Weighted Majority Game

A weighted majority game is a simple game in which each player can be assigned some numerical weight, and a numerical quota can be fixed, such that a coalition is winning if and only if the total weight of its members equals or exceeds the quota. The concept is useful for analyzing how voting power is shared among members of an assembly on the basis of possible coalitions that may be formed between them. These members (or players) could be shareholders, representatives of political parties or countries, and so on, each one assigned a certain weight (in terms of shares, votes, seats, etc.). A fundamental finding is that voting power in general is not proportional to voting weight. The formal concept of a weighted majority game goes back to the key book on game theory by John von Neumann and Oskar Morgenstern in 1944.

Let N ={1, 2,…, n} be the set of members of an assembly. Each member i is assigned a non negative weight wi. Let q be the “majority quota” that defines a winning coalition—that is, any coalition S with total weight of q or greater is authorized to take action, for example, make a collective decision, enact a law, and so on. A weighted majority game is defined by these two elements (though sometimes additional information may be taken into consideration, such as a priori unions and different probabilities of coalition formation). In its basic form, a weighted majority game is defined by the formula [q, w] = [q; w1,…, wn]. Let us call the total weight of coalition S wS. The quota q cannot exceed the total weight wN, because in this case no decision can be made, and normally it cannot equal or fall below wN/2. (If q £wN/2, the game is said to be improper.) If q = wN, we have a unanimity game, inasmuch as any decision needs the approval of all assembly members. If a simple majority (i.e., anything greater than wN/2) is sufficient, we have a simple majority game.