Absorbing outcomes in 2 × 2 games

Abstract

This paper addresses the question of the stability, or absorbing properties, of conceptual systems involving two significant players. “Absorbing outcomes” are defined in the 41 2 × 2 ordinal games without nonmyopic equilibria (i.e., long-term stable outcomes) and classified into four mutually exclusive categories. These categories are based on the movements rational players—who may be individuals, groups, or higher-level living systems—would make from worst, next-worst, and next-best outcomes in these games, allowing for the deletability of some movements. Absorbing outcomes, at some level, are shown to exist in all 41 games, but not all are equally defensible as stable outcomes. Conditions for determining them are generally applicable to identifying nonmyopic equilibria in the 37 2 × 2 games that contain them, though they would not show up the cooperative outcomes in the games of prisoners' dilemma and chicken. Absorbing outcomes do not necessarily coincide with the concept of a Nash equilibrium, or a myopically stable outcome, which also has an interpretation based on the movements of rational players in a 2 × 2 game. Nevertheless, in games without nonmyopic equilibria, absorbing outcomes do seem to capture the idea of long-term stability better than Nash equilibria when alternating, sequential moves by players are allowed.