Abstract
Voting power theories measure the ability of voters to influence the outcome of an election under a given voting rule. In general, each theory gives a different evaluation of power, raising the question of their appropriateness, and calling for the need to identify classes of rules for which different theories agree. We study the ordinal equivalence of the generalizations of the classical power concepts–the influence relation, the Banzhaf power index, and the Shapley–Shubik power index–to multi-choice organizations and political rules. Under such rules, each voter chooses a level of support for a social goal from a finite list of options, and these individual choices are aggregated to determine the collective level of support for this goal. We show that the power theories analyzed do not always yield the same power relationships among voters. Thanks to necessary and/or sufficient conditions, we identify a large class of rules for which ordinal equivalence obtains. Furthermore, we prove that ordinal equivalence obtains for all linear rules allowing a fixed number of individual approval levels if and only if that number does not exceed three. Our findings generalize all the previous results on the ordinal equivalence of the classical power theories, and show that the condition of linearity found to be necessary and sufficient for ordinal equivalence to obtain when voters have at most three options to choose from is no longer sufficient when they can choose from a list of four or more options.
Highlights
We study the ordinal equivalency of voting power theories for multi-choice organizations
We study the ordinal equivalence of the generalizations of the Shapley–Shubik and Banzhaf power indices and the influence relation for multi-choice organizations and political rules
Which voting power theory should we use to evaluate the ability of individuals to affect the outcome of a vote in an organization? each theory gives a different evaluation of power in general, our study shows that the generalizations of three classical power concepts – the global influence relation, the Banzhaf power index, and the Shapley–Shubik power index – ordinally coincide for a large class of multi-choice organizations and political rules
Summary
We study the ordinal equivalency of voting power theories for multi-choice organizations. In reaction to the inconsistent evaluation of political power by different theories, scholars have attempted to identify classes of voting rules for which different theories induce the same ordinal structure in the set of voters (see, e.g., [17,18,11,19,16] and the references therein). Despite the popularity of these rules, it has been argued that they are restrictive and do not capture reallife political decision making, as more than two levels of individual and collective approval are generally observed [26,8,9,27,28,29,30,31] This convincing argument motivates our analysis of the ordinal equivalence of power theories for multi-choice organizations and political rules
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