Power Indices Essay ⋆ Political Science Essay Examples ⋆ EssayEmpire

Since power is perhaps the most important concept in political science, it is natural that political scientists have been looking for correlates of power and, in that effort, found it important to measure it. However, the dispositional nature of the intuitive concept of power makes the measurement difficult. The difficulties encountered are akin to those faced within the study of causal relationships. Given this difficulty, some scholars have focused on the relatively clear-cut settings provided by voting bodies with unambiguous rules—decision rules—for determining which groups of voters form a winning majority. The first assumption made in these studies is that only winning groups—coalitions—of voters have power in the sense of influencing the voting outcomes. The second assumption is that these coalitions all have an equal amount of power.

One way to look at a voter’s power in a voting body is to observe how often the voter is a member of a winning coalition. After all, it is the winning coalitions that determine the policies adopted by the body. The more often a voter is present in a winning coalition, the larger influence—it is assumed— the voter has on the policies. Before any coalitions have been formed, it is impossible to tell which kinds of winning coalitions will emerge and, consequently, in how many of them any given voter is a member. In the a priori voting power indices, the concept of swing plays an important role. A voter has a swing in coalition S, for instance, if S is winning when the voter is its member, but no winning when the voter is not a member. The Banzhaf indices equate voting power of a voter with the number of the voter’s swings when all coalitions are considered. The absolute Banzhaf index, also known as the Penrose-Banzhaf index, divides the number of the voter’s swings by 2n-1, while the normalized Banzhaf index uses the sum of all voters’ swings as the divisor.

The Shapley-Shubik index, in turn, focuses on permutations of voters, i.e. ordered sequences of them. The total number of all possible sequences of n voters is given by n! = n(n-1)(n-2)…1. Among these, a voter’s power index value is obtained as the number of such sequences in which the voter has a swing when the winning coalition is formed by adding voters one at the time from the beginning of the sequence. This is the same as giving each swing of a voter in a coalition S with s members the weight (s!)(n-s)!/n! and summing these numbers over all coalitions in which the voter has a swing

The two Banzhaf indices and the Shapley-Shubik index are the best-known indices of a priori voting power, but not the only ones. Another index, the public goods index shares the basic rationale of the Banzhaf indices, but instead of swings in winning coalitions, the number of swings in minimal winning coalitions is counted. Minimal winning coalitions differ from winning ones in that all members in them have a swing.

More recent indices are based on spatial voting games (i.e., they assume voter ideal points in policy space). A voter’s power, according to these indices, is measured by the distance of (game-theoretic) equilibrium outcomes and the voter’s ideal point.

Bibliography:

Banzhaf, John F. “Weighted Voting Doesn’t Work: A Mathematical Analysis.” Rutgers Law Review 19 (1965): 317–343.
Brams, Steven J. Game Theory and Politics. New York: Free Press, 1975.
Felsenthal, Dan S., and Moshe Machover. The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes. Cheltenham, U.K.: Edward Elgar, 1998.
Holler, M. J., ed. Power,Voting and Voting Power. Würzburg, Ger.: Physica Verlag, 1982.
Laruelle, Annick, and Federico Valenciano. “Inequality in Voting Power.” Social Choice and Welfare 22, no. 2 (2004): 413–431.
Napel, Stefan, and Mika Widgrén. “Power Measurement as Sensitivity Analysis: A Unified Approach.” Journal of Theoretical Politics 16, no. 4 (2004): 517–538.
Penrose, Lionel. “The Elementary Statistics of Majority Voting.” Journal of the Royal Statistical Society 109, no. 1 (1946): 53–57.
Roth, Alvin, ed. The Shapley Value. Cambridge: Cambridge University Press, 1988.
Shapley, L. S., and Martin Shubik. “A Method for Evaluating the Distribution of Power in a Committee System.” American Political Science Review 48, no. 3 (1954): 787–792.
Steunenberg, Bernard, Dieter Schmidtchen, and Christian Kobold. “Strategic Power in the European Union.” Journal of Theoretical Politics 11, no. 3 (1999): 339–366.

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