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Open Questions in Utility Theory

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Mathematical Topics on Representations of Ordered Structures and Utility Theory

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 263))

Abstract

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature.

In honour of G. B. Mehta on occasion of his 75th birthday.

Also dedicated to the memory of our colleague Gerhard Herden.

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Notes

  1. 1.

    In other quite different and multidisciplinary contexts, this is sometimes called an isotony, an entropy function, a measurement, a score or an order-preserving map. (See e.g. [8, 10, 17, 18, 30, 31, 43, 49, 77]).

  2. 2.

    This property is also known as the translation-invariance of the total order \(\precsim \) as regards the binary operation \(\circ \). Notice that, in particular, a totally ordered semigroup is always cancellative, namely \(s \circ u = t \circ u \Leftrightarrow s = t \Leftrightarrow u \circ s = u \circ t \ \ (s,t,u \in S).\)

  3. 3.

    Despite we are working with totally ordered semigroups, it can be proved that we could actually be working with a totally preordered semigroup, where \(\precsim \) is a total preorder but not necessarily a linear order (i.e.: the binary relation \(\precsim \) could fail to be antisymmetric). When this happens, we might pass to be working with a quotient space \(S/\sim \) whose elements are the indifference classes of the elements of S with respect to \(\sim \). That is, given \(s \in S\), its corresponding class is the set \(\lbrace t \in S: t \sim s \rbrace \). Provided that there is a compatibility between the total preorder \(\precsim \) and the binary operation \(\circ \) such that \(s \precsim t \Leftrightarrow s \circ u \precsim t \circ u \Leftrightarrow u \circ s \precsim u \circ t\) holds for every \(s,t,u \in S\), it is easy to see that \(S/\sim \) inherits a structure of totally ordered semigroup by considering in a natural way that the binary operation \(\circ \) as well as \( \precsim \) directly act on the indifference classes that \(\sim \) induces on S.

    By this reason, in what follows we will be working with totally ordered semigroups, instead of just totally preordered semigroups, unless otherwise stated.

  4. 4.

    In this setting, a mapping f with these properties is said to be an additive utility function.

  5. 5.

    Here on \(S \times S\) we will consider the product topology coming from \(\tau \) on S.

  6. 6.

    Notice that this is, so-to-say, a theorem about “automatic continuity”. It actually states that on a totally ordered group \((G,\circ ,\precsim )\), both the operation \(\circ \) and the unary operation of taking an inverse are, a fortiori, continuous as regards the order topology \(\tau _{\precsim }\).

  7. 7.

    A total preorder \(\precsim \) on \((X,+,\cdot _{\mathbb {R}},*)\) is said to be non-zero provided that there are \(\bar{x}, \bar{y}\in X\) such that \(\mathbf{0}\prec \bar{x}*\bar{y}\).

  8. 8.

    It is usual that the map \(\mu _X\) and the corresponding fuzzy set X are used interchangeably if this does not give rise to confusion.

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Acknowledgements

This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE).

We are grateful to two reviewers for their valuable suggestions and comments.

Thanks are also given to the organizers and participants in the congress SUMTOPO 2019, 34th Summer Conference on Topology and its Applications, University of the Witwatersrand, Johannesburg (South Africa) 1–4 July 2019, for their helpful discussions and comments on the contents of our contribution, a substantial part of which was presented there.

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  1. Departamento de Estadística, Informática y Matemáticas, Inarbe (Institute for Advanced Research in Business and Economics), Universidad Pública de Navarra, 31006, Pamplona, Spain

    María J. Campión

  2. Departamento de Estadística, Informática y Matemáticas, InaMat (Institute for Advanced Materials), Universidad Pública de Navarra, 31006, Pamplona, Spain

    Esteban Indurain

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  1. María J. Campión
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  1. Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Università degli Studi di Trieste, Trieste, Italy

    Gianni Bosi

  2. Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra, Pamplona, Navarra, Spain

    María J. Campión

  3. Facultad de Economía y Empresa, Departamento de Análisis Económico, Universidad de Zaragoza, Zaragoza, Spain

    Juan C. Candeal

  4. Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra, Pamplona, Spain

    Esteban Indurain

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Campión, M.J., Indurain, E. (2020). Open Questions in Utility Theory. 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_3

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