Fair Mixing

We consider a setting in which agents vote to choose a fair mixture of public outcomes. The agents have dichotomous preferences: Each outcome is liked or disliked by an agent. We discuss three outstanding voting rules. The Conditional Utilitarian rule, a variant of the random dictator, is strategyproof and guarantees to any group of like-minded agents an influence proportional to its size. It is easier to compute and more efficient than the familiar Random Priority rule. We show, both formally and by numerical experiments, that its inefficiency is low when the number of agents is low. The efficient Egalitarian rule protects individual agents but not coalitions. It is excludable strategyproof : An agent does not want to lie if she cannot consume outcomes she claims to dislike. The efficient Max Nash Product rule offers the strongest welfare guarantees to coalitions, which can force any outcome with a probability proportional to their size. But it even fails the excludable form of strategyproofness.