Neutral freedom and freedom as control

To say that we should be neutral about the value of alternatives is not to say that no value judgments are involved. See Sect. 3.3. Neutrality is conditioned on certain normative presuppositions.

In his work on measuring freedom, Carter (1999) emphasizes the distinction between the value of specific freedoms and value of overall freedom, and proposes a measure of overall freedom that abstracts from the values of specific freedoms.

Sen (1985, 1991) also discussed the relation of freedom to flexibility.

This is an exaggeration.

The version of the axioms I present here are slightly different than that presented in Dekel et al. (2001). The monotonicity axiom is the same as in Dekel et al. (2001), but the continuity and independence axioms differ. The independence axiom of Dekel et al. (2001) differs in that (1) it is formulated with respect to strict preference and (2) it contains only the \(\Rightarrow\) direction of Axiom 4 below. In conjunction, the axioms presented here characterize the same preferences as the axioms in Dekel et al. (2001). The reason for the difference is that my main result, Theorem 1, appeals to a representation theorem for preference for flexibility with incomplete preferences due to Kochov (2007) and Galaabaatar (2010), and so I use the versions of independence and monotonicity used in those papers. The differences only matter when preferences are incomplete.

The formulation of the axiom is different than in Dekel et al. (2001). See footnote 6.

The theorem continues to hold despite the slightly different formulation of the axioms. See footnote 6.

The representation (3) is not unique—multiple probability measures p correspond to the same preferences \(\preccurlyeq\). Dekel et al. (2001) restrict attention to a subset of utility functions U satisfying certain normalizations and thereby attain a unique representation. Uniqueness is not important for my purposes.

This example previously appeared in Sher (2018a).

I mention this last fact just to emphasize that in Scenarios A and B below, if the agent goes to prison, it will be the same prison in both scenarios.

In this example we identify the options c and s with the degenerate lotteries that put probability one on each of these two options.

It is not entirely obvious because one might say that both menus allow no freedom, and so are equivalent in that sense. A counterargument would be that bliss is much more choiceworthy than suffering; there are many more situations in which an agent would or should choose bliss rather than suffering if given the choice. Moreover, even if one accepts that the singleton menus \(\left\{ \textit{suffering}\right\}\) and \(\left\{ \textit{bliss}\right\}\) are equivalent in terms of freedom in that they allow no freedom, one might not accept the stronger implication of neutrality that it is equivalent in terms of freedom to interchange suffering and bliss in every menu. In arguing against the cardinality order of Pattanaik and Xu (1990), Sen (1990) argues similarly that the choiceworthiness of alternatives is relevant to the freedom allowed by a menu.

Sher (2015) is an earlier working paper version of the current paper. The weakenings of the neutrality axiom can be found in Sect. 6 of that version. The paper can be found at https://​drive.​google.​com/​file/​d/​18ORLZcVTlKnGHYy​nSNLvGkkfYmoRJkG​Q/​view?​usp=​sharing.

Nehring and Puppe (2008) also applies the attribute approach to freedom of choice.

Sher (2018a) does not explicitly consider applying neutrality within each category of freedom; the paper focuses rather on assigning different values to different freedoms (either in the same or in different categories). The approach would however be consistent with applying neutrality within categories.

For example, consider the 27 possible ways of assigning the utilities \(-1,0,\) and 2 to the different candidates allowing that different candidates may be assigned the same utility. If each of these 27 assignments is viewed as equiprobable, that would amount to a different situation of symmetric uncertainty.

For the formal definition of what it means to take a weighted average of menus, see (2) in Sect. 2.2, and the discussion surrounding (2).

Let us consider two particular neutral judges who disagree about these two menus. When neutral judge 1’s specific preferences are realized, judge 1 assigns utility 1 to one alternative and utility 0 to the other two alternatives. For each of the three alternatives, there is a \(\frac{1}{3}\) probability that judge 1 will assign utility 1 to that alterntative. When neutral judge 2’s specific preferences are realized, judge 2 assigns utility 0 to one alternative and utility 1 to the other two alternatives. For each of the three alternatives, there is a \(\frac{1}{3}\) probability that neutral judge 2 will assign utility 0 to that alternative. Judge 1 attains an expected utility of \(\frac{2}{3}\) from having access to \(M_1\) and an expected utility of \(\frac{7}{9}\) from having access to \(M_2\). So judge 1 prefers \(M_2\). Judge 2 attains an expected utility of 1 from having access to \(M_1\) and an expected utility of \(\frac{8}{9}\) from having access to \(M_2\). So judge 2 prefers \(M_1\). Judge 1 and judge 2 disagree.

See in particular Sect. 3.2 and Corollary 1.

The repetitions occur because each of the two lotteries contain only two distinct probabilities—in this case 0 and 1. If one of the two lotteries contained three distinct probabilities, as in the menu \(M=\left\{ \left( 1,0,0\right) ,\left( \frac{1}{2},\frac{1}{3},\frac{1}{6}\right) \right\}\), there would be no repetitions.

Some red points on the right hand side can be arrived at in multiple ways. This is a general feature which is not particular to the menu in this example because for any menu M and all \(\ell \in M\), \(\frac{1}{n!} \sum _{\pi \in \Pi } \ell ^\pi = \left( \frac{1}{n},\frac{1}{n},\ldots ,\frac{1}{n}\right)\).

It follows that if one rejects some comparison made by the symmetry order, one must reject one of the axioms characterizing freedom.

The symmetry order is referred to as the grading order in Sher (2015).

The cardinality order ranks one deterministic menu above another if and only if the former contains more elements.

A preorder \(\preccurlyeq\) is a reflexive and transitive relation.

A preorder \(\preccurlyeq _0\) is coarser than a preorder \(\preccurlyeq _1\) if for all \(M,M' \in {\mathscr {M}}, M' \preccurlyeq _0 M \Rightarrow M' \preccurlyeq _1 M\).

For example, if \(M_2\) contains two lotteries and \(M_3\) contains three lotteries, but the convex hull of \(M_3\) is a proper subset of the convex hull of \(M_2\), then \(M_2\) allows more freedom than \(M_3\) according to the symmetry order.

\(z \le z'\) means that for all i, \(z_i \le z'_i\).

In the case that the voting mechanism is deterministic (i.e., \(\alpha (z) \in \{0,1\}, \forall z\)) , it is obvious that (9) represents the probability of being pivotal. In the Appendix, I explain why we can also regard (9) as the probability of being pivotal in stochastic mechanisms.

In Sher (2018a), I present a general framework for evaluating freedom in games.

In this paper, I have assumed that a menu is a closed set of lotteries over alternatives. In (12), I allow all subsets of \(\left[ 0,1\right]\). If I had here restricted to closed subsets of \(\left[ 0,1\right]\), the \(\sup\) in (12) could be replaced by \(\max\).

The critical thing here is not the menus contain two lotteries, but rather that the lotteries in the menus are over two outcomes.

Strictly speaking the condition is that all expected utility maximizers prefer P to Q given any prior beliefs.

Since the preordered flexibility axioms imply that a menu is indifferent to its convex hull, nothing substantive would change if we substituted the set of randomized strategies for the set of pure strategies \(\Sigma\) used in the above construction.

See Sect. 6.

Specifically, only the \(\Leftarrow\) direction employs the separating hyperplane theorem. If \(S\left( M\right) \not \subseteq S\left( M'\right)\), then there exists \(\ell \in S\left( M\right) {\setminus } S\left( M'\right)\), and then the strong separating hyperplane theorem implies that \(\ell\) and \(G(M')\) can be strongly separated.

It is straightforward to confirm that when \(p^\pi\) is defined in this way, then \(p^\pi\) indeed belongs to \(\Delta ^*\left( U\right)\).