Paradox Of Redistribution Research Articles

Note on a Paradox/Anmerkungen zu einem Paradoxon

In recently published articles by Dreyer and Schotter (1980), Fischer and Schotter (1978), and Schotter (1979) a surprising phenomenon was found within the analysis of voting power. This phenomenon was titled the of Redistribution. Paradox exists when, given a vote distribution w = (wi,..., wn) and a voting rule d, we can construct a new vote distribution w' = (w{,..., w„) which gives at least one voter a smaller proportion of the vote yet gives him more power, when power is measured by either the Banzhaf or the Shapley-Shubik power (Fischer and Schotter, 1978, p. 52). In this note I shall explain why this phenomenon be neither surprising nor paradoxical. I shall hereby refer to Fischer and Schotter's (1978) paper The inevitability of the paradox of redistribution in allocation of voting weights, for this is the most formai presentation of the Paradox. Moreover, I shall focus the discussion on the Banzhaf power index Fischer and Schotter's analysis shows that the pro positions deduced therefrom also hold in essence for the Shapley-Shubik power index. inevitability of the Paradox, according to Fischer and Schotter (F & S), depends on the number of voters (η) and the decision rule (d) that is applied to the vote distribution (w).

Power Relationships in the International Monetary Fund: The Consequences of Quota Changes

THERE is a widespread agreement that the International Monetary (IMF) has played a central role in fostering a dynamic and open international economy in the postwar period. As the world economy evolved, the consensus has also emerged, however, that substantial changes were required in the Fund's rules so as to make them better reflect the changing power relationships among its members. In an attempt to make these changes, the Proposed Second Amendment to the Articles of Agreement of the International Monetary Fund (IMF, 1976) was offered to the IMF's members for ratification along with a change in the distribution of the members' quotas. The new rules became effective as of April 1, 1978, when the amendment was ratified by the required 60% of members with 80% of the total vote. It is the purpose of this paper to demonstrate, using some simple game theoretical tools, that in many cases the framers of these amendments and quota changes have achieved a result that may be exactly the opposite of their intentions. More specifically, we will show that 1) Under the proposed changes 38 countries in the IMF have their percentage of the total vote decreased and yet their voting power within the organization increased when power is measured by the Banzhaf power index. 2) With the new vote weight distribution and voting rules, four major countries-Belgium, Holland, West Germany and Japan have their percentage of the total vote increased and yet have their percentage of the total power decreased. Paradoxical results of this variety were discussed theoretically by Fischer and Schotter (1978) and labeled the Paradox of Redistribution.' 1 3) Under the previous distribution of votes and voting rules, smaller countries had voting powers that were out of proportion to their voting weights, and the newly introduced changes would generally aggravate this disproportion. 4) The power of the United States within the increases substantially on issues where most countries vote through their Executive Directors (as groups), as opposed to issues where they vote through their Governors (individually). 5) Under both the previous and current voting system, significant diminishing returns to voting weights exist and the tendency is more pronounced under the new system. In most voting situations power is a concave function of voting weight with large linear segments. 6) Although these results are in many instances not quantitatively substantial, qualitatively they indicate a noticeable discrepancy between what one would think the consequences of the voting changes would be and what they actually are. In this paper we will proceed as follows: Section I will present some background material about the voting procedures at the IMF. Section II will discuss the index that we are using to measure power. Section III will present the results of our calculations and discuss their significance. Finally, section IV will offer some conclusions and consider the relevance of our findings.

The inevitability of the ?paradox of redistribution? in the allocation of voting weights

In voting bodies, when voting weights are reallocated, it may be observed that the voting power of some members, as measured by the Shapley-Shubik and Banzhaf power indices, increases while their voting weight decreases. By a simple constructive proof, this paper shows that such a “paradox of redistribution” can always occur in any voting body if the number of voters, n, is sufficiently large. Simulation results show that the paradox is quite frequent (up to 30 percent) and increases with n (at least for small n). Examples are given where the Banzhaf and Shapley-Shubik indices are not consistent in demonstrating the paradox.