A unified continuous framework for two-player two-strategy games: Game-theoretic analysis of interactions

Games with two players and two strategies are basic models used to investigate strategic interactions in the real world. These fundamental games, such as the prisoner's dilemma, have shaped our understanding of cooperation and conflict, which are universal across both social and biological systems. However, players in these games are limited to binary choices, a simplification that fails to capture the continuous nature of real-world interactions. Moreover, social and biological interactions lack a mathematical description, and 2 $\times$ 2 games appear to be independent of one another. In this paper, we propose a unified continuous formulation for 2 $\times$ 2 games, which we term Linear Dualism Games (LDG). The strategy space of the LDG is divided into the cooperation domain and the defection domain. Supported by LDG, we present mathematical descriptions for three main types of interactions: competition, mutualism, and exploitation, and establish connections among 2 $\times$ 2 games. Interactions can vary across the different domains of the LDG. For example, in the prisoner's dilemma, the interaction is competition in both the cooperation and defection domains, while in the stag hunt game, the interaction is mutualism in the cooperation domain and competition in the defection domain. Furthermore, the prisoner's dilemma can be changed into the hawk-dove game by altering the interaction type in the defection domain from competition to mutualism. We also identify new 2 $\times$ 2 games that have not been discussed before, such as the battle of the allies, the battle of the married couple, and the converted gang game.