Weighted Voting

Most generally, weighted voting is a decision-making procedure in which different voters cast unequal numbers of votes. Controversially, some analysts have portrayed the varying weights (numbers of votes) as connoting different levels of power for the players (voters). Denote the voters as 1 through n and with weights w1 through wn. Then for a weighted voting system, for at least some pair of voters i and j, wi > wj. Though not strictly required, formal discussions of weighted voting typically assume some quota q and designate as winning coalitions those whose total weight meets or exceeds the quota.

Theorists denote voting weighted voting systems using the notation {q: w1, w2,…,wn}. Weighted voting systems provide the most interesting application of such voting power measures as those due to Lionel Penrose, John Banzhaf, and Lloyd Shapley and Martin Shubik. Interestingly, they all show that the voting power is generally not proportional to voting weights. Consider, for example, the system {7: 5, 4, 3}. Clearly, none of the three players can win the vote alone, yet a coalition of any two players will garner a victory. As another example, consider the scheme {50: 44.5, 7.0, 36.6, 6.4, 5.4} that describes party control of the German Bundestag immediately after the 1998 federal election. The Social Democrats, with 44.5% of the seats, needed the Greens, who had just 7%, to form a coalition government, so in some sense these parties with highly unequal voting weights had equal voting power.

A second general result is that the voting power of players varies with quota size, even while voter weights remain constant; in practice, the direction of the power change can be counterintuitive and it is straightforward to construct examples wherein less powerful players become more powerful as the quota is raised. Broadly speaking, power differentials among the players tend to vary inversely with quota size. In the limit case, weighted voting makes no difference under unanimity rule.

Theorists can use the concept of weighted votes to model much more complicated systems. A voting rule not ordinarily described as a weighted voting system may in fact be equivalent to a weighted majority game. For example, the U.N. Security Council has 15 members: 5 permanent members with veto power and 10 rotating members without veto power. Without loss of generality, we can assign a weight of 1 to each of the 10 rotating members. Then the voting system must be [Page 711]{q: x, x, x, x, x, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}. By the rules of the institution, to pass a motion all five permanent members plus four nonpermanent members must agree. Hence 5x +43q. On the other hand, only 1 permanent member is needed to veto a motion, so 4x+ 10 < q. If we solve 4x+ 10 < q £ 5x +4 for q and x for we obtain {39: 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}. That is, the Security Council arrangement is equivalent to a weighted voting scheme with a quota of 39 and a weight of 7 for the permanent members.

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