Measuring autonomy freedom

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.

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, $\forall A \in {\user1{{\wp }}}{\left( {\text{X}} \right)}$, $\forall \Pi _{i} \in {\user1{{\wp }}}{\left( \Pi \right)}$,

$${\left( {A,\Pi _{i} } \right)} \sim {\left( {\max _{i} {\left( A \right)},\Pi _{i} } \right)}.$$

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,

$${\left( {{\left\{ {a_{1} } \right\}} \cup \widehat{A},\Pi _{i} } \right)} \sim {\left( {{\left\{ {a_{1} } \right\}},\Pi _{i} } \right)}$$

and

$${\left( {{\left\{ {a_{2} } \right\}} \cup \widehat{A},\Pi _{i} } \right)} \sim {\left( {{\left\{ {a_{2} } \right\}},\Pi _{i} } \right)}.$$

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,

$${\left( {{\left\{ {a_{1} } \right\}} \cup {\left\{ {a_{2} } \right\}} \cup \widehat{A},\Pi _{i} } \right)} \sim {\left( {{\left\{ {a_{1} } \right\}} \cup {\left\{ {a_{2} } \right\}},\Pi _{i} } \right)}.$$

By considering successively a ₃,a ₄,...,a _g , and applying INF and COM repeatedly, we finally obtain

$${\left( {\max _{i} {\left( A \right)} \cup \widehat{A},\Pi _{i} } \right)} \sim {\left( {\max _{i} {\left( A \right)},\Pi _{i} } \right)}$$

or

$${\left( {A,\Pi _{i} } \right)} \sim {\left( {\max _{i} {\left( A \right)},\Pi _{i} } \right)}.$$

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

$${\left| {max_{i} {\left( A \right)}} \right|} = {\left| {max_{j} {\left( B \right)}} \right|} \Rightarrow {\left( {A,\Pi _{i} } \right)} \sim {\left( {B,\Pi _{j} } \right)}.$$

(1)

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:

$${\left( {{\left\{ {a_{1} ,a_{2} } \right\}},\Pi _{i} } \right)} \sim {\left( {{\left\{ {b_{1} ,b_{2} } \right\}},\Pi _{j} } \right)}.$$

By INF, ({a ₃},Π _i )∼({b ₃},Π _j ); by COM,

$${\left( {{\left\{ {a_{1} ,a_{2} ,a_{3} } \right\}},\Pi _{i} } \right)} \sim {\left( {{\left\{ {b_{1} ,b_{2} ,b_{3} } \right\}},\Pi _{j} } \right)}$$

and so on. Finally we have:

$${\left( {{\left\{ {a_{1} , \ldots ,a_{g} } \right\}},\Pi _{i} } \right)} \sim {\left( {{\left\{ {b_{1} , \ldots ,b_{g} } \right\}},\Pi _{j} } \right)},$$

i.e.,

$${\left( {max_{i} {\left( A \right)},\Pi _{i} } \right)} \sim {\left( {max_{j} {\left( B \right)},\Pi _{j} } \right)}$$

(2)

Since ⪰ satisfies INF and COM, we can apply Lemma 6.1 and obtain

$${\left( {A,\Pi _{i} } \right)} \sim {\left( {max_{i} {\left( A \right)},\Pi _{i} } \right)}$$

(3)

and

$${\left( {B,\Pi _{j} } \right)} \sim {\left( {max_{i} {\left( B \right)},\Pi _{j} } \right)}$$

(4)

Now, Eqs. (2), (3), (4), and transitivity of ⪰ imply (A, Π _i )∼(B, Π _j ).

Now we show that

$${\left| {\max _{i} {\left( A \right)}} \right|} > {\left| {\max _{j} {\left( B \right)}} \right|} \Rightarrow {\left( {A,\Pi _{i} } \right)} \succ {\left( {B,\Pi _{j} } \right)}.$$

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),

$${\left( {{\left\{ {a_{1} , \ldots ,a_{g} } \right\}},\Pi _{i} } \right)} \sim {\left( {B,\Pi _{j} } \right)}$$

(5)

Now, max _i ({a ₁,...,a _g+1})={a ₁,...,a _g+1}. By ARA,

$${\left( {{\left\{ {a_{1} , \ldots ,a_{{g + 1}} } \right\}},\Pi _{i} } \right)} \succ {\left( {{\left\{ {a_{1} , \ldots ,a_{g} } \right\}},\Pi _{i} } \right)}$$

and, by (5) and transitivity of ≻,

$${\left( {{\left\{ {a_{1} , \ldots ,a_{{g + 1}} } \right\}},\Pi _{i} } \right)} \succ {\left( {B,\Pi _{j} } \right)}.$$

By adding a _g+2,...,a _g+t successively, and by using ARA repeatedly, we have

$${\left( {{\left\{ {a_{1} , \ldots ,a_{{g + t}} } \right\}},\Pi _{i} } \right)} \succ {\left( {B,\Pi _{j} } \right)}$$

i.e.,

$${\left( {max_{i} {\left( A \right)},\Pi _{i} } \right)} \succ {\left( {B,\Pi _{j} } \right)}.$$

(6)

We know from Lemma 4.1 that

$${\left( {A,\Pi _{i} } \right)} \sim {\left( {max_{i} {\left( A \right)},\Pi _{i} } \right)}.$$

Clearly, (A, Π _i )∼(max _i (A),Π _i ), (6) and transitivity of ⪰ imply (A, Π _i )≻(B,Π _j ). Q.E.D.