Probabilistic models of rating systems

The talk is concerned with the two rating systems: Elo and TrueSkill. The process of variation of a player rating in an infinite series of games with the same adversary is concerned in the Elo model. This process is shown to have a stationary distribution for all values of the parameters; its median is found in the case the players have the same skill. For the Bayesian rating system TrueSkill, we analyze some properties of the model, find the exact formula for the posterior distribution of the skill level in the case of two players and obtain estimates of its approximation accuracy.