Extract

In Ex. A.1 we illustrate the calculation of the Bz measure and Bz index, using a fairly simple but non-trivial WVG with four voters. In Ex. A.2 we illustrate the calculation of the Shapley value, using a game with four players. In Ex. A.3 we return to the WVG of Ex. A.1 and calculate the S-S index for it. In Ex. A.4 we calculate the D-P and Js indices for the same WVG. A.1 Example (Bz Measure and Index) Let W be a WVG with assembly N = {a, b, c, d} that is isomorphic, in alphabetic order, to [5; 3, 2, 1, 1]. To compute the Bz power of each of the voters, you must first list the coalitions of W and their critical members. Fortunately, you do not have to list all 16 coalitions: a coalition need be listed only if it has at least one critical member. Such a coalition is said to be vulnerable. Here are the five vulnerable coalitions of W, listed in order of size, with their critical members underlined: {a, b}, {a, b, c}, {a, b, d}, {a, c, d}, {a, b, c, d}. By Def. 3.2.2, the Bz power, β x , of voter x is equal to the number of [vulnerable] coalitions in which x is critical, divided by 2n−1 . Here n = 4, so in our WVG we have: βa = 5, βb = 3, βc = βd = 1. 8 8 8 Note that you need not repeat the calculation separately for c and d: these...