Rating competitors in games with strength-dependent tie probabilities

TL;DR

This paper introduces a new Bayesian rating system for head-to-head games that accounts for the dependence of tie probabilities on player strength, improving the accuracy of strength estimates.

Contribution

It develops a Bayesian dynamic model that explicitly incorporates strength-dependent tie probabilities, addressing limitations of traditional rating systems.

Findings

The new system provides more reliable strength estimates in chess.

It demonstrates improved accuracy on a large correspondence chess dataset.

The model allows efficient Bayesian updates with a single Newton-Raphson iteration.

Abstract

Competitor rating systems for head-to-head games are typically used to measure playing strength from game outcomes. Ratings computed from these systems are often used to select top competitors for elite events, for pairing players of similar strength in online gaming, and for players to track their own strength over time. Most implemented rating systems assume only win/loss outcomes, and treat occurrences of ties as the equivalent to half a win and half a loss. However, in games such as chess, the probability of a tie (draw) is demonstrably higher for stronger players than for weaker players, so that rating systems ignoring this aspect of game results may produce strength estimates that are unreliable. We develop a new rating system for head-to-head games based on a model by Glickman (2025) that explicitly acknowledges that a tie may depend on the strengths of the competitors. The approach uses a Bayesian dynamic modeling framework. Within each time period, posterior updates are computed in closed form using a single Newton-Raphson iteration evaluated at the prior mean. The approach is demonstrated on a large dataset of chess games played in International Correspondence Chess Federation tournaments.