ARROWS PARADOX AND MATHEMATICAL-THEORY OF DEMOCRACY

Two measures, the weight of coalitions and the probability of situatio ns in decision making, are used to characterize the representativeness , i.e. the capability of individuals to represent the social preferenc e. 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 conce pt of dictator has been refined to a dictator in a proper sense, Arrow 's axioms become consistent. The idea of optimal representation is ext ended to limited groups of representatives which make decisions on beh alf of the whole society. We consider the cabinet (named by analogy wi th the cabinet of ministers) which consists of a few representatives w ith delimited domains of competence, and the council which makes decis ions by means of voting. It is shown that the representativeness of op timal cabinets and councils tends to 100% of maximally possible values as the number of their members increases, independently of the size o f the society. We suggest a geometric interpretation of optimal repres entatives, cabinets, and councils, based on approximation formulas for the indicators of representativeness derived for the model with a lar ge number of independent individuals. Finally, for cabinets and counci ls we establish the consistency of different concepts of optimality wi th respect to different indicators of representativeness. Our consider ation is applicable to multicriteria decision making. An appointment o f a cabinet or a council corresponds to selecting a few partial criter ia. Therefore, the obtained results can be used for reducing the set o f partial criteria to a certain sufficient minimum. The concepts of di ctator, cabinet, and council can be understood as models of president, government, and parliament, respectively. Thus our results justify re ducing social choice to individual choice or small group choice. Altho ugh it is in use in all democratic systems, its acceptability is not e vident at all. In other words, we justify the demoncraticity of such f orms of political power as th president, the parliament, and the gover nment.