On swap-distance geometry of voting rules

  • Authors:

  • Svetlana Obraztsova;Edith Elkind;Piotr Faliszewski;Arkadii Slinko

  • Affiliations:

  • National Technical University of Athens, Athens, Greece & Steklov Institute of Mathematics, St. Petersburg, Russia;Nanyang Technological University, Singapore, Singapore;AGH University of Science and Technology, Krakow, Poland;University of Auckland, Auckland, New Zealand

  • Venue:
  • Proceedings of the 2013 international conference on Autonomous agents and multi-agent systems
  • Year:
  • 2013

Quantified Score

Hi-index 0.00

Visualization

Abstract

Axioms that govern our choice of voting rules are usually defined by imposing constraints on the rule's behavior under various transformations of the preference profile. In this paper we adopt a different approach, and view a voting rule as a (multi-)coloring of the election graph - the graph whose vertices are elections over a given set of candidates, and two vertices are adjacent if they can be obtained from each other by swapping adjacent candidates in one of the votes. Given this perspective, a voting rule F is characterized by the shapes of its "monochromatic components", i.e., sets of elections that have the same winner under F. In particular, it would be natural to expect each monochromatic component to be convex, or, at the very least, connected. We formalize the notions of connectivity and (weak) convexity for monochromatic components, and say that a voting rule is connected/(weakly) convex if each of its monochromatic components is connected/(weakly) convex. We then investigate which of the classic voting rules have these properties. It turns out that while all voting rules that we consider are connected, convexity and even weak convexity are much more demanding properties. Our study of connectivity suggests a new notion of monotonicity, which may be of independent interest.