A 1979 revision of SUNY-Stony Brook WP No. 184,1977. (A copy of the original working paper is available from the author or from the Economics Department of the State University of New York at Stony Brook.)
This paper formalized the "pregame" framework with a finite number of player types and with side payments. There were numerous examples of pregames in the prior literature, including the Shapley-Shubik glove game. Essentially, for the case of this paper, a pregame is a function assigning a worth to every possible groups of players, where the worth of a group depends on the numbers of players of each type in the group. (Mathematically, if there are T types of players, a pregame is simply a function from the T-fold Cartesian product of the non-negative integers to the real numbers.) The essential condition is exhaustion of gains to scale, that is, all gains to collective activities can be realized by groups of players bounded in size of membership (now also called strict small group effectiveness). A number of theorems which have become well known are demonstrated for sequences of games with a fixed distribution of player types:
(1) The equal treatment property For all sufficiently large games (those that exhaust gains to scale), every payoff in the core treats all players of the same type identically.
(2) Nonemptiness of approximate cores Approximate cores are nonempty for all sufficiently large games.
(3) Approximate core convervence to equal treatment payoffs. Approximate cores converge to equal treatment payoffs and to the core of a continuum limit game where the continuum limit game has finite coalitions but an Aumann-type feasibility condition.
(4) Strict convergence. Under a stronger condition of exhaustion of gains to scale, approximate cores converge to cores of balanced cover games (or to cores of subsequences of games with nonempty cores). Analogues and extensions of most of the results of this paper have now been published, primarily in the 1994 ertens-Sorin volume.