Dictionaries
This is one of a series on conjectures and unsolved problems in social choice theory. It concerns dictionaries for social choice procedures. Such dictionaries are formally related to wreath products of permutation groups. Many results in social choice theory are derived from counterexamples in which a society is constructed for which the social choice procedure of interest generates an outcome which violates some desideratum. The dictionary for a social choice procedure is simply the range of the function defined by applying the social choice procedure to all possible societies. Thus we could generate (counter-) examples for a social choice procedure simply by inspecting its dictionary. Formally, let S be a finite universal set of alternatives. The set of feasible sets \(L\subset 2^ S\) is nonempty, with \(n(t)\equiv | S_ t| >1\) for each \(S_ t\in L\). A word is a list of (weak) orderings, one for each feasible set. A preference profile is a list of strong orderings of S. A profile may have arbitrarily high positive cardinality. A social choice procedure is a function assigning a word to each preference profile. The dictionary of the procedure is the range of this function. To determine the social optimum for a given profile and feasible set, then, we first use the procedure to generate the corresponding word for the profile, then the word determines an ordering of the given feasible set. The optimum is a maximal element of the feasible set according to the ordering thus determined. \textit{D. Saari} [``The optimal ranking method is the Borda count'', mimeo (1985)] has established that the Borda rule has the smallest dictionary in a class of procedures called weighted elections. Furthermore, he has established a binary inclusion property (BIP) on L (``A dictionary for voting paradoxes'', mimeo (1987)], a converse of which guarantees that L's dictionary is full for every set of weights. In particular, the theorem implies that the dictionary for simple majority voting is full. That is there exist Condorcet cycles. The author suggests that several generalizations of this framework would be of interest. Perhaps the BIP can be weakened. Permitting more general procedures, in particular allowing words to be choice functions rather than lists of orderings, is of interest. Individuals should be permitted weak orderings. Finally, the effects of restrictions on the number of individuals permitted in a profile should be examined.
scientific article; zbMATH DE number 4145614