Symmetric solutions of some production economies

Abstract

A symmetricn-person game (n, k) (for positive integerk) is defined in its characteristic function form byv(S)=[¦S¦/k], where ¦S¦ is the number of players in the coalitionS and [x] denotes the largest integer not greater thanx, (i.e., anyk 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 ofp=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 otherk−1 payoffs equal too are proved; other cases are discussed thereafter.