such probabilities despite the fact that B and D never played each other.

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack provides the following array functions where R1 is a square array of the results of head-to-head matches (as B2:E5 of Figure 1) and R2 is a column array with the rankings of the competitors (as AC2:AC5 of Figure 3).

BT_MODEL(R1, prec, iter): returns a column array with the rankings of the competitors whose pairwise wins are shown in R1. A maximum of iter iterations is performed (default 100), although the iterations stop when the sum of the ranks (before normalization) is less than or equal to 1+prec (prec defaults to .0000001).

PairwiseRanks(R1): returns a square array with the pairwise probabilities based on the sample in R1 (as shown in Figure 2).

PairwiseRanks(R2): returns a square array with the pairwise probabilities based on the population rankings in R2 (as shown in Figure 5).

We see that =BT_MODEL(B2:E5) produces results similar to those shown in range AC2:AC5 of Figure 3. =PairwiseRanks(B2:E5) produces the results shown in range AF8:AI11 of Figure 2 and =PairwiseRanks(AC2:AC5) produces the results shown in range AL8:AO11 of Figure 5.

In fact, we can obtain the results shown in Figure 5 via the array formula =PairwiseRanks(BT_MODEL(B2:E5)).

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Wikipedia (2023) Bradley-Terry model
https://en.wikipedia.org/wiki/Bradley%E2%80%93Terry_model

Fleischhaker, D. D. (2019) Modelling outcomes in Canadian professional football via generalized Bradley-Terry models
https://www.semanticscholar.org/paper/Modelling-Outcomes-in-Canadian-Professional-via-Fleischhaker/e778edff4806655492e1d46b1628fa844e23eb71/figure/0