Collective decision-making is a familiar feature of our social, political, and economic lives. It ranges from the relatively trivial (e.g. the choice of the next family car) to the globally significant (e.g. whether or not a country should go to war). Yet, whether trivial or globally significant, such decisions involve a number of challenging problems. These problems arise in the standard social choice setting, where individuals differ in their preferences. They also arise in the standard decision-making setting, where individuals share the same preferences, but differ in their decisional...