How Should Votes Be Weighted to Reflect the Existing and “Calculated” Distribution of Voting Power of Weighted Voting Organizations Integrating Different Majority Requirements?

Abstract

Voting weight and voting power are not necessarily equal. The former represents the number of votes allocated to each member while the latter represents the ability of a member to influence voting outcomes. In this paper, we observe that, in general, ‘calculated’ voting powers, measured by the normalized Banzhaf index, tend to be linearly linked to voting weight. However, there are key exceptions; larger countries or ‘outliers’ have powers far less or more than proportional to their weight and their powers vary with majority requirements. First, based on a sample of weighted voting organizations [(African Development Bank (AfDB), International Bank for Reconstruction and Development (IBRD), International Fund for Agricultural Development (IFAD) and International Monetary Fund (IMF)], we ask, ourselves, how the votes should be weighted to reflect the existing and ‘calculated’ distribution of voting power, or the potential ‘calculated’ voting powers a larger country could expect with its ‘existing’ voting weight if proportionately between weight and voting power is the one observed for all other smaller countries and is the desired one. In this last case we offer an estimation of the opportunity cost of cooperation in the international organization in terms of loss of power but at the same time an estimation of the minimum implicit gains which cover these costs.

A.1 Appendix: Banzhaf Indices

The pioneering work of Penrose (1946, 1952) on measuring voting power was ignored by mainstream social choice theorists. His/her main idea was so natural and simple: the more powerful a voter is, the more often will the outcome go the way s/he votes. This means that a more powerful voter is more able to influence the outcome and is more often on the winning side of a division (Felsenthal and Machover, 2004, p. 5).

Without knowing of Penrose’s work, Banzhaf (1965), an American jurist, was the first, among many other scholars, who reinvented some of his/her ideas. Banzhaf addressed the problem of measuring the voting power in much the same way as Penrose. However, he/she was not interested in absolute voting power but in relative voting power, in other words, in the ratio of one voter’s power to another’s. The Banzhaf index is represented by ß.

The Banzhaf index considers all coalitions Ti as equiprobable, such that voters are arranged randomly and in no particular order. The Banzhaf index for a member i represents the number of swings for that member divided either by the total number of coalitions of other members measuring in that case the probability of a swing or by the total number of swings for all members measuring thus the member’s relative capacity to swing. The number of swings for a member i is ${\eta}_i={\displaystyle {\sum}_{\tau_i}1}$

The non-normalized Banzhaf index for a member i is the probability of a swing, denoted by ß_i’, which includes the total number of coalitions except i, that is, 2^n-1, as denominator. Actually, each subset of N that does not take into account i represents the voting outcome of the remaining n-1 voters. Consequently, 2^n-1 represents the maximum number of swings for voter i. Hence, the non-normalized Banzhaf index can be represented as follows:

$$ {\beta_i}^{\prime }={\displaystyle {\sum}_{\tau_i}1/{2}^{n-1}=}\ {\eta}_i/{2}^{n-1},\kern1em i=1,2,\dots, n $$

The non-normalized Banzhaf index measures the absolute voting power of each voter and illustrates relative voting powers of different members but without giving a direct interpretation of power distribution (Leech, 2002a, p. 11).

For that reason, the normalized Banzhaf index, ß_i, is used to measure the relative voting power among members. It represents the number of swings for member i as a fraction of the total number of swings for all members. Hence, the Banzhaf index can be written as follows:

$$ {\upbeta}_{\mathrm{i}}={\eta}_i/{\displaystyle \sum \eta } $$

For example, suppose a voting body represented by the vector v = [60: 40, 30, 20] where 60 is the decision rule, in other words, a decision requires 60 votes to pass, and 40 represents voting weight of member state A, 30 is the voting weight of member state B, and 20 is the voting weight of member state C. The winning coalitions, with swing voters (i.e., pivotal voters who change the coalition from a losing to a winning one) in bold:

AB: both member states are decisive voters since the coalition loses if either member state leaves.

AC: both member states are decisive voters since the coalition loses if either member state leaves.

ABC: A is only pivotal voter since the coalition wins even if B leaves or C leaves but not if A leaves.

Thus, the total number of swing votes is 5, where A is decisive three times, B is decisive one time, and C is decisive one time. Therefore, the power of member states measured by the Banzhaf index is divided as follows:

$$ \mathrm{A}=3/5;\kern2em \mathrm{B}=1/5\kern1em \mathrm{and}\kern1em \mathrm{C}=1/5. $$

We should note that the Banzhaf index ß_i is just a normalized version of the Coleman index (as we will see below) and is only used for comparing the voting powers of members under the same decision rule and is not consistent under two different decision rules. In the latter case, the Penrose index should be used (Felsenthal and Machover, 2004, pp. 6–7).