Coalitional strategic games

Abstract

In pursuit of games played by groups of individuals (each group itself being a player), we develop a theory of strategic games in which each player is rational in the sense of expected utility theory, except that her preferences may fail to be transitive. Two natural solution concepts are defined, Nash equilibrium and the equilibrium in beliefs, depending on the interpretation of mixed strategies. We provide sufficient conditions for the existence of both equilibrium concepts. For instance, it turns out that an equilibrium is sure to exist if each player possesses two pure strategies (and may have cyclic preferences across pure and mixed strategy profiles), without any further qualifications. To go beyond equilibrium existence, we use the coalitional expected utility representation by Hara et al. (2019) and characterize the set of Nash equilibria in terms of this representation. We also study rationalizability in such games (without transitivity), as well as some equilibrium refinements, and compare our findings with those of standard game theory. Our investigation is meant to be a step toward understanding the nature of strategic interaction across groups of individuals and clarifying the role of transitivity in game theory.