A generalization of the Bradley–Terry model for draws in chess with an application to collusion

In inference problems where the dataset comprises Bernoulli outcomes of paired comparisons, the Bradley–Terry model offers a simple and easily interpreted framework. However, it does not deal easily with chess because of the existence of draws, and the white player advantage. Here I present a new generalization of Bradley–Terry in which a chess game is regarded as a three-way competition between the two players and an entity that wins if the game is drawn. Bradley–Terry is then further generalized to account for the white player advantage by positing a second entity whose strength is added to that of the white player. These techniques afford insight into players’ strengths, response to playing black or white, and risk-aversion as manifested by probability of drawing. The likelihood functions arising are easily optimised numerically. I analyse a number of datasets of chess results, including the infamous 1963 World Chess Championships, in which Fischer accused three Soviet players of collusion. I conclude, on the basis of a dataset that includes only the scorelines at the event itself, that the Candidates Tournament (Curaçao 1962) exhibits evidence of collusion (p<10−5), in agreement with previous work. I also present scoreline evidence for the effectiveness of such a drawing cartel: noncollusive games are less detrimental to future play than collusive games (p<10−5).