Farsighted stability in prisoner's dilemma

We study players' behavior in the prisoner's dilemma by using two stability notions: the stable set of von Neumann and Morgenstern with indirect domination and the largest consistent set defined by Chwe. Both notions assume possibility of sequential deviations and farsightedness of players. When players use only pure strategies, these two stability concepts provide us with the same outcome: {(Cooperation, Cooperation), (Defect, Defect)} when players behave independently; and {(Cooperation, Cooperation)} when players' joint, but not binding, moves are also considered. In the prisoner's dilemma with many alternatives such as the mixed extension of the prisoner's dilemma, the two notions produce completely different outcomes. In the stable set, every individually rational outcome could be stable in the former case; and only a Pareto efficient outcome is essentially stable in the latter case. The largest consistent set consists of all individually rational outcomes in either case.