Maximin, minimax, and von Neumann–Morgenstern farsighted stable sets

•vNM farsighted stable sets are analyzed in strategic form games.•Pareto efficient and strictly individually rational outcomes are supported.•Pareto inefficient and strictly enforceable outcomes cannot be included.•The above results hold only for two-person games with coalitional deviations.

In this note, we investigate the relationship between the classical concepts of maximin and minimax, which were originally defined in the context of zero-sum games in von Neumann and Morgenstern (1953), and the von Neumann–Morgenstern (vNM) farsighted stable set using the indirect domination defined in Chwe (1994). We show two main results for two-player games: an existence result and an almost-uniqueness result. Under a mild assumption, we show that any strategy profile that is Pareto efficient and strictly individually rational–that is, strictly above each player’s maximin value–is generically a singleton vNM farsighted stable set. Moreover, there does not exist a vNM farsighted stable set that includes a strategy profile that is strictly individually rational and yields a payoff greater than the minimax value for a player, but not Pareto efficient.