Arrow's paradox and mathematical theory of democracy (Q1315391)

By mathematical theory of democracy we understood the results on the capacity of single individuals and limited groups of individuals to make decisions on behalf of the whole society. As in mathematical economics, where it is shown that agents can contribute to social welfare by optimizing their interests, in mathematical theory of democracy it is proved that the social choice can be realized by the individuals who are motivated by their own preferences. This way the potential existence of representatives is established, which is the basis for all democratic systems. The principal innovation of our approach is the formalization of the idea of representativeness, with which we characterize single individuals (president, deputies), and collective representations (parliament and government). The representativeness is measured by the weight of the coalition whose preference is satisfied in a certain situation of decision making. These situations are considered as simple events in a probability model, implying the representativeness to be a random variable defined on the set of situations and with the values expressed in per cents of the population satisfied. It is proved that for arbitrary probability measures there always exist individuals who, on the average, represent a majority (Theorem 1). This result is applied to Arrow's paradox about the inevitability of a dictator (Theorem 2). It follows that we can distinguish between dictator-representatives and dictators in a proper sense. After the concept of dictator has been refined to the dictator in a proper sense, Arrow's paradox becomes solvable: There always exists a social welfare function with no dictator (in a proper sense). Besides individual representatives, we consider two types of collective representation: cabinet (named by analogy with the cabinet of ministers) which consists of a few representatives with delimited domains of competence, and council which makes decisions by means of voting. We estimate the average representativeness of so called optimal cabinets and councils and show that it depends on the number of their members but not on the size of the society. We also prove that this indicator tends to some maximal values as the number of cabinet and council members increases (Theorems 7 and 8). We propose a geometric interpretation of optimal representatives, cabinets, and councils which is based on approximation formulas for the average representativeness valid for the model with a large number of independent individuals. Each representation is identified with a certain vector, and it is shown that the best representation is the one whose characteristic vector provides the greatest projection on the society's characteristic vector (Theorems 3, 9, 10). We consider another indicator of representativeness, majority representativeness, similar to the average representativeness, with which some special political situations are modeled (Theorems 4 and 5). We formulate the conditions of equivalence of the two indicators of representativeness (Theorems 6, 11, and 12). Finally, we outline applications to multicriteria decision making, where partial criteria correspond to representatives in a social choice model. The model is used for reducing the set of partial criteria to a certain sufficient minimum, for selecting most representative ones, and for estimating the probability of correct decisions based on the criteria selected.