Symmetric solutions of some production economies

A symmetric n-person game (n, k) (for positive integer k) is defined in its characteristic function form by v(S)=[|S|/k], where |S| is the number of players in the coalition S and [x] denotes the largest integer not greater than x, (i.e., any k players, but not less, can "produce" one unit). It is proved that in any imputation in any symmetric von Neumann-Morgenstern solution of such a game, a blocking coalition of p=n-k+1 players who receive the largest payoffs is formed, and their payoffs are always equal. Conditions for existence and uniqueness of such symmetric solutions with the other k-1 payoffs equal too are proved; other cases are discussed thereafter.