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

Abstract

Abstract 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 (Curacao 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 ).