The phase space of symmetric 2x2 games

<p>Symmetric 2x2 games are a cornerstone of game theory and its applications in biology, economics, and social sciences, yet no unified visual overview of their full parameter space exists in the literature. We present a minimal but exhaustive phase space of all symmetric 2x2 games in a two-dimensional payoff plane, leveraging symmetries and using parametrizations consistent with the cooperation and Prisoner's Dilemma literature. Within this phase space, we superimpose the boundaries of all standard strategic properties (Nash equilibria, Pareto efficiency, maximin strategies, and social optimum) across all 12 ordinal game classes but also considering cardinal payoffs. We highlight how the strategic challenges of games in each region originate from their unique combination of these standard properties. This unified representation reveals the continuous relationships between game archetypes: the phase space indeed situates well-known games (Prisoner's Dilemma, Stag Hunt, Hawk-Dove, (anti-)coordination) alongside understudied (Second Best, Harmony) regions. It thereby provides an immediate reading of any symmetric 2x2 game’s strategic properties, and makes explicit how they shift as payoffs vary within and across ordinal classes. We then review each ordinal class in detail, covering its history, defining characteristics, and within-class variations. We also exhaustively review the 26 symmetric games with payoff ties, name them, and place them on the same phase space. This work, and in particular the phase space visual, is intended as both a reference tool for experienced researchers and a pedagogical entry point.</p>