Compromise in combinatorial vote (Q2171851)

This paper examines collective choice problems in which the set of social outcomes is the Cartesian product of a finite number of finite sets. Each individual is assigned a two-level preference, which is defined as a pair consisting of a vector of strict rankings of elements in each set and a strict ranking of social outcomes. Concepts of product stability and weak product stability are proposed: 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. Product stability and weak product stability are investigated for the class of compromise solutions involving q-approval rules, for \(q\) ranging from 1 to the number \(I\) of voters (fallback bargaining solution). Also proposed is a concept of preference domain: a preference domain is a subset of two-level preferences that are mutually related in a restricted manner. It is demonstrated that neutral two-level preference domains that enable weak product stability of \(q\)-approval rules exist only for \(q=I\) or in some rare circumstances. Neutral two-level preference domains for which the fallback bargaining solution is weakly product stable are characterized.