Abstract

\textbf{How can balance be quantified in game settings?} This question is crucial for game designers, especially in player-versus-player (PvP) games, where analyzing the strength relations among predefined team compositions—such as hero combinations in multiplayer online battle arena (MOBA) games or decks in card games—is essential for enhancing gameplay and achieving balance. We have developed two advanced measures that extend beyond the simplistic win rate to quantify balance in zero-sum competitive scenarios. These measures are derived from win value estimations, which employ strength rating approximations via the Bradley-Terry model and counter relationship approximations via vector quantization, significantly reducing the computational complexity associated with traditional win value estimations. Throughout the learning process of these models, we identify useful categories of compositions and pinpoint their counter relationships, aligning with the experiences of human players without requiring specific game knowledge. Our methodology hinges on a simple technique to enhance codebook utilization in discrete representation with a deterministic vector quantization process for an extremely small state space. Our framework has been validated in popular online games, including \textit{Age of Empires II}, \textit{Hearthstone}, \textit{Brawl Stars}, and \textit{League of Legends}. The accuracy of the observed strength relations in these games is comparable to traditional pairwise win value predictions, while also offering a more manageable complexity for analysis. Ultimately, our findings contribute to a deeper understanding of PvP game dynamics and present a methodology that significantly improves game balance evaluation and design.