## Consistency without neutrality in voting rules |

**William S. Zwicker**

March 6, 2008

10:45 am

Bailey Hall 207

Scoring rules are voting systems for which voters assign points to candidates according to how high they appear on that voter’s ranking of all candidates. The candidate(s) with the greatest point total wins. Smith (1973) and Young (1974, 1975) characterized this important class of voting systems in terms of four axioms; of these, the consistency axiom is arguably most critical. In 1995, Myerson dropped the requirement that a vote consist of a ranking of candidates, extending this characterization to a broader context that includes approval voting. However, Myerson's result still relies on a form of the neutrality axiom, by requiring that the objects to which points are assigned (traditionally, the individual candidates) be treated with complete symmetry by the voting rule. Thus his result excludes yet other voting rules for which the points that add up to the total score are awarded to rankings, to sets of candidates, to final grades in a course, etc. Examples of such generalized scoring rules (henceforth, “GSR”) include the Kemeny rule, certain grading systems, and some systems for electing committees; GSR encompasses many additional “natural” voting rules. Absent neutrality, something else is required to characterize GSR. Our candidate is the Strong Proportional Crossing Point Principle, Strong PCP. We’ll discuss the intuition behind this new axiom, and show that in the absence of neutrality it is strictly stronger than consistency. The idea for PCP rests on an equivalence between GSR and the class of polyhedral voting rules, suggested by Juan Enrique Martinez Legaz.

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