Ordinal Potential-based Player Rating

A two-player symmetric zero-sum game is transitive if for any pure strategies $x$, $y$, $z$, if $x$ is better than $y$, and $y$ is better than $z$, then $x$ is better than $z$. It was recently observed that the Elo rating fails at preserving transitive relations among strategies and therefore cannot correctly extract the transitive component of a game. Our first contribution is to show that the Elo rating actually does preserve transitivity when computed in the right space. Precisely, using a suitable invertible mapping $\varphi$, we first apply $\varphi$ to the game, then compute Elo ratings, then go back to the original space by applying $\varphi^{-1}$. We provide a characterization of transitive games as a weak variant of ordinal potential games with additively separable potential functions. Leveraging this insight, we introduce the concept of transitivity order, the minimum number of invertible mappings required to transform the payoff of a transitive game into (differences of) its potential function. The transitivity order is a tool to classify transitive games, with Elo games being an example of transitive games of order one. Most real-world games have both transitive and non-transitive (cyclic) components, and we use our analysis of transitivity to extract the transitive (potential) component of an arbitrary game. We link transitivity to the known concept of sign-rank: transitive games have sign-rank two; arbitrary games may have higher sign-rank. Using a neural network-based architecture, we learn a decomposition of an arbitrary game into transitive and cyclic components that prioritises capturing the sign pattern of the game. In particular, a transitive game always has just one component in its decomposition, the potential component. We provide a comprehensive evaluation of our methodology using both toy examples and empirical data from real-world games.