Generalization of the Elo system for interactions between many players

Анотація

The paper proposes a generalization of the classical Elo rating system for multi-player environments and employs optimization techniques to identify the model’s optimal configuration. The proposed approach represents multi-agent interactions as a set of pairwise comparisons and formulates the rating estimation problem as the minimization of a logistic loss function. This formulation enables the derivation of analytical expressions for rating updates based on gradients used to search for optimal hyperparameters, resulting in smoother and more stable learning dynamics compared to the classical Elo system. An additional adaptive update coefficient is introduced, which depends on the number of interactions and the number of agents involved in each event. Such normalization helps prevent excessive rating drift in large datasets and mitigates overly dampened updates when the amount of data or the number of agents is small. The paper also examines principles for initializing the ratings of new agents and for dynamically adjusting the learning rate based on the accumulated information for each agent, allowing the system to converge more rapidly to accurate rating levels and improving overall stability and interpretability. Gradient descent is employed to search for the optimal hyperparameter values by minimizing the loss function and automatically selecting appropriate parameter settings. This enables the system to produce more accurate rating estimates and, consequently, achieve better predictions of future interactions. The proposed system easily adapts to environments with multi-party interactions while remaining fully compatible with binary-comparison scenarios. When necessary, it can be extended with additional hyperparameters to account for the specifics of a given domain. The model illustrates the advantages of combining classical rating methodology with modern optimization techniques and is applicable to sports, gaming, and educational systems where accuracy, adaptability, and interpretability are essential.

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