A Bradley-Terry Artificial Neural Network Model for Individual ...

An ANN Model For Individual Ratings in Group Competitions 7
winner and the question asked is always what is the probability that team
A will win, the input for w A will always be 1 and the input for w B will
always be −1. The equation for the model can be written as:
1
Output = Pr(A def B) =
. (4)
1 + e−(wA−wB) which is the same as (3), except w A and w B are substituted for θ A and θ B .
The model is updated applying the delta rule which is written as follows:
∆w i = ηδx i (5)
where η is a learning rate, δ is the error measured with respect to the output,
and x i is the input for w i . The error, δ is usually measured as:
δ = Target Output − Actual Output. (6)
For the ANN Bradley-Terry model, the target output is always 1 given the
assumption that A will defeat B. The actual output is the output of the
single node ANN. Therefore, the delta rule error can be rewritten as:
δ = 1 − Pr(A def. B) = 1 − Output, (7)
and the weight updates can be written as:
∆w A = η(1 − Output) (8)
∆w B = −η(1 − Output) (9)
Here, x i is implicit as 1 for w A and −1 for w B , again since A is assumed to
be the winning team. It is well-known that a single-layer ANN trained with
the delta rule will converge to the global minimum as long as the learning