Classifying Limited-Move Stability Cycles in 2 × 2 Games

The 2×2 game is the simplest non-trivial model of strategic interaction: there are two players, each has two strategies, and each has a strict preference ranking over the four possible outcomes. For models of play that depend only on the ranking of the outcomes, the catalog of 2×2 games permits many useful comparisons and contrasts. By interpreting a 2×2 game as a graph model, we obtain new data on the properties of limited-move (Lh) stability. Specifically, for each 2×2 strict ordinal game, we determine the Lh-stable outcomes; show how stability depends on the horizon, h; and find the lengths of cycles and the numbers of moves until cycling begins. We then compare our observations with other classifications of these games and with the values of the conflict and harmony indices.