On the Justice of Decision Rules

Which decision rules are the most efficient? Which are the best in terms of maximin or maximax? We study these questions for the case of a group of individuals faced with a collective choice from a set of alternatives. A key message from our results is that the set of optimal decision rules is well defined, particularly simple, and well known: the class of scoring rules. We provide the optimal scoring rules for the three different ideals of justice under consideration: utilitarianism (efficiency), maximin, and maximax. The optimal utilitarian scoring rule depends crucially on the probability distribution of the utilities. The optimal maximin (respectively maximax) scoring rule takes the optimal utilitarian scoring rule and applies a factor that shifts it towards negative voting (respectively plurality voting).