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Social Welfare Functionals and Invariance Transformations
In this section, we will investigate various degrees of interpersonal comparability though admissible set of transformations. In the Arrovian framework, there is no room for in- terpersonal comparability among individuals, or in other words it is completely ordinal. Given a utility profile, any monotonic transformation applied to this profile will be in- formationally equivalent to the first one in this framework. We will see that by different subsets of monotonic transformations, various degrees of comparability can be formal- ized. It is worthwhile to note that there is an inverse relationship with the size of these dmissible subsets of transformations and the degree of comparable information. For this purpose, first we will extend our analysis to a finite set N of individuals confronting a finite non-empty set A of alternatives. For the rest of this section, we will mainly use the analysis of D’Aspremont and Gevers [10]. For all i ∈ N and for all a ∈ A, we denote by u(a, i) the individual i’s utiliy level for the alternative a. By defining utility function on A × N , we can now compare the welfare of an individual i at alternative a, to the welfare of an individual j at alternative b. We denote a profile of size n utility functions by U = (u(·, 1), . . . , u(·, n)) and we write U for the set of all possible utility profiles. Next, we define a social welfare functional as a mapping F : D → O where ∅ = D ⊆ U is the set of admissible profiles and O is the set of all orderings on A. Rep-resentation of the informational environment will be established through the partition of the set of admissible profiles into information sets, similar to our analysis in section 3. Finally, for every U 1, U 2 ∈ U, we write RU1 = F (U 1) and RU2 = F (U 2).
Measurability and Comparability Axioms
For each of the following cases, an information-invariance condition requires F to be constant on each of the information sets. In other words, two profiles U 1 and U 2 are informationally equivalent, if F (U 1) and F (U 2) are identical.
Preference-Approval framework
So far we have seen various informational frameworks used in collective decision making problems and discussed their implications for some aggregation rules. In recent literature of Social Choice Theory, there are many proposals of voting rules which call for new informational frameworks. (see, for example, Hillinger [32], Aleskerov et al. [33], Balinski and Laraki [18], Smith [19]) In particular Approval Voting, ( [15, 16, 17]) requires a qualification profile of “ap- proved” or “disapproved” alternatives and the ones which are approved by the highest number of individuals are the AV winners.10As a recent study (Sanver [35]) suggests, common meaning assumption for these two qualifications can be interpreted as the existence of a real number, say 0, whose meaning as a utility measure is common to all individuals. As an example, “being self-matched” in matching theory models (for ex- ample, see Roth and Sotomayor [36]) can be interpreted as the common zero of that framework.
In the context of this chapter, “existence of a zero for an individual” leads to the next invariance condition by using the the terminology of section 3. We denote by Φ∗ the set of monotonic transformations with a fixed point, which is generated by successively all pairs of u, v ∈ U in the following condition: Information sets of preference-approval framework for an individual: For any u, v ∈ U, u ∼Φ∗ v if and only if v = φ ◦ u, for some φ ∈ Φ∗, which is monotonically increasing and φ(0) = 0. The next lemma shows that preference-approval framework leads to finer partitions than the ones generated by the set of monotonic transformations, ΦM, which is generated by successively all pairs of u, v ∈ U.
Measuring consensus for preference-approvals
We now introduce our proposal for measuring consensus in the context of preference- approvals. As we have discussed previously, richer information content of the preference approval structures can be desirable to distinguish some threshold levels as well as the rankings in collective decision problems. On the other hand, analyzing the homogene-ity level in such profiles asks for an extension of the standard measures of consensus in the literature. We show that the consensus measure introduced by Garc´ıa-Lapresta and Perez´-Roman´ [51, 60] for weak orders is robust to the additional approval information for ordinal preferences when the metric proposed by Proposition 3.2.6 is used as an input.
Table of contents :
1 Introduction
2 Informational Frameworks in Collective Decision Making
2.1 Introduction
2.2 Basic Notions
2.3 Information Sets for an Individual
2.4 Social Welfare Functionals and Invariance Transformations
2.4.1 Measurability and Comparability Axioms
2.5 Utilitarianism and Rawlsian Models
2.6 Preference-Approval framework
2.7 Concluding Remarks
3 Measuring Consensus in a Preference-Approval Context
3.1 Introduction
3.2 Preliminaries
3.2.1 Preference-approval structures
3.2.2 Distances and metrics
3.3 Consensus measures
3.3.1 Basic notions
3.4 Measuring consensus for preference-approvals
3.4.1 Some results
3.4.2 An Illustrative example
3.4.3 Replications
3.5 Concluding remarks
4 On Manipulabilition from an Unacceptable Social Choice to an Acceptable one
4.1 Introduction
4.2 Basic notions and definitions
4.2.1 Preference-approval framework
4.2.2 Preference-approval aggregation
4.2.3 Circular domains
4.3 Results
4.4 Concluding remarks
Bibliography
