Arrow's paradox and mathematical theory of democracy

Abstract

Two measures, the weight of coalitions and the probability of situations in decision making, are used to characterize the representativeness, i.e. the capability of individuals to represent the social preference. It is proved that there always exists an individual who represents a majority on average, and an individual who represents a majority in most cases. This result is applied to Arrow's social choice model. It follows that there always exists a dictator who is a representative of the society rather than a dictator in a proper sense. After the concept of dictator has been refined to a dictator in a proper sense, Arrow's axioms become consistent. The idea of optimal representation is extended to limited groups of representatives which make decisions on behalf of the whole society. We consider the cabinet (named by analogy with the cabinet of ministers) which consists of a few representatives with delimited domains of competence, and the council which makes decisions by means of voting. It is shown that the representativeness of optimal cabinets and councils tends to 100% of maximally possible values as the number of their members increases, independently of the size of the society. We suggest a geometric interpretation of optimal representatives, cabinets, and councils, based on approximation formulas for the indicators of representativeness derived for the model with a large number of independent individuals. Finally, for cabinets and councils we establish the consistency of different concepts of optimality with respect to different indicators of representativeness. Our consideration is applicable to multicriteria decision making. An appointment of a cabinet or a council corresponds to selecting a few partial criteria. Therefore, the obtained results can be used for reducing the set of partial criteria to a certain sufficient minimum. The concepts of dictator, cabinet, and council can be understood as models of president, government, and parliament, respectively. Thus our results justify reducing social choice to individual choice or small group choice. Although it is in use in all democratic systems, its acceptability is not evident at all. In other words, we justify the demoncraticity of such forms of political power as the president, the parliament, and the government.