# Stochastic Extensions of the Elo Rating System
## 1 Introduction  
Rating systems track and predict the performance of competitors in pairwise zero-sum games. They were initially developed to objectively measure the strength of chess players, with the first successful system proposed by [PERSON] in 1960 and adopted by the United States Chess Federation (USCF) to replace the more problematic Harkness rating system. The International Chess Federation (FIDE) also started publishing ratings of its players in 1970 and continues to publish them today. The Elo rating system, or variants of it, are also used by most chess websites, where users are automatically matched against other players of similar strength, as well as by federations of Go and Scrabble players, and several video games.  
The motivation, mathematical basis, and problems of the Elo rating system are explained in detail in [PERSON]'s book [1], and the academic literature is summarized in [2]. The system gives each player a real parameter called their rating and defines an algorithm to update it with the result of each new game. Winning a game increases the player's rating, and losing decreases it, but the increase depends on the rating of the rival. This allows for the ratings to change as the strength of the players also does. More sophisticated systems like _glicko_, proposed by [PERSON] [3], include two parameters for each player, representing the strength and its variability, to allow for players gaining strength more quickly than others.  
In his thesis, [PERSON] also extends Elo ratings to sports like American football to make predictions about the score (not just the result), and more recently, the idea of applying Elo ratings to team sports has gained popularity. For instance, [[https://fivethirtyeight.com/features/how-we-calculate-nba-elo-ratings/](https://fivethirtyeight.com/features/how-we-calculate-nba-elo-ratings/)]([https://fivethirtyeight.com/features/how-we-calculate-nba-elo-ratings/](https://fivethirtyeight.com/features/how-we-calculate-nba-elo-ratings/)) (accessed on 19 July 2024) has been computing the ratings of NBA teams since 2014, and other websites like www.eloratings.net (accessed on 19 July 2024) do the same for national soccer teams.  
There is an extensive literature on modeling the score of sports with stochastic processes, either Brownian motion [4] or other continuous processes such as the gammaprocess [5]. In order to account for ties, discrete processes can be used, typically Poisson processes [6; 7], at the expense of more convoluted prediction and update formulas. Logistic models have also been used to predict the score during the game [8; 9].  
Our contributions are a more formal study of the model underlying the Elo system, with a few results regarding unbiased estimators for the ratings that are used in our computational analysis of league games. We derive how the natural extension of the Elo model for games involving a scoreboard can be used to predict the result during the game and develop a systematic setup for evaluating different systems.  
Finally, we show how a discrete stochastic process can be used to model the score and integrated into a novel type of Elo rating system. For comparison, we implement the rating system behind [PERSON]'s soccer-specific SPI ratings [10] and show that our system has similar predicting power for the result of the games, despite only tracking one parameter for each team.  
The remainder of this paper is organized as follows: Section 2 recalls the main aspects of the (static) Elo rating system. The proposed stochastic extensions of the Elo system are then introduced in Section 3. Their performance at predicting the results of team sports, namely, basketball and soccer, is analyzed through the computational study presented in Section 4. Finally, Section 5 is devoted to shed conclusions and expose possible lines of future work.