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An Extension to the Thurstone-Mosteller Model for Chess

1992; Wiley; Volume: 41; Issue: 5 Linguagem: Inglês

10.2307/2348921

2517-6153

R. J. Henery,

Experimental Behavioral Economics Studies

The Thurstone-Mosteller model for paired comparisons has been modified to take account of ties (draws in the case of chess) by Batchelder & Bershad (1979, Journal of Mathematical Psychology, 19, 39-60). The present paper describes a simple extension which allows for widely differing proportions of ties (or draws). The model is applied to some data of Keene & Divinsky (1989, Warriors of the Mind) relating to games between 64 of the greatest chess players. In addition, some more recent chess data is examined, and it is shown that the length of a chess game has an important bearing on the proportion of wins and draws. Keene & Divinsky (1989) gather together the results of all known games between 64 of the world's greatest chess players, past and present. To answer the question of who is (was) the greatest player, they fitted a Bradley-Terry model to their data, with a draw counting as half of a win. Their treatment of draws introduces serious bias since the proportion of draws varies widely among the chosen players and there is a systematic increase in the proportion of draws from the earliest games to the present day. Also, the use of the Bradley-Terry model is a little at variance with the elaborate rating system run by the international chess body FIDE. The FIDE ratings are based on a Thurstone-Mosteller model as described by Elo (1978). Although, as Keene & Divinsky (1989) and Elo (1978) agree, the precise model is not too important when comparing players of nearly similar abilities, and, although the Bradley-Terry model is computationally simpler, we believe that the Thurstone-Mosteller model should be preferred so as to facilitate comparisons with the internationally accepted Elo system. This paper introduces a modification of the Thurstone-Mosteller model which takes into account the differing draw proportions of the players and this model fits the Keene- Divinsky data reasonably well. Also, by analysing another set of data from ChessBase, it appears that the length of game is an important factor that should be taken into account when judging the relative strengths of players. Both factors should be taken into account in any complete model, but neither dataset is complete enough for this purpose, and in any case there are other factors, such as the ages of the players, that would be required before calling any model complete. Therefore we content ourselves with demonstrating (1) the usefulness of the draw model on the Keene-Divinsky dataset and (2) the necessity of allowing for length of game using the ChessBase dataset. Given the right dataset, it would be a simple matter to set up a single Thurstone-Mosteller model which made due allowance for all factors mentioned.