Optimal dictatorial and multi-dictatorial choice in Arrow’s model with applications to multicriteria decision making

Abstract

The capability of a given individual to represent the preference of other individuals in a collective choice model is characterized by two quantitative indicators of representativeness. We prove that there always exists an individual who, on average, represents a majority, and an individual who represents a majority in more than 50% of all possible cases.

We apply this result to Arrow’s collective choice model and revise Arrow’s paradox. It follows that there always exists a dictator who is a representative of the collective rather than a dictator in a proper sense. The refinement of the concept of a dictator leads to the consistency of Arrow’s axioms.

Besides single representatives, we consider the cabinet (named by analogy with a cabinet of ministers) which consists of a few dictators with delimited domains of competence. We show that the representativeness of optimal cabinets tends to 100%-values with the increase in their size, not depending on the size of the collective. We suggest a geometric interpretation of optimal dictators and cabinets. It is based on the approximation formulas for the indicators of representativeness derived for the model with independent individuals. Finally, for cabinets we establish the consistency of different concepts of optimality, resulting from the use of different indicators of representativeness.

Our model has applications to multicriteria decision making. The appointment of a cabinet corresponds to the selection of a few partial criteria. Therefore, our results can be used for the reduction of the set of criteria to a certain sufficient minimum.