Compromise in combinatorial vote

We 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 outcomes are related in a neutral way, and we investigate the existence of neutral two-level 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.