A Lipschitz–Wasserstein Framework for Modeling Reaction-Time Distributions in Video Game Design

We present a novel framework for modeling reaction time distributions in the context of video games aimed at providing a performance tool to support the design of new levels. Modern games generate rich behavioral telemetry (including reaction times, success rates, and interaction patterns) that can be leveraged to understand player behavior and inform adaptive game design. Given a set of general parameters describing a newly designed level, the framework predicts the corresponding reaction time distribution, offering actionable insight during the design process. To address this problem, we employ a combination of statistical fitting via maximum likelihood estimation, weighted approximations, and Lipschitz-based estimators in Wasserstein space. This mathematical framework establishes the groundwork for future AI-based extensions, using metric-space learning to predict distributions for unseen level configurations. The methodology provides theoretical guarantees under mild mathematical assumptions, ensuring bounded estimation errors through the assumption of Lipschitz continuity. Three approaches are proposed, all grounded in a Lipschitz characterization of the metric model parameters, which embeds the vector representation of levels into the 1-Wasserstein space of reaction time distributions. The practical applicability of the framework is demonstrated on a dataset of 480 gameplay observations across 24 participants and 20 distinct trials, testing all three fitting procedures on a set of representative examples.