On the self-(in)stability of weighted majority rules

A voting rule f is self-stable (Barberà and Jackson, 2004) if any alternative rule g does not have sufficient support in the society to replace f, where the decision between f and g is based on the rule f itself. While Barberà and Jackson focused on anonymous rules in which all agents have the same voting power, we consider here the larger class of weighted majority rules. Our main result is a characterization of self-stability in this setup, which shows that only few rules of a very particular form satisfy this criterion. This result provides a possible explanation for the tendency of societies to use more conservative rules when it comes to changing the voting rule. We discuss self-stability in this latter case, where a different rule F may be used to decide between f and g.