Comparing influence theories in voting games under locally generated measures of dissatisfaction

Abstract

The Isbell desirability relation (I), the Shapley-Shubik index (SS) and the Banzhaf-Coleman index (BC) are power theories that grasp the notion of individual influence in a yes-no voting rule. Also, a yes-no voting rule is often used as a tool for aggregating individual preferences over any given finite set of alternatives into a collective preference. In this second context, Diffo Lambo and Moulen (DM) have introduced a power relation which ranks the voters with respect to how ably they influence the collective preference. However, DM relies on the metric d that measures closeness between preference relations. Our concern in this work is: do I, SS, BC and DM agree when the same yes-no voting rule is the basis for collective decision making? We provide a concrete and intuitive class of metrics called locally generated (LG). We give a characterization of the LG metrics d for which I, SS, BC and DM agree on ranking the voters.

Round-robin political tournaments: Abstention, truthful equilibria, and effective power

“…Given N and A with |N | ≥ 3 and |A| ≥ 2, a necessary and sufficient condition under which for all voting rule W , all preference profile R, the truthful vote profile s * (R) is a Nash equilibrium for the associated strategic form game Γ R = (N, Z, U ) is that α = λ. 13 Note that all our findings go through if we replace 1…”

“…This class of metrics, which generalizes the well-known Kemeny distance, was introduced in Diffo Lambo et al (2012). 12 We recall below the definition of a locally generated metric in Diffo Lambo et al (2012).…”

“…Diffo Lambo et al (2012) show that the mapping d α,β,λ is a metric if and only if the following property Δ α,β,λ is satisfied:…”

“…) can be viewed as the utility that he derives from that ranking. 13 It logically follows, by summing up utilities over all the pairs of candidates, that…”

“…Laruelle and Valenciano (2005) propose a model for measuring a voter's success and decisiveness in yes-no voting games and assess different power indices on the basis of this model (see also Laruelle et al (2006)). Closer to our work is Diffo Lambo et al (2012) who introduce the notion of locally generated metrics which they use to study the ordinal equivalence between the effective power relation and the classical notions of voting power. They however restrict their analysis to yes-no voting games and they do not study equilibrium existence.…”