Equilibrium-Invariant Embedding, Metric Space, and Fundamental Set of 2×2 Normal-Form Games

Equilibrium solution concepts of normal-form games, such as Nash equilibria, correlated equilibria, and coarse correlated equilibria, describe the joint strategy profiles from which no player has incentive to unilaterally deviate. They are widely studied in game theory, economics, and multiagent systems. Equilibrium concepts are invariant under certain transforms of the payoffs. We define an equilibrium-inspired distance metric for the space of all normal-form games and uncover a distance-preserving equilibrium-invariant embedding. Furthermore, we propose an additional transform which defines a better-response-invariant distance metric and embedding. To demonstrate these metric spaces we study $2\times2$ games. The equilibrium-invariant embedding of $2\times2$ games has an efficient two variable parameterization (a reduction from eight), where each variable geometrically describes an angle on a unit circle. Interesting properties can be spatially inferred from the embedding, including: equilibrium support, cycles, competition, coordination, distances, best-responses, and symmetries. The best-response-invariant embedding of $2\times2$ games, after considering symmetries, rediscovers a set of 15 games, and their respective equivalence classes. We propose that this set of game classes is fundamental and captures all possible interesting strategic interactions in $2\times2$ games. We introduce a directed graph representation and name for each class. Finally, we leverage the tools developed for $2\times2$ games to develop game theoretic visualizations of large normal-form and extensive-form games that aim to fingerprint the strategic interactions that occur within.

💡 Research Summary

The paper introduces a systematic way to view normal‑form games as points in a metric space that respects equilibrium concepts. It begins by recalling that Nash, correlated, and coarse‑correlated equilibria are invariant under affine transformations of payoffs (adding a constant to each player’s payoff and scaling by a positive factor). These “equilibrium‑invariant transforms” form a group that leaves the set of equilibria unchanged. By endowing the space of all n‑player games with an L₂‑type distance on the payoff tensors, the authors show that this distance is preserved under the transform group, giving a well‑defined metric on the space of games modulo equilibrium‑invariant equivalence.

Focusing on two‑player, two‑strategy (2×2) games, which are fully described by eight real numbers, the authors construct a dramatic dimensionality reduction. They apply a linear change of basis followed by an arctangent mapping to obtain two angles, θ₁ and θ₂, each ranging from –π to +π. Geometrically, each angle corresponds to a point on the unit circle: θ₁ captures the relative incentive for one player to switch rows, while θ₂ captures the analogous incentive for the other player to switch columns. The pair (θ₁, θ₂) constitutes the equilibrium‑invariant embedding. Because the embedding is distance‑preserving, games that are close in the (θ₁, θ₂) plane have similar equilibrium structures, and many strategic properties (zero‑sum component, coordination vs. anti‑coordination, presence of cyclic best‑response dynamics, etc.) can be read off directly from the location on the circle.

The paper then removes redundancies due to player and strategy permutations. By applying “equilibrium‑symmetric” transformations (swapping rows/columns, reflecting signs, etc.) the authors shrink the embedding region to one‑eighth of its original area, yielding a canonical fundamental domain.

A further refinement is the best‑response‑invariant transform, which preserves only the ordering of best‑responses rather than the full equilibrium set. Under this weaker equivalence, many distinct points in the (θ₁, θ₂) plane collapse onto the same representative. Remarkably, the authors prove that all non‑trivial 2×2 games collapse into exactly 15 equivalence classes. These classes coincide with the classic classification by Borm (1987) but are now situated within a continuous geometric space, allowing one to measure distances between classes and to visualize transitions. Each class is illustrated by a directed graph where nodes are pure strategy profiles and edges indicate a player’s best‑response move; the graphs are given intuitive names such as “Cycle”, “Anti‑Cycle”, “Coordination”, “Competition”, etc.

Beyond the 2×2 setting, the authors demonstrate how the same pipeline can be applied to larger normal‑form games and even extensive‑form games. By embedding high‑dimensional payoff tensors into low‑dimensional spaces using the same invariant transforms, and then applying standard dimensionality‑reduction tools (PCA, t‑SNE, UMAP), they produce visual “fingerprints” of strategic interaction for entire game families. Such visualizations can aid in mechanism design, policy analysis, and the interpretation of multi‑agent reinforcement‑learning outcomes.

In summary, the contributions are threefold: (1) definition of equilibrium‑invariant and best‑response‑invariant metrics on the space of normal‑form games; (2) a two‑angle embedding that reduces 2×2 games from eight to two parameters while preserving strategic distances; (3) a rigorous derivation of a fundamental set of 15 game classes, each equipped with a graph‑theoretic representation. The work bridges abstract game‑theoretic concepts with concrete geometric intuition and provides practical tools for analyzing and visualizing strategic interactions in both small and large games.