Optimality and equilibrium for binary decision problems in a committee

We consider a committee facing a binary decision under uncertainty. Each member holds some private information. Members may have different preferences and initial beliefs, but they all agree which decision should be taken in each of the two states of the world. We characterize the optimal anonymous and deterministic voting rule and provide a homogeneity assumption on preferences and beliefs under which sincere voting is a Nash equilibrium for this rule. We also provide a necessary and sufficient condition for sincere voting to be an equilibrium under any deterministic majoritarian voting rule. We show that a class of slightly randomized majoritarian voting rules make sincere voting a strict and unique pure-strategy equilibrium. A slight deontological preference for sincere voting, or ex post revelation of individual votes - transparency - combined with a concern for esteem, has the same effect.