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

An ANN Model For Individual Ratings in Group Competitions 5
(Bradley & Terry, 1952; Davidson & Farquhar, 1976; David, 1988; Hunter,
2004). This paper differs from previous research in two ways. First, it proposes
a method to determine individual ratings and rankings when the subjects
being compared are groups. (Huang et al., 2005) proposed a similar
extension, however their method of fitting the model assumes that “each
individual forms a winning or losing team in some competitions which together
involve all subjects” In situations involving thousands of players on
teams that may never compete against each other, their method is not guaranteed
to converge. In addition, they could not prove the method of fitting
would converge to the global minimum. The following method, however, will
always converge to the global minimum because it is an ANN trained with
the delta rule—a method whose convergence properties are well-known. Perhaps
more importantly, the model proposed by (Huang et al., 2005) fits each
parameter simultaneously, which is both impractical when the amount of
competitions and individuals is large, and undesirable if the model needs to
be updated after every competition. This is because new individuals may
enter the model at any point, and therefore it is impossible to know when
to stop and fit all the individual parameters. This leads to the second additional
contribution of this paper.
The more common methods for fitting a Bradley-Terry model require
that all of the comparisons to be considered involving all of the competitors
have already taken place so the model can be fit simultaneously. While this
can be more theoretically accurate, it is intractable in situations involving