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Equilibrium solution concepts of normal-form games, such as Nash equilibria,correlated equilibria, and coarse correlated equilibria, describe the jointstrategy 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. Wedefine an equilibrium-inspired distance metric for the space of all normal-formgames and uncover a distance-preserving equilibrium-invariant embedding.Furthermore, we propose an additional transform which defines abetter-response-invariant distance metric and embedding. To demonstrate thesemetric spaces we study $2\times2$ games. The equilibrium-invariant embedding of$2\times2$ games has an efficient two variable parameterization (a reductionfrom eight), where each variable geometrically describes an angle on a unitcircle. 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 gameclasses is fundamental and captures all possible interesting strategicinteractions in $2\times2$ games. We introduce a directed graph representationand name for each class. Finally, we leverage the tools developed for$2\times2$ games to develop game theoretic visualizations of large normal-formand extensive-form games that aim to fingerprint the strategic interactionsthat occur within.

Equilibrium-Invariant Embedding, Metric Space, and Fundamental Set of $2\times2$ Normal-Form Games