Plackett-Luce modeling with trajectory models for measuring athlete strength

It is often the goal of sports analysts, coaches, and fans to predict athlete performance over time. Models such as Bradley-Terry and Plackett-Luce measure athlete skill based on results of competitions over time, but have limited predictive strength without making assumptions about the nature of the evolution of athletic skill. Growth curves are often applied in the context of sports to predict future ability, but these curves are too simple to account for complex career trajectories. We propose a non-linear, mixed-effects trajectory to model the ratings as a function of time and other athlete-specific covariates. The mixture of trajectories allows for flexibility in the estimated shape of career trajectories between athletes as well as between sports. We use the fitted trajectories to make predictions of an athlete's career trajectory through a model of how athlete performance progresses over time in a multi-competitor scenario as an extension to the Plackett-Luce model. We show how this model is useful for predicting the outcome of women's luge races, as well as show how we can use the model to compare athletes to one another by clustering career trajectories.