The statistical analysis of a thurstonian model for rating chess players

This paper formalizes and provides static and dynamic estimators for a scaling model for rating chess players. The model was suggested by the work of Arpad Elo, the inventor of the chess rating system in current use by both the United States and international chess federations. The model can be viewed as a Thurstone Case V model that permits draws (ties). A related model based on a linear approximation is also analyzed. In the chess application, possibly changing ability parameters are estimated sequentially from sparse data structures that often involve many fewer than $\text{M(M ? 1)}\text{2}$ observations on the M players to the rated. In contrast, psychological applications of paired-comparison scaling generally use models with no draw provision to estimate static parameters from a systematically obtained data structure such as a replicated “round robin” involving all M entities to be scaled. In the paper, both static and sequential estimators are provided and evaluated for a number of different data structures. Sampling theory for the estimators is developed. The application of rating systems to track temporally changing ability parameters may prove useful in many areas of psychology.