minimumwinning

Minimum winning, in political science and cooperative game theory, refers to coalitions that are just sufficient to achieve a win. The term is often used in two related senses: minimal winning coalitions and minimum-size winning coalitions. A clear understanding requires a formal setup: in a weighted voting game, each player i has a weight w_i and there is a quota q. A coalition S is winning if the sum of its members’ weights is at least q. S is a minimal winning coalition if S is winning and removing any member makes it losing. Among all winning coalitions, a minimum-size winning coalition is a winning coalition with the smallest possible number of members.

These concepts help analyze power and bargaining in institutions such as legislatures, committees, or boards. A minimal winning coalition has no slack: every member is essential for maintaining a win. As a result, members of such coalitions often have greater leverage in negotiations, since their presence is pivotal to achieving a majority. The notion of minimum-size coalitions highlights which combinations of members can form a viable government or passing bloc with the fewest participants.

In practice, minimum-winning coalitions are used to study voting power indices, cabinet formation, and legislative bargaining. Power is not determined solely by total voting weight; the structure of minimal coalitions reveals how pivotal different actors are when coalitions form. Analysts may distinguish between the set of all minimal winning coalitions and the subset that are of minimum size, since these can imply different implications for stability and distribution of influence.

See also: coalition game, voting power index, Banzhaf power index, Shapley-Shubik power index, weighted voting system.