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Public Choice

Erschienen in:

01.10.2013

Elections with partially ordered preferences

verfasst von: Michael Ackerman, Sul-Young Choi, Peter Coughlin, Eric Gottlieb, Japheth Wood

Erschienen in: Public Choice | Ausgabe 1-2/2013

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Abstract

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 partially ordered sets. We identify some specific methods that could be applied in the second step. In addition, by replacing the indirectly specified preferences in a suitable way, we suggest a technique that can use any positional, pairwise, or other voting method that accepts totally (or “completely”) ordered inputs to tally ballots. We also describe another way to potentially compute pairwise rankings from partially ordered alternatives and discuss some practical and theoretical difficulties associated with our approach.

Vorheriger Artikel Honey, I shrunk the kids’ benefits—revisiting intergenerational conflict in OECD countries

Nächster Artikel A political economy model of market intervention

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Our proposal has some features that are also present in the following description from Bassett and Persky (1994: 1076) of a method that has been used to obtain placements in an “original program” event at skating competitions: “a judge gives two cardinal marks … for required elements and presentation”; “for each judge an ordinal ranking of skaters is determined from total marks, the sum of the two subcomponent cardinal scores. These ordinal ranks and not the raw scores become the basis for determining placements”. The common features are: (1) there is more than one stage and (2) the last stage aggregates derived orderings on the alternatives. However, the stages that come before the last stage differ. Each characteristic we consider is one where it is known whether it is present or absent in a particular alternative. In our proposal, a voter initially expresses his preferences on such characteristics and those preferences on characteristics are used to obtain his derived ordering on the alternatives. Alternatively, at the skating competition, a judge initially evaluates the extent to which a skater exhibits certain characteristics and those evaluations are used to obtain his derived ordering. For a survey of other methods that use individuals’ judgements about characteristics of alternatives as inputs for group decision making, see Bose et al. (1997).

The mathematical properties of posets have been studied in great depth and breadth. See, for instance, Stanley (1997).

Our formulation of the basic features of a voter’s derived preferences on alternatives matches with “preference structures” as described in Sects. 2.1–2.3 of Roubens and Vincke (1985). Our formulation of the commonly used approach of using totally ordered sets to represent voter preferences in situations where no two distinct alternatives are equivalent matches with “total order structures” as described in Sect. 3.2 of Roubens and Vincke (1985). Our approach for situations where there are no v, S, and T so that S∼_v T and where S∥_v T for some v, S, and T matches with “partial order structures” as described in Sect. 3.6 of Roubens and Vincke (1985).

In a strict partial order, if \(S, T \in\mathcal{P}\) and if S<T, then not(T<S). Hence, a strict partial order vacuously satisfies the antisymmetry property.

Because of the way that indifference is defined, this approach implicitly assumes that the preference relation “is as at least as good as” satisfies reflexivity and completeness.

In Theorem 2.2 of Aleskerov and Monjardet (2002: 26), it is proved that < is a strict bucket order on \(\mathcal{P}\) if and only if < satisfies

For all \(S, T \in\mathcal{P}\), if S<T, then \(T \not< S\) (asymmetry).

For all \(R, S, T \in\mathcal{P}\), if \(R \not< S\) and \(S \not< T\), then \(R \not< T\) (negative transitivity).

The term “weak order” has been used for a strict bucket order in some references where this alternative formulation appears; see, for instance, Fishburn (1973b: 75; 1973a: 460), Fishburn and Gehrlein (1975: 93), Aleskerov and Monjardet (2002: 22). If an individual’s strict preferences are a strict bucket order and the “alternative approach” to modeling preferences (described in the previous section) is used, then the preference relation “is as at least as good as” is transitive, complete, and reflexive (see, for instance, the first line after Eq. (7.6) in Fishburn (1973b: 76). In some references, the term “weak order” and the closely related term “weak ordering” have (alternatively) been used for such preference relations (that is, ones which are transitive, complete, and reflexive)—see, for instance, Arrow (1963: 12), Sect. 3.6 of Roubens and Vincke (1985), or Merlin and Valognes (2004: 344). In what follows, we will not use the term “weak order” for a strict bucket order—in order to keep readers from mistakenly thinking that we are referring to preference relations that are transitive, complete, and reflexive.

There are, of course, various other issues about which voters might be concerned. For example, there may be two candidates who voters believe would serve well (or poorly) on the committee together. For some other specific examples of issues about which voters might wish to have input, see Ratliff (2006: 351).

Antisymmetry and transitivity prohibit cycles of more than one element in posets. Directed graphs, a.k.a. digraphs, with multiple or weighted edges would be a more appropriate model for such an approach. See, e.g., Bondy and Murty (1976) for more on digraphs.

The interpretation of preference digraphs has been explored in the context of pairwise voting (see Taylor (1968) or Reid (2004), for example).

By a subposet of a poset (P, ≤_P) we mean a subset Q of P together with a partial order ≤_Q on Q such that, for all x,y∈Q, x≤_Q y if and only if x≤_P y.

The basic idea behind the linear extension method is similar to that of the simple Copeland (1951) method, a more than half-century old voting procedure based on pairwise comparisons. By considering linear extensions for each preference poset, the linear extension method incorporates the magnitude of pairwise comparisons, lack of which was a major argument against the simple Copeland method by Merlin and Saari (1996, 1997).

This property does not hold for all voting methods. See Fishburn (1977) for example.

For a given poset (P, ≤_P), the poset (P, \(\leq_{P}'\)) where \(\leq_{P}'= \{(x,y):(y,x) \in\, \leq_{P}\}\) is called the dual of (P, ≤_P). Using this terminology, this step can be described more precisely as obtaining the duals of the posets under consideration.

Reversing a poset can change the definition of a concept that depends on its order. For such a concept, a second definition that is obtained by reversing all of the comparisons in the poset is called its dual concept. For instance, for a poset, upper bound is the dual concept of lower bound. Analogously, reversing a list of preference posets can change the definition of a concept that depends on their orders. For such a concept, a second definition that is obtained by reversing all of the comparisons in the posets can (similarly) be called its dual concept. The result that we’ve stated is based on the observation that, for a list of preference posets, Condorcet winner is the dual concept of Condorcet loser when the pairwise linear extensions method is being used.

For some specific metrics that could be used to measure “closeness”, see Fagin et al. (2004, 2006).

Whereas NP-complete problems ask a yes/no question, #P-complete problems (pronounced “sharp-P”) are the functional analogue. A number must be determined, such as how many linear extensions of a poset there are.

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Titel: Elections with partially ordered preferences
verfasst von: Michael Ackerman
Sul-Young Choi
Peter Coughlin
Eric Gottlieb
Japheth Wood
Publikationsdatum: 01.10.2013
Verlag: Springer US
Erschienen in: Public Choice / Ausgabe 1-2/2013
Print ISSN: 0048-5829
Elektronische ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-012-9930-3

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