Paradox Of New Members

The paradox of new members, discovered and named by Steven J. Brams, pertains to weighted voting games and measures of a priori voting power defined for these games. Consider a voting body consisting of n existing members, each endowed with a fixed voting weight or number of votes. Suppose now that the body is enlarged by admitting new members with their own fixed voting weights or votes. The decision rule—simple majority, two-thirds majority, or whatever fixed proportion of the total voting weight—remains the same before and after the enlargement. We now compute the a priori voting power index values of (1) the existing members of the original voting body and (2) of all members of the enlarged body. The paradox of new members occurs if it turns out that some existing member of the voting body has a larger power index value in the enlarged body than in the original one. The paradox boils down to the possibility that a decrease in a voter's share of votes is accompanied with an increase in the voter's influence over the voting outcomes, while the decision rule is kept constant vis-à-vis the relative voting weights.

The following example illustrates the paradox. Let the voting body consist of three members A, B, and C with votes 3, 2, and 2, respectively, and let the decision rule be the simple majority, that is, 4 out of 7 votes. Then the standardized Banzhaf index value of each member is 1/3 because every member is critically present in two winning coalitions out of six: that is, the removal of the member would make the winning coalition nonwinning. The same holds for the Shapley—Shubik Index as each member is pivotal in two out of six permutations. (Indeed, this voting body is effectively a simple majority unweighted voting system.) But now suppose that the body is enlarged by one member, D, with one vote. The simple majority now becomes 5 out of 8. There are six winning coalitions where at least one member is critical. A has 5, B has 3, C has 3, and D has 1 swing in those coalitions. Hence, the standardized Banzhaf index values are, respectively, 5/12, 3/12, 3/12, and 1/12. Thus A's Banzhaf index value has increased with the entrance of D, and the Shapley—Shubik Index gives exactly the same values in this example. Perhaps more striking is that A's absolute Banzhaf voting power, that is, A's probability of casting a decisive vote, also increases from .5 to .625 with the enlargement of the voting body, despite A's reduced share of the total votes. The essential point is that the enlargement means that A's extra vote relative to B and C now gives A possibilities that B and C do not have to gain power by forming coalitions.

A particularly interesting special case of the paradox occurs when a dummy player is empowered, that is, when a member with no swings in a voting body becomes a critical member in at least one winning coalition of the enlarged body. Brams and Paul Affuso give empirical examples from the history of the European Union. From 1958 until 1973 the Union (then the European Community) consisted of six members: France, Germany, Italy, Belgium, the Netherlands, and Luxembourg, with [Page 472]voting weights 4, 4, 4, 2, 2, and 1, respectively, in the Council of Ministers. With the decision rule of 12 out of 17, Luxembourg was not critical in any winning coalition. Hence, Luxembourg was a dummy, with a Banzhaf value of 0. When Denmark, Ireland, and the United Kingdom joined the Community in 1973, with 3, 3, and 10 votes respectively, the voting weights of the existing members were modified so that the 4-vote members got 10, the 2-vote members 5, and the 1-vote member 2 votes, with a new decision rule of 41 out of 58 (which, in relative terms was essentially equal to the previous 12/17 rule). Thus the vote shares of all six existing members, including Luxembourg, diminished. Yet, the power index value of Luxembourg increased from 0 to .016 in terms of the standardized Banzhaf index and from 0 to .010 when measured by the Shapley—Shubik Index. The dummy member Luxembourg was thus empowered.

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