Abstract
In a menu-choice framework with a compact metric space of alternatives, I prove a representation of preference for flexibility that features continuous ex-post utility functions as well as a continuous aggregator over ex-post utility levels. Continuity of the ex-post utility functions ensures that each function attains its maximum on any compact menu of alternatives. This, in turn, supports the standard interpretation that relates the value of flexibility to the maximization of ex-post preferences. I also show that an incomplete menu-preference is characterized by the Pareto order induced by a collection of aggregators, in place of a single one.
I wrote the first draft of this paper in 2007, when I was a Ph.D. student at New York University, which circulated under the title “An Ordinal, Continuous Representation Theorem for Preference for Flexibility.” I owe special thanks to Efe Ok, who was my advisor at the time. I also thank Barton Lipman, Pietro Ortoleva, Clemens Puppe, Debraj Ray, Gil Riella, two anonymous referees and the editors of this volume for useful discussions and suggestions.
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- 1.
To mention a few examples, the proof of Gul and Pesendorfer’s [20] representation appeals to the von Neumann–Morgenstern representation theorem, whereas Dekel, Lipman and Rustichini [8] build upon some classical results in functional analysis such as the Hahn-Banach extension theorem and the Riesz representation theorem.
- 2.
For instance, the independence axiom of Dekel, Lipman and Rustichini [8] combined with the monotonicity axiom implies Kreps’ [24] submodularity property [8, Footnote 21]. While the independence axiom becomes irrelevant in an ordinal setup, submodularity is still indispensable, which arguably makes the latter more substantive as regards the notion of preference for flexibility.
- 3.
Representations of this sort arise in the analysis of preference for flexibility and related notions such as liquidity demand for money (see, e.g., [17, 18]). Multi-preference approach to the measurement of value of freedom of choice is another related area of research (see, e.g., [2]). Bleichrodt and Quiggin [3] provide a more recent application to health economics.
- 4.
“ If ...\(\dot{\succsim }\) is represented ...with \(U(\cdot ,s)\) continuous for each s..., then \(\dot{\succsim }\) is continuous on \(X^{c}\). The converse seems reasonable: If \(\dot{\succsim }\) is continuous on \(X^{c}\), then a representation with continuous \(U(\cdot ,s)\) and u is possible. But I am unable to supply a proof of the converse—U as constructed in the proof of the theorem will be lower semi-continuous only” (Kreps [24, p. 575]). Here, \(X^{c}\) stands for the collection of menus, \(\dot{\succsim }\) is the DM’s preference relation over menus, u denotes the aggregator, and \(U(\cdot ,s)\) is a state-dependent utility function.
- 5.
- 6.
In Kochov’s model, a prior refers to a probability distribution over subjective states, or state-dependent utility functions. Each prior defines an expectation operator over ex-post utility levels, which corresponds to an additive aggregator in the present setup (see Footnote 8).
- 7.
The Hausdorff distance between \(A,B\in \mathcal {K} (\mathcal {X})\) equals the maximum of \(\max _{a\in A}\min _{b\in B}\sigma (a,b)\) and \(\max _{b\in B}\min _{a\in A}\sigma (a,b),\) where \(\sigma \) stands for the metric on \(\mathcal {X}\).
- 8.
For a finite set of utility functions U, a (monotonic) additive aggregator \(\varphi \) is defined by nonnegative numbers \(\alpha _{u} (u\in U)\) such that \(\varphi \left( \max _{A}U\right) =\sum _{u\in U}\alpha _{u}\max _{a\in A}u(a)\) for every \(A\in \mathcal {K} (\mathcal {A})\). When U is infinite, the summation operator is replaced by the expectation operator induced by a probability measure on U.
- 9.
More generally, Gorno [19] shows that any complete and transitive preference relation on nonempty subsets of a finite set can be represented with an additive aggregator over state-dependent utility levels. The distinctive feature of Kreps’ version is that each state-dependent utility function has a nonnegative weight, in line with the monotonicity axiom (see Footnote 8).
- 10.
Specifically, for every \(\varphi \in \Phi \), we can find a finite capacity \(\eta _{\varphi }\) on the set U such that for every \(A\in \mathcal {K}(\mathcal {A}),\) \(\varphi \left( \max _{A}U\right) =\int _{0}^{\infty }\eta _{\varphi }\left( \left\{ u\in U:\max _{a\in A}u(a)\ge \alpha \right\} \right) d\alpha .\) (A finite capacity \(\eta \) on U is a real valued function on the collection of all subsets of U such that \(\eta (\emptyset )=0,\) \(\eta (U)<\infty \), and \(\eta (V)\le \eta (W)\) whenever \(V\subseteq W\subseteq U.\))
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Evren, Ö. (2020). Preference for Flexibility: A Continuous Representation in an Ordinal Setup. In: Bosi, G., Campión, M., Candeal, J., Indurain, E. (eds) Mathematical Topics on Representations of Ordered Structures and Utility Theory. Studies in Systems, Decision and Control, vol 263. Springer, Cham. https://doi.org/10.1007/978-3-030-34226-5_14
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