An approach to generalizing some impossibility theorems in social choice

In social choice theory, voting methods can be classified by invariance properties: a voting method is said to be C1 if it selects the same winners for any two profiles of voter preferences that produce the same majority graph on the set of candidates; a voting method is said to be pairwise if it selects the same winners for any two preference profiles that produce the same weighted majority graph on the set of candidates; and other intermediate classifications are possible. As there are far fewer majority graphs or weighted majority graphs than there are preference profiles (for a bounded number of candidates and voters), computer-aided techniques such as satisfiability solving become practical for proving results about C1 and pairwise methods. In this paper, we develop an approach to generalizing impossibility theorems proved for C1 or pairwise voting methods to impossibility theorems covering all voting methods. We apply this approach to impossibility theorems involving "variable candidate" axioms--in particular, social choice versions of Sen's well-known $\gamma$ and $\alpha$ axioms for individual choice--which concern what happens when a candidate is added or removed from an election. A key tool is a construction of preference profiles from majority graphs and weighted majority graphs that differs from the classic constructions of McGarvey and Debord, especially in better commutative behavior with respect to other operations on profiles.