We consider effectivity functions for finitely many players and alternatives. We assume that players have incomplete information—with private values—about the preferences of the other players. Our main result is the characterization of effectivity functions which have an ex post Nash consistent representation, i.e., there is a game form such that (i) the distribution of power among coalitions of players is the same as in the effectivity function and (ii) there is an ex post Nash equilibrium (in pure strategies) for any preference profile.
Peters, H. J. M., Schröder, M. J. W., & Vermeulen, A. J. (2015). On existence of ex post Nash consistent representation for effectivity functions. Social Choice and Welfare, 45(2), 287-307. https://doi.org/10.1007/s00355-015-0891-6