Dictionaries
type: Item
instance of: scholarly article
title: Dictionaries
zbMATH Open document ID: 0698.90006
DOI: 10.1007/BF00295863
author: Jerry S. Kelly
published in: Social Choice and Welfare
publication date: 1989-01-01T00:00:00Z
review text: 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 votin…
Mathematics Subject Classification ID: 91B14
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 4145614
zbMATH Keywords: social choice procedures
zbMATH Keywords: wreath products of permutation groups
zbMATH Keywords: dictionary
zbMATH Keywords: preference profile
zbMATH Keywords: Borda rule
zbMATH Keywords: weighted elections
zbMATH Keywords: Condorcet cycles
reviewed by: Stephen J. Turnbull
MaRDI profile type: Publication
cites work: Q5668657
cites work: Inverted orders for monotone scoring rules
cites work: The ultimate of chaos resulting from weighted voting systems
cites work: Symmetry, voting, and social choice
full work available at URL: https://doi.org/10.1007/bf00295863
OpenAlex ID: W4254080570
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description: scientific article; zbMATH DE number 4145614