Lipschitz-Based Reinforcement Learning for Response-Time Distributions in Video-Game Design

This study proposes a mathematical framework for predicting complete response-time distributions associated with parametric video-game configurations. Each configuration is encoded as a point in a normalized metric space, and the statistical descriptors of its response-time distribution (median, mean, selected quantiles, interquartile range, and Skewness) are treated as real-valued Lipschitz functions on that space. Predictions for unseen configurations are obtained through McShane–Whitney extension formulas, which provide geometrically controlled upper and lower bounds compatible with the empirical Lipschitz regularity of the observed data. To handle the sequential incorporation of new observations, a regularization mechanism is introduced that replaces raw descriptors violating Lipschitz continuity constraints with a convex combination of the observed value and a weighted historical estimate. In the extended version of the method, the coefficient of this combination is selected through a one-pass online Q-learning-inspired procedure that selects, for each descriptor and instability regime, a data-dependent trade-off between fidelity and geometric regularity. The final output is a continuous Log-Normal density fitted by nonlinear least squares to the predicted descriptors, together with a Wasserstein-type uncertainty band derived from the Lipschitz bounds. The framework is validated on a controlled experiment with 24 participants across 20 game levels. Results show that the predicted distributions shift systematically with the input configuration and that, in the illustrative comparison, the adaptive mechanism produces feature-specific smoothing decisions that differ from those obtained with a fixed coefficient. The method also provides interpretable predictions from small experimental datasets without requiring fully data-driven models.