In the context of single-winner ranked-choice elections between m candidates, we explore the tradeoff between two competing goals in every democratic system: the majority principle (maximizing the social welfare) and the minority principle (safeguarding minority groups from overly bad outcomes). To measure the social welfare, we use the well-established framework of metric distortion subject to various objectives: utilitarian (i. e. , total cost), α-percentile (e. g. , median cost for α= 1/2), and egalitarian (i. e. , max cost). To measure the protection of minorities, we introduce the -mutual minority criterion, which requires that if a sufficiently large (parametrized by) coalition T of voters ranks all candidates in S lower than all other candidates, then none of the candidates in S should win. The highest for which the criterion is satisfied provides a well-defined measure of mutual minority protection (ranging from 1 to m). Our main contribution is the analysis of a recently proposed class of voting rules called k-Approval Veto, offering a comprehensive range of trade-offs between the two principles. This class spans between Plurality Veto (for k=1) - a simple voting rule achieving optimal metric distortion - and Vote By Veto (for k=m) which picks a candidate from the proportional veto core. We show that k-Approval Veto has minority protection at least k, and thus, it accommodates any desired level of minority protection. However, this comes at the price of lower social welfare. For the utilitarian objective, the metric distortion increases linearly in k. For the α-percentile objective, the metric distortion is the optimal value of 5 for α k/ (k+1) and unbounded for α< k/ (k+1). For the egalitarian objective, the metric distortion is the optimal value of 3 for all values of k.
Kızılkaya et al. (Wed,) studied this question.