Theorem: Equal weight implies equal power but not the converse. Equal weight implies equal power • Let’s consider the Shapley-Shubik power index first. • Suppose we have a weighted voting system for n voters given by [ q : w1, w2, w3, …, wn ] . • Suppose voter i, with i {1,2,3,..., n} has weight wi and is pivotal k times. Note that k is any number from 0 to n!. • By definition, k is the number of permutations of all voters in which the sum of the weights of voters preceding voter i is less than q and the sum of the weights of the voters preceding voter i plus the weight of voter i is greater than or equal to q. • By definition, the Shapley-Shubik power index for voter i is k n! . Equal weight implies equal power n voters ___ ___ ___ sum < q sum > q wi ___ ___ ___ ___ If voter i has power k/n! then there are k different permutations of this type. Now consider any other voter j with the same weight as voter i. That is, assume wJ = w i. Because wJ = wi, the number of permutations in which voter j is pivotal is also equal to k. Equal weight implies equal power • In other words, given any voter j, with we may conclude the following: j {1,2,3,..., n} and wJ = wi • k is also the number of permutations of all voters in which the sum of the weights of voters preceding voter j is less than q and the sum of the weights of the voters preceding voter j plus the weight of voter j is greater than or equal to q. • Thus, we may conclude the Shapley-Shubik power index for voter j is also given by k . n! • Finally, we may conclude that any voters with equal weight will have equal power as measured by the Shapley-Shubik index. Equal weight implies equal power • Now, let’s consider the Banzhaf power index. As before, suppose we have a weighted voting system for n voters given by [ q : w1, w2, w3, …, wn ] . • Suppose voter i, with i {1,2,3,..., n} has weight wi with wi > 1 and is critical to k winning coalitions. Note that now k is any integer from 0 to 2n-1. • Remember that to compute the Banzhaf power for voter i we consider only combinations of voters and not permutations. • Because voter i is critical to k winning coalitions, there are k distinct coalitions of voters with weight wc that satisfy the following inequality: q < wc < q + wi – 1. Equal weight implies equal power • Now, consider any other voter j, with j {1,2,3,..., n}. • Suppose that voter i and j have equal weight, that is, assume that w J = wi. • Now consider the number of distinct coalitions of voters with weight wc that satisfy the following inequality: q < wc < q + wJ – 1. • Because wJ = wi then also q + wJ – 1 = q + wi – 1. • Therefore, the number of distinct coalitions satisfying the inequality q < wc < q + wJ – 1 is also equal to k. • This is because, if wJ = wi , then the inequalities q < wc < q + wi – 1 and q < wc < q + wJ – 1 are equivalent. Equal weight implies equal power • We may conclude that any other voter j , with weight wJ = wi, is also critical to k distinct coalitions. • And finally, by definition, both voters i and j have Banzhaf power 2k. • Thus any voters with equal weight will have equal power as measured by the Banzhaf analysis of power. Equal weight implies equal power • Clearly, any two voters with equal weight will have equal nominal power. • Suppose we have a weighted voting system for n voters given by [ q : w1, w2, w3, …, wn ] . Suppose the total weight of the system is w = w1 + w2 + w3 + … + wn. • Suppose two voters have equal weight. That is, assume wJ = wi. • Therefore the nominal power of these voters is equal, that is, because wJ = wi then it is also true that wi w j . w w • Conclusion: Voters with equal weight have equal nominal power. The Converse is Not True • For the Shapley-Shubik, Banzhaf and nominal measures of power for a weighted voting system, we have shown that the following statement is true: • If voters have equal weight, then they have equal power. • The converse of this statement is: If voters have equal power then they have equal weight. • The converse is not true. • To prove a statement is not true, we need only provide a counterexample to the statement. Equal Power Does Not Imply Equal Weight • Consider the statement: If voters have equal power, then they have equal weight. • This can be shown to be false by the following counter-example: • Consider the weighted voting system [ 100 : 60, 39, 1 ]. • The Banzhaf index for this system is ( 2, 2, 2 ) and the ShapleyShubik index is ( 1/3, 1/3, 1/3 ). • Thus, we have an example where power is equal among voters but none have equal weight. • In summary, if voters have equal weight then they have equal power but if they have equal power, they might not have equal weight.