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Quantifying Degrees of E-admissibility in Decision Making with Imprecise Probabilities

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Reflections on the Foundations of Probability and Statistics

Part of the book series: Theory and Decision Library A: ((TDLA,volume 54))

Abstract

This paper is concerned with decision making using imprecise probabilities and looks at extensions and aspects of the criterion of E-admissibility, as introduced by Levi and extensively studied and advocated by Teddy Seidenfeld. In the first part, we introduce a decision criterion that allows for explicitly modeling how far maximal decisions in Walley’s sense are accepted to deviate from E-admissibility. We also provide an efficient and simple algorithm based on linear programming theory for this criterion. In the second part of the paper, we propose two measures for quantifying what we call the extent of E-admissibility of an E-admissible act, i.e. the size of the set of measures for which the corresponding act maximizes expected utility. The first measure is the maximal diameter of this set, while the second one relates to the maximal barycentric cube that can be inscribed into it. Also here, for both measures, we give linear programming algorithms capable to deal with them. Finally, we discuss some ideas in the context of ordinal decision theory. The paper concludes with a stylized application example illustrating all introduced concepts.

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Notes

  1. 1.

    The one exception is the discussion in Sect. 13.4, where we do not assume a cardinal utility representation.

  2. 2.

    See Schervish et al. (2013) for the situation where multiple utilities through different currencies are available and exchange rates have to be taken into account.

  3. 3.

    It should, however, be emphasized that in the context of imprecise probabilistic models (like for instance credal sets or interval probabilities) the relationship between optimal decision functions in terms of prior risk and posteriori loss optimal acts may be more subtle than in the context of precise probability: the main theorem of Bayesian decision theory may fail (cf., e.g., Augustin (2003, Section 2.3)). This failure is in essence a variant of the general phenomenon of potential sequential incoherence in decision making and discrepancy between extensive and normal forms, as investigated in depth by Seidenfeld (e.g., Seidenfeld (1988, 1994)). Immediate counter-examples arise from the phenomenon of dilation, which has intensively been studied by Seidenfeld and co-authors (cf., e.g., Seidenfeld (1994), Seidenfeld and Wassermann (1993), Wassemann and Seidenfeld (1994)), see also, e.g., Hailin (2015).

  4. 4.

    See, however, e.g., Seidenfeld et al. (1989), Wheeler (2012), Majo-Wilson and Wheeler (2016, Section 2), and the references therein, for arguments to consider also non-convex sets of probabilities.

  5. 5.

    Note that if \(\mathcal {M}\) is described by functions (f 1, …, f r) and bounds \((( \underline {b_1}, \overline {b}_1) , \dots , ( \underline {b_r}, \overline {b}_r))\) not meeting this assumptions, we can always equivalently characterize it by functions (f 1 − c 1, …, f r − c r) and bounds \((( \underline {b_1}-c_1, \overline {b}_1-c_1) , \dots , ( \underline {b_r}-c_r, \overline {b}_r -c_r))\), where, for all s = 1, …, r, we set \(c_s= \underline {b}_s \) if \( \underline {b}_s> 0\) and \(c_s=-\overline {b}_s \) if \( \overline {b}_s < 0\) and c s = 0 if \(0 \in [ \underline {b}_s, \overline {b}_s]\). This assumption is technically convenient in the context of the optimization results later in the paper, since it allows to directly specify an admissible solution whenever \(\mathcal {M}\) is interpreted as the set of (potential) admissible solutions of some linear programming problem.

  6. 6.

    See, for instance, Kofler and Menges (1976) and Gilboa and Schmeidler (1989). Many authors denote \(\mathcal {M}\) by Γ, and thus the term Γ-maximin is common as well.

  7. 7.

    This criterion is mainly advocated by Walley (1991) and work following him.

  8. 8.

    Compare also Seidenfeld et al. (2010) and (Vicig and Seidenfeld 2012, Section 3).

  9. 9.

    This criterion is introduced by Levi (1974).

  10. 10.

    Both criteria just discussed are also of high interest in forecasting with imprecise probabilities. While for imprecise probabilities there is no real-valued strictly proper scoring rule, it is possible to formulate an appropriate lexicographic strictly proper scoring rule with respect to \(\mathcal {M}\)-maximinity and E-admissibility, supplemented by \(\mathcal {M}\)-maximinity (Seidenfeld et al. (2012)).

  11. 11.

    Note that considering criteria additional to E-admissibility seems very natural, since, by introducing the criteria of P- and S-admissibility, this was already done in the original work of Levi (1974).

  12. 12.

    Note that, due to standard results from linear programming theory, such an optimal solution always exists since the constraint set is bounded and there always exists an admissible solution since \(a_z \in \mathbb {A}_{\mathcal {M}}\).

  13. 13.

    Another, very prominent, way for proceeding in such situations is working with partially cardinal preference relations as done in Seidenfeld et al. (1995).

  14. 14.

    For a simple example consider the decision problems \((\mathbb {A} ,\Theta ,u)\) and \((\mathbb {A}\cup \{a^*\} , \Theta , \tilde {u})\) given by

    and the prior π on Θ induced by (π({θ 1}), π({θ 2}), π({θ 3})) = (0.2, 0.2, 0.6). Here we have that \(D^{\mathbb {A}}_{\pi }(a_1)=0.6 >0.4= D^{\mathbb {A}}_{\pi }(a_2)\) but \(D_{\pi }^{\mathbb {A}^*}(a_2)=0.4 > 0 = D_{\pi }^{\mathbb {A}^*}(a_1)\).

  15. 15.

    The calculation was performed with the rcdd package (see Geyer and Meeden (2019)), which provides an interface for using Fukuda (2018)’s cdd library in the R statistical computing environment (see R Core Team (2020)).

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Acknowledgements

We are grateful to two referees for their very helpful comments and to Sebastien Destercke for valuable discussions about decision criteria. Moreover, we want to express our gratitude to Jean Baccelli, Seamus Bradley, Stephan Hartmann, Arthur Merin, Jürgen Landes, Gregory Wheeler and many other (former) members of the Munich Center for Mathematical Philosophy for stimulating talks and discussions on decision theory and imprecise probabilities during the last years, and to Teddy Seidenfeld for inspiring lectures and discussions not only during his visits to Munich.

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  1. Department of Statistics, Ludwig-Maximilians-University of Munich, Munich, Germany

    Christoph Jansen, Georg Schollmeyer & Thomas Augustin

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  1. Christoph Jansen
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Correspondence to Christoph Jansen .

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Editors and Affiliations

  1. Department of Statistics, Ludwig-Maximilians-University of Munich, Munich, Germany

    Thomas Augustin

  2. Universidade de São Paulo, São Paulo, Brazil

    Fabio Gagliardi Cozman

  3. Frankfurt School of Finance & Management, Frankfurt am Main, Germany

    Gregory Wheeler

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Jansen, C., Schollmeyer, G., Augustin, T. (2022). Quantifying Degrees of E-admissibility in Decision Making with Imprecise Probabilities. In: Augustin, T., Cozman, F.G., Wheeler, G. (eds) Reflections on the Foundations of Probability and Statistics. Theory and Decision Library A:, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-031-15436-2_13

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