Consider a set of \(K\) items. Let the items be nodes in a graph and let there be a directed edge \((i, j)\) when \(i\) has won against \(j\) at least once. We call this the comparison graph of the data, and denote it by \(G_W\). Assuming that \(G_W\) is fully connected, the Bradley-Terry model states that the probability that item \(i\) beats item \(j\) is
$$p_{ij} = \frac{\pi_i}{\pi_i + \pi_j},$$
where \(\pi_i\) and \(\pi_j\) are positive-valued parameters representing the skills of items \(i\) and \(j\), for \(1 \le i, j, \le K\). The function btfit
can be used to find the strength parameter \(\pi\). It produces a "btfit"
object that can then be passed to btprob
to obtain the Bradley-Terry probabilities \(p_{ij}\).
If \(G_W\) is not fully connected, then a penalised strength parameter can be obtained using the method of Caron and Doucet (2012) (see btfit
, with a > 1
), which allows for a Bradley-Terry probability of any of the K items beating any of the others. Alternatively, the MLE can be found for each fully connected component of \(G_W\) (see btfit
, with a = 1
), and the probability of each item in each component beating any other item in that component can be found.