The Power of Voting Blocs: An Example

How does one measure power in a voting situation? A mathematical power index proposed by Lloyd Shapley and Martin Shubik in [3] has found wide favor among mathematicians and social scientists. In this note, I wish to use this index and some elementary game theory to analyze a particular voting situation, illustrative of a class of voting problems. The Shapley-Shubik power index is calculated as follows. Assume that voters one by one join a coalition in support of a proposal. The voter who, upon joining the coalition, changes the coalition from a losing coalition into a winning coalition is called pivotal. A voter's Shapley-Shubik power index is simply his probability of being pivotal, assuming in the absence of other information that all orders of coalition formation are equally likely. The Shapley-Shubik index is thus intuitively pleasing, and easy to calculate at least in simple cases. It is also supported by an elegant axiomatic characterization (see [1]). The problem to which I wish to apply the index concerns the distribution of power in the county government of Rock County, Wisconsin. Rock County, in southern Wisconsin, is dominated by the two cities of Janesville (population 46,000) and Beloit (population 36,000). There is often considerable rivalry between the two cities. The remainder of the county consists of small towns and rural areas. The county is governed by a Board of County Supervisors consisting of 40 members, elected from districts roughly equal in population. 11 supervisors are from the city of Beloit, 14 are from Janesville, 15 from town and rural areas. Historically, the supervisors from each city have been quite independent of one another in philosophy and voting behavior. Bloc voting by the supervisors from Beloit, or from Janesville, has not developed. I do not know about the situation in Janesville, but in Beloit there has been considerable unhappiness among city officials over this lack of cohesiveness. Surely Beloit would wield more influence if its delegation would agree-to vote as a bloc. We can analyze this situation by using the Shapley-Shubik power index to calculate the total power of Beloit's eleven supervisors if they vote independently, and if they organize as a bloc. In the first case, each supervisor will have 1/40 of the total power, and Beloit's eleven together will have 11/40 or 27M%. In the second case, there are effectively 30 voters: 29 independent supervisors, and the Beloit bloc casting 11 votes. Since 21 votes are needed to pass a measure, Beloit will pivot if it joins a coalition 11th through 21st, i.e., 11/30 of the time. Beloit will have 11/30 or 362% of the power, a considerable increase. Janesville's supervisors, who had 14/40 = 35% of the power before Beloit organized, would have 14/29 19/30 = 30'% of the power after Beloit organizes. The problem with this scenario is, of course, that if Beloit's supervisors organized as a bloc, there would be considerable pressure for Janesville's supervisors to organize also. If that happened, we would have a game of 17 voters: 15 casting a single vote, Beloit casting 11, and Janesville casting 14. We can compute the Shapley-Shubik power indices for this game by counting the lattice points in the regions labeled B, J and 0 in FIGURE 1. A point (x, y) in this figure represents a voting coalition whose xth member in order of joining is Beloit and whose yth member is Janesville. The regions B, J, 0 represent points where Beloit, Janesville or others, repectively, are pivotal. This figure shows that