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.

该研究使用适当的可逆映射，将博弈应用于 Elo 评分，然后返回原始空间来提取博弈的可转移组件，并将其与已知的 sign-rank 概念相联系。