A Research Note on the Size of Winning Coalitions

This paper is concerned with the problem of relating aggregate coalition payoffs to the winnings of an individual player, so that some theoretical foundation might be developed for dealing with problems of coalition formation. Professor William H. Riker was concerned with this problem in his Theory of Political Coalitions; in that book he developed a formulation holding that in several common political situations players would strive to form only minimum winning coalitions. Riker based that formulation on his derivation in game theory of “the size principle,†which held that in zero-sum games among rational players with perfect information, only minimum winning coalitions would occur. The first part of this research note shows that there is a difficulty in Riker's derivation of the “size principle,†presents counter-examples to that principle, and shows that it is unsound in general. The second part of this note develops the maximum number of positive gainers principle, which shows that in the kind of games being examined there is a maximum upper limit to the number of players who will positively gain; but this does not insure that the winning coalition will be minimal.