Dynamic Stochastic Models for Time-Dependent Ordered Paired Comparison Systems

When paired comparisons are made sequentially over time as for example in chess competitions, it is natural to assume that the underlying abilities do change with time. Previous approaches are based on fixed updating schemes where the increments and decrements are fixed functions of the underlying abilities. The parameters that determine the functions have to be specified a priori and are based on rational reasoning. We suggest an alternative scheme for keeping track with the underlying abilities. Our approach is based on two components: a response model that specifies the connection between the observations and the underlying abilities and a transition model that specifies the variation of abilities over time. The response model is a very general paired comparison model allowing for ties and ordered responses. The transition model incorporates random walk models and local linear trend models. Taken together, these two components form a non-Gaussian state-space model. Based on recent results, recursive posterior mode estimation algorithms are given and the relation to previous approaches is worked out. The performance of the method is illustrated by simulation results and an application to soccer data of the German Bundesliga.