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Theory and Decision

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04.12.2019

The two faces of independence: betweenness and homotheticity

verfasst von: Daniel R. Burghart

Erschienen in: Theory and Decision | Ausgabe 4/2020

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Abstract

This paper shows that expected utility belongs to the intersection of models satisfying betweenness and a homotheticity condition for risky choice. Betweenness models can accommodate variable risk attitudes, originally highlighted by the Allais paradox, by restricting indifference curves to be linear while allowing non-parallelism. Homotheticity, in contrast, restricts indifference curves to be parallel while permitting non-linearities, such as those highlighted by inverse-S probability weighting. Data from an experiment indicate that approximately 2/3s of participants satisfied homotheticity. Of this group, about half satisfied the expected utility hypothesis. The other half violated expected utility and therefore betweenness. These results indicate that homothetic preferences, and in particular those violating betweenness, can be empirically relevant for decision making under risk.

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Having the worst outcome serve as the origin and homothetic center has strong appeal. In traditional microeconomic analysis, the worst commodity bundle available is the origin (i.e., \( \mathbf {0} \in \mathbb {R}^L_+\)) and is also the homothetic center.

As discussed more thoroughly in Sect. 1.4, Diecidue et al. (2009), Safra and Segal (1998), and Grant and Kajii (1998) all examine models that invoke some form of homotheticity, but not betweenness. To the extent that citation counts are indicative of the attention that this class of models have received, Google Scholar (as of early July, 2017) indicates that these three papers have jointly accumulated only 165 citations (49, 93, and 23, respectively). In contrast, disappointment aversion theory (Gul 1991) and weighted utility (Chew 1983), which are in the class of models that satisfy betweenness but not homotheticity, have garnered more than 1100 and 600 citations, respectively. In the class of non-betweenness and non-homothetic models, two popular examples are rank-dependent utility (Quiggin 1982) and cumulative prospect theory (Tversky and Kahneman 1992). These two papers have received more than 2700 and 10,500 citations, respectively.

Specifically, the experimental design in this paper included lotteries only in the strict interior of a specific subdomain of the probability space. See Sect. 2 for details.

The proof for this proposition was generously provided by the editor.

Note that such a pair \(\alpha \) and \(\beta \) can always be found with at most one of them different from unity.

Appendix B.5 contains specific details about the experimental design including an exhaustive listings of all the lotteries available in each choice situation, general construction details, and the instructional packet that was given to participants.

Two notable studies are by Hey and Orme (1994) and Camerer and Ho (1994).

At the time of the experiment 1 CHF was equivalent to 1.05 USD.

The details of the synthetics’ joint sampling distribution of HM-IA are in Appendix B.4.

The nature of joint sampling distributions and the discrete values taken on by the HM-IA gives rise to a critical value of 12 for HM-IA when considering marginal distributions.

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Titel: The two faces of independence: betweenness and homotheticity
verfasst von: Daniel R. Burghart
Publikationsdatum: 04.12.2019
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2020
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-019-09735-2

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