Power To Initiate Action And Power To Prevent Action

[Page 525]These terms, which pertain to the general topic of voting power measures, were introduced by James S. Coleman in 1971. Coleman observed that the most commonly used measure of voting power at the time, the Shapley-Shubik power index, is based on cooperative game theory and assumes that players seek to form a winning coalition whose members divide up some fixed pot of spoils. But, he pointed out, that usually decisions made by committees or other collectivities do not have that character, rather the spoils are indivisible. For example

Because a voter's expected share of the spoils itself measures voting power, Shapley-Shubik power within any group necessarily sums to 1. But, Coleman observed, the power of collectivities to act may vary greatly. A body that uses a unanimity rule to initiate action often fails to act because unanimity is difficult to achieve, while one that uses simple majority rule is in some sense as likely to act as not. While voting power within each body is equal, in a meaningful sense all members of the majority-rule group are more powerful than those in the unanimity-rule group. This kind of example led Coleman to distinguish between two apparently distinct but closely interrelated faces of individual voting power: the power to initiate action and the power to prevent action.

In describing how Coleman formalized these concepts, it is useful to have at hand a specific example of a five-member weighted voting body. By voting yes or no, five voters can partition themselves into 16 complementary subsets (or coalitions), so altogether there are 32 distinct coalitions (including the coalition of all and its empty complement), and each voter belongs to half of them. Suppose that voter A has 3 votes, B has 2 votes, and C, D, and E have 1 vote each and that a quota of 5 votes is required to initiate action (from which it follows that 4 votes is sufficient to prevent action). It can be checked that there are five minimal winning coalitions, each of which has the required 5 votes to initiate action but would fall below the quota if any member left the coalition: {A, B}, {A, C, D}, {A, C, E}, {A, D, E}, and {B, C, D, E}. It can be further checked that eight additional coalitions have more than 5 votes, giving a total of 13 winning coalitions. A voter i is critical to a coalition S if i belongs to S and S is winning but S - {i} is not (or, equivalently, if i does not belong to S and S is not winning but S - {i} is). Every member of a minimum winning coalition is critical to it, but some or all members of more inclusive coalitions fail to be critical. It can be checked that A is critical for 11 winning coalitions, B for 5, and C, D, and E for 3 each.

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