k-ApprovalVeto: A Spectrum of Voting Rules Balancing Metric Distortion and Minority Protection

In the context of single-winner ranked-choice elections between 𝑚 candidates, we explore the tradeoff between two principles that are essential to constitutional democracies: 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 𝑘-Droop minority criterion, which requires that if a sufficiently large (parametrized by 𝑘) coalition 𝑇 of voters ranks all candidates in 𝑆 at the bottom (in any order), then none of the candidates in