Identifying and Clustering Counter Relationships of Team Compositions in PvP Games for Efficient Balance Analysis

TL;DR

This work has developed two advanced measures that extend beyond the simplistic win rate to quantify balance in zero-sum competitive scenarios and presents a methodology that significantly improves game balance evaluation and design.

Abstract

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 Age of Empires II, Hearthstone, Brawl Stars, and 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.

Figures (17)

• Figure 1: Radar chart comparison of two team compositions across matchups with six different opponents. The left panel illustrates a scenario with no domination, where both compositions exhibit their strengths against specific opponents. The right panel shows a case where Comp1 dominates Comp2, achieving higher win rates against all opponents, illustrating clear dominance in overall performance.
• Figure 2: Architecture of the Neural Counter Table $C_\theta$. The diagram illustrates the process of estimating residual win values between team compositions. Team Comp A and Team Comp B are encoded into latent representations $z_e(c_A)$ and $z_e(c_B)$ through shared encoder weights. These latent codes are then quantized into embedding vectors $z_q(c_A)$ and $z_q(c_B)$ using the nearest neighbor search. The embedding vectors are classified into counter categories A and B. The decoded quantized vectors $[z_q(c_A), z_q(c_B)]$ and $[z_q(c_B), z_q(c_A)]$ are fed into fully connected (FC) layers with tanh activation functions to produce intermediate values $x_1$ and $x_2$. The residual win value prediction is calculated as the average of the differences between these intermediate values, providing an estimation of the residual win value $W_{res}$. The dashed layer implies shared weights.
• Figure 3: This diagram illustrates the learning procedure for deriving the Neural Rating Table $R_\theta$ and the Neural Counter Table $C_\theta$. The process begins with team compositions (Team Comp A and Team Comp B) and their corresponding match outcomes (win-lose results). In the first stage, team compositions are processed through a shared rating encoder to obtain composition ratings (Comp Rating A and Comp Rating B). These ratings are then utilized in the Bradley-Terry model to predict win values, forming the Neural Rating Table $R_\theta$. In the second stage, these compositions are further processed through a shared category encoder to determine counter categories. The residual win value predictor uses these categories to refine win value predictions, accounting for cyclic dominance, thus forming the Neural Counter Table $C_\theta$.
• Figure 4: An example of extending the classical Rock-Paper-Scissors to more complex cases.
• Figure 5: Demonstration of codebook utilization improvement using VQ Mean Loss.

Theorems & Definitions (9)

• Definition 2.1
• Proposition 2.2
• Proposition 5.2
• Definition 5.3
• Lemma 5.6
• Proposition 5.8
• Proposition 5.9
• Proposition 5.10
• Lemma 5.11