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Ordinal Potential-based Player Rating

Nelson Vadori, Rahul Savani

TL;DR

This work addresses the failure of traditional Elo ratings to preserve transitivity in general-sum games by introducing Hyperbolic Elo, which applies an invertible basis transformation $\varphi_{\beta}(x)=\tfrac{1}{\beta}\tanh(\beta x)$ before computing Elo and then mapping back. Building on this, it characterizes transitive games as a weak separable ordinal potential and connects transitivity to sign-rank, enabling a disk-based decomposition of any game into a transitive component and cyclic components. A neural network architecture learns this decomposition by optimizing a sign-preserving loss over a disk-space representation, decoupling sign-structure learning from amplitude reconstruction via basis functions. Empirical results show improved sign accuracy over prior methods on toy and real-world games, validating the approach and its potential for robust, sign-aware game analysis and rating. The work provides a concrete framework for decomposing hybrid games and offers avenues for scalable online updates and advanced architectures in future research.

Abstract

It was recently observed that Elo ratings fail at preserving transitive relations among strategies and therefore cannot correctly extract the transitive component of a game. We provide a characterization of transitive games as a weak variant of ordinal potential games and show that Elo ratings actually do preserve transitivity when computed in the right space, using suitable invertible mappings. Leveraging this insight, we introduce a new game decomposition of an arbitrary game into transitive and cyclic components that is learnt using a neural network-based architecture and that prioritises capturing the sign pattern of the game, namely transitive and cyclic relations among strategies. We link our approach to the known concept of sign-rank, and evaluate our methodology using both toy examples and empirical data from real-world games.

Ordinal Potential-based Player Rating

TL;DR

This work addresses the failure of traditional Elo ratings to preserve transitivity in general-sum games by introducing Hyperbolic Elo, which applies an invertible basis transformation before computing Elo and then mapping back. Building on this, it characterizes transitive games as a weak separable ordinal potential and connects transitivity to sign-rank, enabling a disk-based decomposition of any game into a transitive component and cyclic components. A neural network architecture learns this decomposition by optimizing a sign-preserving loss over a disk-space representation, decoupling sign-structure learning from amplitude reconstruction via basis functions. Empirical results show improved sign accuracy over prior methods on toy and real-world games, validating the approach and its potential for robust, sign-aware game analysis and rating. The work provides a concrete framework for decomposing hybrid games and offers avenues for scalable online updates and advanced architectures in future research.

Abstract

It was recently observed that Elo ratings fail at preserving transitive relations among strategies and therefore cannot correctly extract the transitive component of a game. We provide a characterization of transitive games as a weak variant of ordinal potential games and show that Elo ratings actually do preserve transitivity when computed in the right space, using suitable invertible mappings. Leveraging this insight, we introduce a new game decomposition of an arbitrary game into transitive and cyclic components that is learnt using a neural network-based architecture and that prioritises capturing the sign pattern of the game, namely transitive and cyclic relations among strategies. We link our approach to the known concept of sign-rank, and evaluate our methodology using both toy examples and empirical data from real-world games.
Paper Structure (19 sections, 6 theorems, 36 equations, 12 figures, 5 tables)

This paper contains 19 sections, 6 theorems, 36 equations, 12 figures, 5 tables.

Table of Contents

  1. INTRODUCTION
  2. ORDINAL POTENTIAL-BASED PLAYER RATING: FROM ELO TO POTENTIAL GAMES
  3. LEARNING TO DECOMPOSE AN ARBITRARY GAME
  4. EXPERIMENTS
  5. CONCLUSION AND FUTURE RESEARCH
  6. EXPERIMENTAL DETAILS AND ADDITIONAL EXPERIMENTS
  7. Loss function
  8. Game of Figure 1 and additional examples of transitive games
  9. Game of Figure 2
  10. Game of Figure 3
  11. Stability of ratings
  12. Architecture, compute and game data
  13. Baselines
  14. Experiments in Table 1
  15. PROOFS AND TECHNICAL COMMENTS
  16. ...and 4 more sections

Key Result

Proposition 1

(The normal decomposition and $m$-Elo do not preserve transitivity) Let $P$ be a transitive game, and let $\mathcal{T}:=Disk(u^\mathcal{T}, v^\mathcal{T})$ be the transitive component of the normal decomposition of $P$. Then, we can have $\mathop{\mathrm{sign}}\limits(\mathcal{T}) \neq \mathop{\math

Figures (12)

  • Figure 1: Transitive game of order two of polynomial type, $n=30$. (left) $Y$-axis: ordinal potential player scores $\Phi_i$; $X$-axis: player index; (right) game $P$ and its learnt basis functions as a function of its disk decomposition, cf. Section \ref{['seclearn']}: $X$-axis: disk space $\mathcal{D}$, $Y$-axis: payoff space $P$. We are able to learn the two generating polynomial functions and that the player scores $\Phi_i$ are evenly spaced (details are provided in Appendix \ref{['app:fig1']}).
  • Figure 2: AlphaStar, $n=20$. (Left) ordinal potential-based ratings of the $n$ players in decreasing order; (middle) sign of the game $P$; (right) sign of the potential component $\Phi_i-\Phi_j$. Red is positive, blue is negative, white is zero. We are able to learn correctly the 9 rating "levels": first, 7 players each with their own rating, then a large cycle where players share the same rating, and finally a player who loses against everyone.
  • Figure 3: Cyclic game of sign-rank $2$ and order $M=2$. (Top left) ours, 2 basis functions; (top right) ours, 1 basis function; (bottom) normal decomposition. Black dots represent the true game $P$; red crosses indicate the true game when there is a mistake on the sign. All methods learn $K=1$ disk component. Contrary to the baseline, our method is able to learn the sign of $P$, as well as the two functions generating the game. $X$-axis: disk space $\mathcal{D}$, $Y$-axis: payoff space $P$.
  • Figure 4: Kuhn-poker, $n=25$ (cyclic game). (Left) true game; (mid) ours; (right) normal decomposition; (top) sign of the game; (bottom) the game in disk space ($X$-axis: disk space $\mathcal{D}$, $Y$-axis: payoff space $P$). We are able to learn the sign of the game with 3 components, all cyclic as in Theorem \ref{['propcyclic1']}. The normal decomposition cannot learn the correct sign with 3 components, and further one component is transitive, which is counterintuitive for a cyclic game. Red is positive, blue is negative, white is zero.
  • Figure 5: Neural Architecture to learn our game decomposition.
  • ...and 7 more figures

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Theorem 1
  • Definition 4
  • Theorem 2
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 6 more