Conflict, No Conflict, Common Interests, and Mixed Interests in 2×2 Games

Abstract game theory originated in the analysis of games of pure conflict. By 1958, however, Schelling set out to “enlarge the scope of game theory, taking the zero-sum game to be the limiting case rather than a point of departure”. The other limiting case, in his view, was the coordination game. Classification according to the degree of conflict quickly became standard procedure. In 1966 Anatol Rapoport and Melvin Guyer published a taxonomy of the 2× 2 games in which the largest category was No-Conflict Games. In 1976 Rapoport, Guyer and David Gordon revised the classification, making No-Conflict Games, Mixed-Motive Games and Games of Complete Opposition the primary categories. No satisfactory classification of the so-called Mixed-Motive Games has emerged. We present a systematic classification based on the topology of the 2×2 games. Topological relationships and symmetry both prove useful in identifying games of pure conflict and games of pure common interest. Our topological treatment reveals a class of games, the Type Games, that has not previously been recognized as distinct. In Type Games the players live in different moral universes. One type is never led into temptation; the other is never free of temptation. One needs no moral instruction; the other must be restrained by law. We also provide an improved terminology for discussing conflict in the 2×2 games, a new subdivision of the games, and a pretty map of the relationships among games based on the degree of conflict.