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

2 Joshua Menke, Tony Martinez
1 Introduction
The Bradley-Terry model is well known for its use in statistics for paired
comparisons (Bradley & Terry, 1952; Davidson & Farquhar, 1976; David,
1988; Hunter, 2004). It has also been applied in machine learning to obtain
multi-class probability estimates (Hastie & Tibshirani, 1998; Zadrozny,
2001; Huang, Lin, & Weng, 2005). The original model says given two subjects
of strengths λ A and λ B , the probability that subject A will defeat or
is better than subject B in a paired-comparison or competition is:
Pr(A def. B) =
λ A
λ A + λ B
. (1)
Bradley-Terry maximum likelihood models are often fit using methods like
Markov Chain Monte Carlo (MCMC) integration or fixed-point algorithms
that seek the mode of the negative log-likelihood function (Hunter, 2004).
These methods however, “are inadequate for large populations of competitors
because the computation becomes intractable.” (Glickman, 1999). The
following paper presents a method for determining the ratings of over 4,000
players over 3,379 competitions.
Elo (see, for example, (Glickman, 1999)) invented an efficient approach
to iteratively learn the ratings of thousands of players competing in thousands
of chess tournaments. The currently used “ELO Rating System” reparameterizes
the Bradley-Terry model by setting λ A = e θA yielding:
Pr(A def B) =
1
1 + 10 − θ A −θ B
400
. (2)