Ranking of decision rules with random power distribution

In collective decision making, the decisional power assigned to each member of the deciding body may have little relation with that member's expertise level. We consider a concept of effectiveness on the family of all decision rules, adapted to such situations. Namely, we measure the performance of a decision rule, when applied to the decision makers, after these have been permuted randomly. We obtain a necessary and sufficient condition for a rule to be more effective than another in this sense, i.e., for its probability of leading to the correct decision to be larger than that of the other. It is shown that, under certain assumptions, the simple majority rule is the most effective, while the expert rule is the least effective. We also deal with the computational complexity involved in applying our condition.