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1977, Theory and Decision
DEFINITION 1 : ~> is a difference relation on S iff for every x, y, z, w, r, s in S, (1) (x, y) >~ (z, w) or (z,w) >~ (x,y). (2) If (x, y) >~ (z, w), then (w, z) >1 (y, x). 1> is a transitive difference relation on S if also (3) If (x,y.) >/(z, w) and (z, w) i> (r, s), then (x,y) >I (r, s).
Public Choice, 2012
Suppose an organization has a committee with multiple seats, and the committee members are to be elected by a group of voters. For the organization, the possible alternatives are the possible sets of individuals who could serve together. A common approach is to choose from among these alternatives by having each voter cast separate votes on the candidates for each seat. When this type of ballot is used, important characteristics of the set of individuals on the committee (such as what percentage of the members will be female) might not be explicitly considered by the voters. Another approach that has been used is to have each voter cast a ballot which ranks all possible sets of members. However, this approach can require the voters to weigh a relatively large number of alternatives. This paper considers group decisions where it is desirable to: (1) explicitly consider characteristics of alternatives and (2) have a relatively small number of options upon which a voter has to express his preferences. The approach that we propose has two steps: First voters vote directly on pertinent characteristics of alternatives; Then these votes are used to indirectly specify preferences on alternatives. The indirectly specified preferences are ones that are naturally modeled using
Synthese, 1969
The problem of aggregating several persons opinions into a representative group opinion was treated formally by Kenneth J. Arrow [1]. This paper will give a proof of a strengthened version of his well-knoWn result and also of a related result presented in [3]. A and P are supposed to be two finite sets. Their elements are called alternatives and persons" respectively. Each person is required to arrange all alternatives on a ranking list according to his individual preference order. The ordering relation is called apreference relation. Preference relations will in general be denoted by R with a variety of subscripts. From each preference relation two other relations-strict preference and samen e s swill be defined as follows: D1 : xPy= a~ f xRy and not yRx D2: xSy = a~t xRy and yRx R is supposed to be a total preorder, i.e. it fulfils the following axioms: R1 : xRy or yRx R2: if xRy and yRz, then xRz. It follows that R is reflexive and that P is irreflexive, asymmetric and transitive and that S is an equivalence relation. That we only require R to be a preorder and not an order (i.e. xRy and yRx may both be true for distinct x and y) means that ties are allowed in the perference orders. P's and S's derived from a certain R will always carry the same subscript as that R. x, y, z and other lower case letters from the last part of the alphabet will be used as variables ranging over A, the set of alternatives, i, j, k, etc. will range over P, the set of persons. A complete set of individual preference relations, i.e. a set consisting of one preference relation for each person, will be called a preference situa
Social Choice and Welfare, 2020
A social choice rule (SCR) is monotonic if raising a single alternative in voters' preferences while leaving the rankings otherwise unchanged is never detrimental to the prospects for winning of the raised alternative. Monotonicity is rather weak but well-known to discriminate against scoring elimination rules, such as plurality with a run o and single transferable vote. We dene the minimal monotonic extension of an SCR as its unique monotonic supercorrespondence that is minimal with respect to set inclusion. After showing the existence of the concept, we characterize, for every non-monotonic SCR, the alternatives that its minimal monotonic extension must contain. As minimal monotonic extensions can entail coarse SCRs, we address the possibility of rening them without violating monotonicity provided that this renement does not diverge from the original SCR more than the divergence prescribed by the minimal monotonic extension itself. We call these renements monotonic adjustments and identify conditions over SCRs that ensure unique monotonic adjustments that are minimal with respect to set inclusion. As an application of our general ndings, we consider plurality with a runo, characterize its minimal monotonic extension as well as its (unique) minimal monotonic adjustment. Interestingly, this adjustment is not coarser than plurality with a runo itself, hence we suggest it as a monotonic substitute to plurality with a runo. JEL Classications: D71, D79.
Studies in Fuzziness and Soft Computing, 2008
In the classical theory of social choice, there exist many voting procedures for determining a collective preference on a set of alternatives. The simplest situation happens when a group of individuals has to choose between two alternatives. In this context, some voting procedures such as simple and absolute special majorities are frequently used. However, these voting procedures do not take into account the intensity with which individuals prefer one alternative to the other. In order to consider this situation, one possibility is to allow individuals showing their preferences through values located between 0 and 1. In this case, the collective preference can be obtained by means of an aggregation operator. One of the most important matter in this context is how to choose such aggregation operator. When we consider the class of OWA operators, it is necessary to determine the associated weights. In this contribution we survey several methods for obtaining the OWA operator weights. We pay special attention to the way the weights are chosen, regarding the concrete voting system we want to obtain when individuals do not grade their preferences between the alternatives.
Theory and Decision, 2012
Journal of Mathematical Psychology, 1985
He is a specialist in issues of representation and reapportionment and has written on a variety of topics dealing with collective decision-making. His most recent co-edited books are Information Pooling and Group Decision-Making, 1985, and Electoral Laws and Their Political Consequences,
2013
Axioms that govern our choice of voting rules are usually defined by imposing constraints on the rule's behavior under various transformations of the preference profile. In this paper we adopt a different approach, and view a voting rule as a (multi-)coloring of the election graph - the graph whose vertices are elections over a given set of candidates, and two vertices are adjacent if they can be obtained from each other by swapping adjacent candidates in one of the votes. Given this perspective, a voting rule F is characterized by the shapes of its "monochromatic components", i.e., sets of elections that have the same winner under F. In particular, it would be natural to expect each monochromatic component to be convex, or, at the very least, connected. We formalize the notions of connectivity and (weak) convexity for monochromatic components, and say that a voting rule is connected/(weakly) convex if each of its monochromatic components is connected/(weakly) convex. ...
Let X be a set of social alternatives, and let V be a set of 'votes' or 'signals'. (We do not assume any structure on X or V). A variable population voting rule F takes any number of anonymous votes drawn from V as input, and produces a nonempty subset of X as output. The rule F satisfies reinforcement if, whenever two disjoint sets of voters independently select some subset Y ⊆ X , the union of these two sets will also select Y. We show that F satisfies reinforcement if and only if F is a balance rule. If F satisfies a form of neutrality, then F satisfies reinforcement if and only if F is a scoring rule (with scores taking values in an abstract linearly ordered abelian group R); this generalizes a result of .
Mathematical Social Sciences, 2010
A fair spanning tree of a graph maximizes the minimum satisfaction among individuals given their preferences over the edges of the graph. In this note we answer an open question about the computational complexity of determining fair spanning trees raised in Darmann et al. [8]. It is shown that the maximin voter satisfaction problem under choose-t-election is N P-complete for each fixed t ≥ 2.
Scandinavian Political Studies, 1981
The article focuses on the problem of choosing the ‘best’ voting procedure for making collective decisions. The procedures discussed are simple majority rule, Borda count, approval voting, and maximin method. The first three have been axiomatized while the maximin method has not yet been given an axiomatic characterization. The properties, in terms of which the goodness of the procedures is assessed, are dictatorship, consistency, path independence, weak axiom of revealed preference, Pareto optimality, and manipulability. It turns out that the picture emerging from the comparison of the procedures in terms of these properties is most favorable to the approval voting.
Public Choice, 1977
2012
Prima di cominciare la trattazione di questa tesi vorrei ringraziare il mio relatore Prof. Vito Fragnelli (o, più amichevolmente, Franco), per avermi dato l'opportunità di svolgere questo lavoro durante il di dottorato e per tutti gli argomenti interessanti che mi ha proposto. Ringrazio il Prof. Guido Ortona, con il quale ho avuto il piacere di discutere di alcuni degli argomenti trattati nella tesi e non solo, e il Dott. Stefano Gagliardo, mio compagno di avventura e collaboratore in questi tre anni di dottorato. Un grazie alla mia famiglia, ai miei genitori in primis, per tutto il sostegno che non mi hanno mai fatto mancare, a mia sorella Manu, a Luca e in particolare ad Alice, che mi fa capire come tutto questo lavoro sia modesto. Nel tempo che io ho impiegato a mettere insieme un po' di conti e di parole lei è riuscita a trasformarsi da un mucchietto di cellule in una personcina cantante, saltellante, amorevole e che conta fino a 20! Un grazie anche a mia cugina Joan Chessa Hollingsworth e a sua figlia Jo, per il loro prezioso aiuto. Grazie a tutti i miei amici, di cui non elenco i nomi perchè fortunatamente sono davvero tanti, ma che amo tutti quanti alla follia. Ognuno di loro conosce il proprio ruolo nella mia vita e le mille ragioni per cui una sola pagina di ringraziamenti comunque non basterebbe. Non posso però fare a meno di nominare almeno Chiara, Monica, Tania e Maura. Merci a toi, Arnaud, parce que tu seras pour moi unique au monde. Je serai pour toi unique au monde. Gennaio 2013, Michela CHESSA Preface My career as PhD student at University of Milan started 3 years ago, in January 2010. After a bachelor and a master thesis on Game Theory, I was really interested in continuing my research on this subject, when my PhD supervisor, Prof. Fragnelli, proposed me to start working on voting systems. I immediately found the topic actual and fascinating, particularly because of all the problems about the adopted electoral system and the political scenario my country, Italy, has in these years. It was a very convincing opportunity of working on Game Theory and on Game Practice in the same time, because of the concreteness of the problems I was going to deal with. Some of the results included in this thesis have been taken from some articles I previously published: the work presented in Chapter 3 contains some results included in "Chessa M., Fragnelli V., Embedding classical indices in the FP family, Czech Economic Review, vol.5, 2011, pp. 289-305", while the work in Chapter 4 has been presented as invited talk at the workshop "Models of Collusion, Games and Decisions for Applications to Judging, Selling and Voting" in Monte Isola (BS), Italy, on June 2012 with the title "The Bargaining Set for Sharing the Power ". Chapter 5 contains part of analysis presented in "Chessa M., Fragnelli V., A quantitative evaluation of veto power,
Fuzzy Sets and Systems, 2013
A common criticism to simple majority voting rule is the slight support that such rule demands to declare an alternative as a winner. Among the distinct majority rules used for diminishing this handicap, we focus on majorities based on difference in support. With these majorities, voters are allowed to show intensities of preference among alternatives through reciprocal preference relations. These majorities also take into account the difference in support between alternatives in order to select the winner. In this paper we have provided some necessary and sufficient conditions for ensuring transitive collective decisions generated by majorities based on difference in support for all the profiles of individual reciprocal preference relations. These conditions involve both the thresholds of support and some individual rationality assumptions that are related to transitivity in the framework of reciprocal preference relations.
Journal of Economic Theory, 1997
Data Envelopment Analysis and Effective Performance Assessment
A key issue in the preferential voting framework is how voters express their preferences on a set of candidates. In the existing voting systems, each voter selects a subset of candidates and ranks them from most to least preferred. The obtained preference score of each candidate is the weighted sum of votes receives in different places. Thus, one of the most important issues for aggregating preferences rankings is the determination of the weights associated with the different ranking places. To avoid the subjectivity in determining the weights, various models based on Data Envelopment Analysis (DEA) have been appeared in the literature to determine the most favorable weights for each candidate. This work presents a survey on models and methods to assess the weights in voting systems. The existing voting systems are divided into two areas. In the first area it is assumed that the votes of all the voters to have equal importance and in the second area voters are classifies into different groups and assumed that each group is assigned a different voting power. In this contribution, some of the most common models and procedures for determining the most favorable weights for each candidate are analyzed.
Journal of Mathematical Economics, 2011
Collective rationality of voting rules, requiring transitivity of social preferences (or quasi-transitivity, acyclicity for weaker notions), has been known to be incompatible with other standard conditions for voting rules when there is no prior information, thus no restriction, on individual preferences Sen, 1970). proposes two restricted domains of individual preferences where majority voting generates transitive social preferences; they are the domain consisting of preferences that have at most two indifference classes, and the domain where any set of three alternatives is partitioned into two non-empty subsets and alternatives in one set are strictly preferred to alternatives in the other set. On these two domains, we investigate whether majority voting is the unique way of generating transitive, quasi-transitive, or acyclic social preferences. First of all, we rule out non-standard voting rules by imposing monotonicity, anonymity, and neutrality. Our main results show that majority rule is the unique voting rule satisfying transitivity, yet all other voting rules satisfy acyclicity (also quasi-transitivity on the second domain). Thus we find a very thin border dividing majority and other voting rules, namely, the gap between transitivity and acyclicity.
Social Choice and Welfare, 1999
An Excess-Voting Function relative to a pro®le p assigns to each pair of alternatives xY y, the number of voters who prefer x to y minus the number of voters who prefer y to x. It is shown that any non-binary separable Excess-Voting Function can be achieved from a preferences pro®le when individuals are endowed with separable preferences. This result is an extension of Hollard and Le Breton (1996). Soc Choice Welfare (1999) 16: 159±167 Many thanks are due to Jean-FrancË ois Laslier and to two referees for their valuable remarks to improve this paper. I would like to thank Basudeb Chaudhuri for his careful reading.
Social Choice and Welfare, 1996
This paper considers the construction of sets of preferences that give consistent outcomes under majority voting. Fishburn [7] shows that by combining the concepts of single-peaked and single-troughed preferences (which are themselves examples of value restriction) it is possible to provide a simple description of the extent of agreement between individuals that allows the construction of sets that are as large as those previously known (for fewer than 7 alternatives) and larger than those previously known (for 7 or more alternatives). This paper gives a characterisation of the preferences generated through these agreements and makes observations on the relation between the sizes of such sets as the number of alternatives increases.
Le Centre pour la Communication Scientifique Directe - HAL - Université de Nantes, 2017
Classical voting rules output a winning alternative (or a nonempty set of tied alternatives). Social welfare functions output a ranking over alternatives. There are many practical situations where we have to output a different structure than a winner or a ranking: for instance, a ranked or non-ranked set of k winning alternatives, or an ordered partition of alternatives. We define three classes of such aggregation functions, whose output can have any structure we want; we focus on aggregation functions that output dominating chains, dominating subsets, and dichotomies. We address the computation of our rules, and start studying their normative properties by focusing on a generalisation of Condorcet-consistency.
2003
I present a general theorem on preference aggregation. This theorem implies, as corollaries, Arrow's Impossibility Theorem, Wilson's extension of Arrow's to non-Paretian aggregation rules, the Gibbard-Satterthwaite Theorem and Sen's result on the Impossibility of a Paretian Liberal.
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