Abstract
In the measurement of autonomy freedom, the admissible potential preference relations are elicited by means of the concept of ‘reasonableness’. In this paper we argue for an alternative criterion based on information about the decision maker’s ‘awareness’ of his available opportunities. We argue that such an interpretation of autonomy fares better than that based on reasonableness. We then introduce some axioms that capture this intuition and study their logical implications. In the process, a new measure of autonomy freedom is characterized, which generalizes some of the measures so far constructed in the literature.
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We thank Martin van Hees, Robert Sugden, Prasanta Pattanaik, the participants at workshops at the CPNSS, London School of Economics, and at the University of Caen and an anonymous referee for their suggestions. The support of the CPNSS is gratefully acknowledged. This paper is part of a research project on “The Analysis and Measurement of Freedom” funded by the Ministero dell’Istruzione, Università e Ricerca (Italy). Its financial support is gratefully acknowledged.
Appendix: the proof of Proposition 4.2
Appendix: the proof of Proposition 4.2
Before proving Proposition 4.2, we state and prove the following lemma.
Lemma 6.1
If ⪰ satisfies INF and COM, then, \(\forall A \in {\user1{{\wp }}}{\left( {\text{X}} \right)}\), \(\forall \Pi _{i} \in {\user1{{\wp }}}{\left( \Pi \right)}\),
Proof
If max i (A)=A, then the result clearly follows. If not, suppose ∣max i (A)∣=g and let max i (A)={a 1,...,a g } and A-max i (A)=Â. Now, max i ({a 1}∪Â)=max i ({a 1})={a 1} and max i ({a 2}∪Â)=max i ({a 2})={a 2}. Hence by INF,
and
Clearly, {a 1}∩{a 2}=∅, max i ({a 1}⋃{a 2})=({a 1}⋃{a 2}), ({a 1}⋃Â)∩({a 2}⋃Â)= and Â∩(max i (({a 1}⋃Â)⋃({a 2}⋃Â)))=∅. Hence we can apply axiom COM and obtain,
By considering successively a 3,a 4,...,a g , and applying INF and COM repeatedly, we finally obtain
or
Q.E.D.
We are now in the position to prove Proposition 4.2.
Proof
Necessity is straightforward. We therefore prove sufficiency. To start with, we show that
Suppose ∣max i (A)∣=∣max j (B)∣=g. It follows that, max i (A)={a 1,...;,a g } and max j (B)={b 1,...,b g }. Using INF, ({a 1},Π i )∼({b 1},Π j ), ({a 2},Π i )∼({b 2},Π j ) and {a 1}∩{a 2}=∅. Now, max i {a 1,a 2}={a 1,a 2}, so we can use axiom COM to yield:
By INF, ({a 3},Π i )∼({b 3},Π j ); by COM,
and so on. Finally we have:
i.e.,
Since ⪰ satisfies INF and COM, we can apply Lemma 6.1 and obtain
and
Now, Eqs. (2), (3), (4), and transitivity of ⪰ imply (A, Π i )∼(B, Π j ).
Now we show that
Suppose ∣max i (A)∣=g+t and ∣max j (B)∣=g. So, max i (A)={a 1,...,a g+t } and max j (B)={b 1,...,b g }. Now, max i ({a 1,...,a g })={a 1,...,a g }. Hence, by (1),
Now, max i ({a 1,...,a g+1})={a 1,...,a g+1}. By ARA,
and, by (5) and transitivity of ≻,
By adding a g+2,...,a g+t successively, and by using ARA repeatedly, we have
i.e.,
We know from Lemma 4.1 that
Clearly, (A, Π i )∼(max i (A),Π i ), (6) and transitivity of ⪰ imply (A, Π i )≻(B,Π j ). Q.E.D.
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Bavetta, S., Peragine, V. Measuring autonomy freedom. Soc Choice Welfare 26, 31–45 (2006). https://doi.org/10.1007/s00355-005-0027-5
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DOI: https://doi.org/10.1007/s00355-005-0027-5