Voting Weights as Power Proxies: Some Theoretical and Empirical Results

Abstract

It is widely known that the voting weights of members in a voting body are not good proxies for their influence or power within the body. The question that arises, however, is how bad a proxy are they?

In other words, if we were to increase (or decrease) a voter’s voting weight within a voting body, would his resulting power within the organization always increase (or decrease), but possible not in proportion to the increase in his weight, or could his influence actually decrease (or increase)?

This note presents the results of two previous studies of this question done by Fischer/Schotter [1978] and Dreyer/Schotter [1978]. This results are clear. Voting weights can be extremely poor proxies for voting power. For instance, it is proven that no matter what voting weight distribution we are observing, it is always possible to find another distribution in which at least one voter has his percentage of the votes decreased and yet his percentage of the voting power increased when power is measured by either the Banzhaf or Shapley-Shubik power index. This type of result is proven to be inevitable for n ≥ 6 in the Banzhaf case and n ≥ 7 in the Shapley-Shubik case.

Finally, this result is illustrated by observing the power relationships at the International Monetary Fund after a recent redistribution in the votes of the organization. There it is shown that 38 countries had their percentage of the vote decreased and yet had their percentage of the organization’s power increased while four major countries were allocated a larger percentage of the vote and yet had their power within the organization decreased.

The author would like to thank Dietrich Fischer and Jacob Dreyer for their previous collaboration on the idea presented here. In addition, the support of the Office of Naval Research, Contract Number NO0014-78-C-0598 is gratefully acknowledged.