Public Goods Index

The Public Goods Index (PGI) measures the power distribution in committees, parliaments, and hierarchical organizations. The index was introduced by Manfred Holler and axiomatized by Holler and Edward W. Packel. In contrast to most other power indices, the PGI explicitly considers political outcomes to be public goods. It measures the potential of an individual member i of a political body N to contribute to the selection of a public good x. As such it builds on Max Weber's definition of power as chance that an “actor within a social relationship will be in a position to carry out his own will despite resistance”: that is, power does not depend on what an actor wants to do but what he or she can do.

The public good perspective implies that (a) every voter will “consume” the same political outcome (i.e., the same level of provision of the public good x), and (b) only decisive sets “matter.” Property (a) does not require that all voters like the chosen political outcome x equally but that each has to consume x and cannot be excluded from its consumption. Nonrivalry in consumption applies, so everyone consumes x irrespective of other members of the society.

[Page 542]In order to clarify property (b) we define S, T, R, and K as subsets of N and call them coalitions. The elements of the decisive sets M(v) are defined such that

Here v(S) = 1 and v(K) = 0 imply that S is a winning coalition and K is a losing one, respectively, thereby defining the simple game (N, v).

M(v) is identical with the set of minimum winning coalitions of (N, v). Each element in M(v) represents a different political outcome. If S is M(v) and S is a strict subset of T, then it is assumed that the political outcome of T is identical to the political outcome of S. Consequently, T can be neglected when it comes to measuring power, because members in T but not in S do not contribute to the determination of x. This argument also applies if both S and R are elements of M(v) and proper subsets of T. Again, T does not count for measuring power. Note that this does not mean that T never forms but, if it forms, it does so on the basis of “luck” (or ideology), not power.

Let ci(v) be the number of coalitions S such that S is a decisive set and i is an element of S. Then the value of the PGI for player i, hi, is given by

Applying this formula to voting game v° = (51; 35, 20, 15, 15, 15), where the decision rule d° = 51 and the vote distribution w° = (35, 20, 15, 15, 15) gives the power distribution h(v°) = (4/15, 2/15, 3/15, 3/15, 3/15). It is apparent that the PGI is not locally monotonic (LM) in voting weights: this example illustrates the “paradox of weighted voting.”

The fact that the PGI does not satisfy LM has been viewed by some as a defect that should lead to rejection of the measure. But a closer analysis shows that game v° induces the violation of LM because it does not guarantee a winning coalition. In the case of five players (or less) PGI does not violate LM if the voting rule is decisive and proper such that a single winning coalition result is assured. Holler and Napel argue that it is a defect in decision rules or, more general, the design of the voting body, not in PGI itself that results in a violation of LM, and it is actually an attraction of the PGI that, contrary to the more popular power measures due to Shapley-Shubik and Banzhaf, it highlights such a peculiarity, perhaps even defect, of the voting body.

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