A Framework for the Quantitative Evaluation of Voting Rules

In order to characterize the set of desirable social choice functions, researchers have proposed axioms that all social choice functions should satisfy. However, it has been shown that achieving these axioms is impossible. Instead of viewing this impossibility result as a limitation, it can be viewed as an opportunity. In this paper, we develop a means of comparing various social choice functions with regard to a desired axiom by quantifying how often the axiom is violated. To this end, we offer a new framework for measuring the quality of social choice functions that builds from and provides a unifying framework for previous research. This framework takes the form of what we call a “violation graph.” Graph properties have natural interpretations as metrics for comparing social choice functions. Using the violation graph we present new metrics, such as the minimal domain restriction, for assessing social choice functions and provide exact and probabilistic results for voting rules including plurality, Borda, and Copeland. Motivated by the empirical results, we also prove asymptotic results for scoring voting rules. First, these results suggest that voting rules based on pairwise comparison (ex: Copeland) are better than scoring rules (ex: Borda count). Second, these results also suggest that although we can never fulfill our desired set of axioms, the frequency of violation is so small that with even a modest number of voters we can expect to never violate our axioms.