Social acceptability of scoring rules under asymmetric thresholds

When voters rank all alternatives from best to worst, an alternative is said to be socially acceptable if the number of voters who rank it among their most preferred half of the alternatives is at least as large as the number of voters who rank it among their least preferred half of the alternatives (Mahajne and Volij, 2018, Social Choice and Welfare 51:223-233). It has also been established that there exists a unique scoring rule that always selects a subset of socially acceptable alternatives. This notion of social acceptability has since been extensively studied in the social choice literature. We study a generalized version of this principle based on asymmetric thresholds: for integers p and q such that 1 ≤p,q ≤⌊m 2 ⌋, an alternative is socially p/q-acceptable whenever the number of voters who rank it among their top p positions is at least as large as the number of voters who rank it among their bottom q positions. We provide a complete characterization of the scoring rules that always select socially p/q-acceptable alternatives. We show that such a rule exists if and only if p ≥q, and that this rule is unique when p= q. For any pair (p,q), we further give a complete characterization of every admissible scoring rule, together with a monotonicity property describing how the admissible class varies with p and q. Finally, we apply our results to several classical scoring rules, including the Borda, plurality, antiplurality, and k-approval rules, identifying the conditions under which they satisfy the proposed acceptability property.