Election of vice-president and adding new members to representative bodies

In Tanguiane (1991, 1993, 1994) we have introduced quantitative indicators of representativeness, with which we have estimated the capacity of individuals and limited groups to represent a collective preference. We have studied three forms of representation: (a) single representative (president); (b) cabinet which consists of representatives personally responsible for certain domains of competence (government); and (c) council which makes collective decisions by means by voting (parliament). In this paper we examine the appointment of president and vice-president. In our model it corresponds to the appointment of a cabinet with two members. We show that it may be impossible to make an optimal appointment successively, finding first the most representative president, and matching the vice-president afterwards. The only way which guarantees their optimal appointment, is choosing them together as a team. We prove, however, that successively chosen president and vice-president, as a cabinet with two members, have the indicators of representativeness greater than or equal to 75% of their maximal value. Besides we investigate a recursive construction of cabinets and councils by optimally adding new members one by one. We prove that the indicators of representativeness of such a recursively constructed cabinet with k members are greater or equal to (1-2-k) · 100% of their maximal value. This estimate has the same exponent as that for the optimal cabinets, meaning that such a recursive construction provides, if not optimal, still rather good results. The recursive construction of representative councils is restricted to particular cases, so that an optimal council should be chosen simultaneously. In conclusion we discuss the applicability of the results obtained to real politics.