Core thresholds of symmetric majority voting games

We consider a committee that consists of n members with one person one vote approving a proposal if the number of affirmative votes from the members reaches threshold k. Which threshold k between 1 and n is “stable” for the committee? We suppose that if a new threshold k' proposed by some committee members obtains k or more affirmative votes, then the new threshold replaces the current one. Assuming that each member has preferences for the set of possible thresholds, we analyze which threshold meets the stability requirements of the core and stable sets. In addition, based on our stability study, we argue that a committee needs to employ two distinct thresholds: one for ordinary issues and another just for threshold changes. To embody this idea, we propose a method, called the constant threshold method, and show that our method always generates a nonempty refinement of the core. We also provide an axiomatic characterization of our method.