Given a set of individuals N, a group is a non-empty subset of N.(A group may

consist of only one person. Also, N itself is a group). Given a PAR, say F, a group G is

a decisive group of F if the following is true for all x, y X and all :

If, for all i G, xPi()y, then xPF ()y.

That is, a group is called a decisive group if, whenever all members of the group prefer x

to y, the society prefers x to y. Loosely speaking, a decisive group has a power to impose

its collective strict preference on the social strict preference. Notice that, depending on

the rule, there may be multiple decisive groups. For example, if N = {1, 2, 3} and F is

the simple majority rule, {1, 2} is a decisive group and {2, 3} is another decisive group.

The set of decisive groups of F, denoted by D(F) is the collection of all decisive groups.

For example, when N = {1, 2, 3}, the set of decisive groups of the simple majority rule

is

D(sm) = {{1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}.

Equivalently, we may write

D(sm) = {G N|#G 2}.

a. Let N = {1, 2, 3, 4, 5} and #X = 4. What is the set of decisive groups of the Borda

rule D(B)?

b. Let N = {1, 2, 3, 4, 5}. Consider the weighted majority rule with the weights w1 = 0.4,

w2 = 0.27, w3 = w4 = w5 = 1.1. What is the set of decisive groups of the weighted

majority rule D(w)?