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 : w 1, w 2, w 3, …, w n ]. Suppose voter i, with has weight w i 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.

Equal weight implies equal power ___ ___ ___ w i ___ ___ ___ ___ n voters sum < q sum > q 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 w J = w i. Because w J = w i, 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 and w J = w i we may conclude the following: 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. 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 : w 1, w 2, w 3, …, w n ]. Suppose voter i, with has weight w i with w i > 1 and is critical to k winning coalitions. Note that now k is any integer from 0 to 2 n -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 w c that satisfy the following inequality: q < w c < q + w i – 1.

Equal weight implies equal power Now, consider any other voter j, with Suppose that voter i and j have equal weight, that is, assume that w J = w i. Now consider the number of distinct coalitions of voters with weight w c that satisfy the following inequality: q < w c < q + w J – 1. Because w J = w i then also q + w J – 1 = q + w i – 1. Therefore, the number of distinct coalitions satisfying the inequality q < w c < q + w J – 1 is also equal to k. This is because, if w J = w i, then the inequalities q < w c < q + w i – 1 and q < w c < q + w J – 1 are equivalent.

Equal weight implies equal power We may conclude that any other voter j, with weight w J = w i, 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 : w 1, w 2, w 3, …, w n ]. Suppose the total weight of the system is w = w 1 + w 2 + w 3 + … + w n. Suppose two voters have equal weight. That is, assume w J = w i. Therefore the nominal power of these voters is equal, that is, because w J = w i then it is also true that 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 counter- example 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 Shapley- Shubik 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.