Extensions to the Gibbard Satterthwaite Theorem, to show that all reasonable voting schemes can generate safe strategic votes

Abstract

The Gibbard Satterthwaite Theorem is one of the most prominent results in social choice theory, and shows that all reasonable voting schemes can generate strategic votes. We draw a distinction between `safe' and `unsafe' strategic votes; if it is possible for too many or too few like-minded voters to cast the same strategic vote, that strategic vote is unsafe. We use practical and theoretical examples to show that the distinction is both interesting and important. We then show that all reasonable voting schemes can generate safe strategic votes. This result extends the Gibbard Satterthwaite Theorem. Next we define `positively safe moves'. These are (from the point of view of the potentially-strategic voter) a particularly attractive type of safe strategic vote. We show that almost all reasonable voting schemes can generate positively safe strategic moves. We also show that there are voting schemes to which Gibbard Satterthwaite applies, but which never generate positively safe moves. We prove additional results showing how voting schemes that return sets rather than singletons can generate positively safe moves. These moves can be made by voters that are electing a committee, and who compare two committees by comparing the most influential member on each committee. We call such voters `GOPs', where GOP stands for `generalised optimism / pessimism'. These results supplement Taylor's version of the Duggan Schwartz Theorem.