Approval Compatible Voting Rules

In a classical voting situation, each voter in a group is asked to report a ranking over a set of alternatives, and a voting rule is applied to determine a winning alternative. But voters may also hold approval preferences, giving rise to an approval winner. If voters with approval preferences are asked to report rankings instead, and assuming that voters are sincere, can an approval winner possibly win the election? Can an approval loser lose the election, or can all approval co-winners be co-winners of the election? These three types of questions lead to different notions of approval compatibility for voting rules, called positive, negative, and uniform approval compatibility. We find that negative approval compatibility is a very weak notion, while uniform positive approval compatibility is a very strong one. We also show that positive approval compatibility as well as uniform negative approval compatibility divide usual voting rules into two significant families: Borda, plurality, plurality with runoff, STV, and Condorcet-consistent rules satisfy these notions, while several positional scoring rules (with Borda and Plurality excepted) violate them