Dindar, HayrullahLaine, Jean2024-07-182024-07-1820220176-17141432-217Xhttps://doi.org/10.1007/s00355-022-01387-6https://hdl.handle.net/11411/7133We consider collective choice problems where the set of social outcomes is a Cartesian product of finitely many finite sets. Each individual is assigned a two-level preference, defined as a pair involving a vector of strict rankings of elements in each of the sets and a strict ranking of social outcomes. A voting rule is called (resp. weakly) product stable at some two-level preference profile if every (resp. at least one) outcome formed by separate coordinate-wise choices is also an outcome of the rule applied to preferences over social outcomes. We investigate the (weak) product stability for the specific class of compromise solutions involving q-approval rules, where q lies between 1 and the number I of voters. Given a finite set X and a profile of I linear orders over X, a q-approval rule selects elements of X that gathers the largest support above q at the highest rank in the profile. Well-known q-approval rules are the Fallback Bargaining solution (q = I) and the Majoritarian Compromise (q = [I/2]). We assume that coordinate-wise rankings and rankings of social out- comes are related in a neutral way, and we investigate the existence of neutral twolevel preference domains that ensure the weak product stability of q-approval rules. We show that no such domain exists unless either q = I or very special cases prevail. Moreover, we characterize the neutral two-level preference domains over which the Fallback Bargaining solution is weakly product stable.eninfo:eu-repo/semantics/closedAccessOstrogorski ParadoxPreference AggregationTheoremArticle2-s2.0-8512476301610.1007/s00355-022-01387-62061Q117559Q4WOS:000755388600001