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Symmetric Solutions of Some Production Economies

Sergiu Hart

**Abstract**

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.

*International Journal of Game Theory* 2 (1973), 1, 53-62