Gilbert Laffond

Composition-consistent tournament solutions and social choice functions

This paper introduces a new axiom for choice in preference profiles and tournaments, called composition-consistency. A social choice function is composition-consistent if it is non-sensitive to the cloning of one or several outcomes. The key feature of the composition consistency property is an operation concept called multiple composition product of profiles. The paper provides a brief overview of some social choice functions studied in the literature. Concerning the tournament solutions, it is proved that the Top Cycle, the Slater and the Copeland solutions are not composition-consistent, whereas the Banks, Uncovered Set, TEQ, Minimal Covering Set are composition-consistent. Moreover, we define the composition-consistent hull of a solution ф as the smallest composition-consistent solution containing ф. The composition-consistent hulls of the Top cycle and Copeland solutions are specified, and we give some hints about the location of the hull of the Slater set. Concerning social choice functions, it is shown that Kemeny, Borda and Minimax social choice functions are not composition-consistent, whereas the Paretian one is composition-consistent. Moreover, we prove that the latter is the composition-consistent hull of the Borda and Minimax functions.

Condorcet choice correspondences: A set-theoretical comparison

Abstract In this paper we provide a set-theoretical comparison of the Condorcet choice correspondences discussed in the literature. We review the results on that issue and state some open problems.

Representation in majority tournaments

Abstract The paper presents a general setting for studying majority-based collective decision procedures where the electorate is divided into constituencies according to an equal-representation principle. It generalizes the well-known Referendum Paradox to the non-dichotomous choice case, and shows that all Condorcet choice rules are sensitive to the design of the apportionment of the electorate, in the sense that final outcomes may entirely differ from those prevailing when there is a single constituency. Direct and representative democratic systems thus lead to mutually inconsistent collective decisions.

Single-switch preferences and the Ostrogorski paradox

The Ostrogorski paradox refers to the possibility for a democratically chosen program involving finitely many binary decisions to be unpopular. It deals with the potential conflict arising between two majority-based choice procedures from a set of alternatives {− 1, 1}N, where N stands for the number of decisions. The first procedure is the simple majority rule applied decision-wise. In the second procedure, voters valuate programs through their symmetric distance to an ideal, and programs are compared according to the simple majority rule. This paper characterizes the preference domain (i.e., the set of ideals) which allows to avoid the paradox for any number of voters and any number of decisions. We prove that such a domain contains all those preference profiles sharing a property called single-switchness, of which we provide alternative interpretations.

Condorcet choice and the Ostrogorski paradox

The Ostrogorski paradox refers to the fact that, facing finitely many dichotomous issues, choosing issue-wise according to the majority rule may lead to a majority defeated overall outcome. This paper investigates the possibility for a similar paradox to occur under alternative specifications of the collective preference relation. The generalized Ostrogorski paradox occurs when the issue-wise majority rule leads to an outcome which is not maximal according to some binary relation φ defined over pairs of alternatives. We focus on three possible definitions of φ, whose sets of maximal elements are respectively the Uncovered Set, the Top-Cycle, and the Pareto Set. We prove that a generalized paradox may prevail for the Uncovered Set. Moreover, it may be avoided for the same issue-wise majority margins as for the Ostrogorski paradox. However, the issue-wise majority rule always selects a Pareto-optimal alternative in the Top-Cycle.

The Copeland measure of Condorcet choice functions

We propose a measure to compare an arbitrary choice function with the Copeland choice function. We compute this measure for the familiar Condorcet choice functions.

HEALTH EDUCATION IMPROVES OLDER SUBJECTS‘ ATTITUDES TOWARD URINARY INCONTINENCE AND ACCESS TO CARE: A RANDOMIZED STUDY IN SHELTERED ACCOMODATION CENTERS FOR THE AGED

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Slaters's winners of a tournament may not be in the Banks set

An exemple is given of a tournament in which none of the Kemeny-Slaters winners of the tournament belong to the Banks set.

Majority voting on orders

We characterize two lexicographic-type preference extension rules from a set X to the set Ψ of all orders on this set. Elements of X are interpreted as basic economic policy decisions, whereas elements of Ψ are conceived as political programs among which a collectivity has to choose through majority voting. The main axiom is called tournament-consistency, and states that whenever majority pairwise comparisons based on initial preferences on X define an order on X, then this order is also chosen by a majority among all other orders in Ψ. Tournament-consistency thus allows to predict the outcome of majority voting upon orders from the knowledge of majority preferences on their components.

Weak covering relations

This paper proposes and justifies a natural way to weaken the concept of covering relation defined on a finite tournament. Various weak covering relations, calledk-covering relations, are introduced. To eachk-covering relation corresponds a strong uncovered set containing all nonk-covered outcomes. It is proved that those strong uncovered sets may be empty. Moreover, the set of all tournaments having an empty strong uncovered set is characterized within two rather large classes of tournaments. Finally, we offer a complete study of the cases where the directed graph defined by ak-covering relation coincides with the initial tournament.