# Distance rationalization of voting rules

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## Abstract

The concept of *distance rationalizability* allows one to define new voting rules or rationalize existing ones via a consensus, i.e., a class of elections that have a unique, indisputable winner, and a distance over elections: A candidate is declared an election winner if she is the consensus candidate in one of the nearest consensus elections. Many classic voting rules are defined or can be represented in this way. In this paper, we focus on the power and the limitations of the distance rationalizability approach. Lerer and Nitzan (J Econ Theory 37(1):191–201, 1985) and Campbell and Nitzan (Soc Choice Welf 3(1):1–16, 1986) show that if we do not place any restrictions on the notions of distance and consensus then essentially all voting rules can be distance-rationalized. We identify a natural class of distances on elections—votewise distances—which depend on the submitted votes in a simple and transparent manner, and investigate which voting rules can be rationalized via distances of this type. We also study axiomatic properties of rules that can be defined via votewise distances.

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