Abstract
We extend the notion of natural extension, that gives the least committal extension of a given assessment, from the theory of sets of desirable gambles to that of choice functions. We give an expression of this natural extension and characterise its existence by means of a property called avoiding complete rejection. We prove that our notion reduces indeed to the standard one in the case of choice functions determined by binary comparisons, and that these are not general enough to determine all coherent choice function. Finally, we investigate the compatibility of the notion of natural extension with the structural assessment of indifference between a set of options.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
This is not an extension of a rejection function defined on a smaller domain \(\mathcal {B}\) to a bigger domain \(\mathcal {Q}_0\). Rather, it is the extension of an assessment, where we do not necessarily know all the rejected options in every option set
in \(\mathcal {B}\) (except for 0).
References
Arrow, K.: Social Choice and Individual Values. Yale University Press (1951)
Uzawa, H.: Note on preference and axioms of choice. Ann. Inst. Stat. Math. 8, 35–40 (1956)
Seidenfeld, T., Schervish, M.J., Kadane, J.B.: Coherent choice functions under uncertainty. Synthese 172(1), 157–176 (2010)
Rubin, H.: A weak system of axioms for “rational” behavior and the nonseparability of utility from prior. Stat. Risk Model. 5(1–2), 47–58 (1987)
Van Camp, A., Miranda, E., de Cooman, G.: Lexicographic choice functions. Int. J. Approx. Reason. 92, 97–119 (2018)
Van Camp, A., de Cooman, G., Miranda, E., Quaeghebeur, E.: Coherent choice functions, desirability and indifference. Fuzzy Sets Syst. 341(C), 1–36 (2018). https://doi.org/10.1016/j.fss.2017.05.019
Aizerman, M.A.: New problems in the general choice theory. Soc. Choice Welfare 2, 235–282 (1985)
Sen, A.: Social choice theory: a re-examination. Econometrica 45, 53–89 (1977)
He, J.: A generalized unification theorem for choice theoretic foundations: avoiding the necessity of pairs and triplets. Economics Discussion Paper 2012–23, Kiel Institute for the World Economy (2012)
Schwartz, T.: Rationality and the myth of the maximum. Noûs 6(2), 97–117 (1972)
Sen, A.: Choice functions and revealed preference. Rev. Econ. Stud. 38(3), 307–317 (1971)
Quaeghebeur, E.: Desirability. In: Augustin, T., Coolen, F.P.A., de Cooman, G., Troffaes, M.C.M. (eds.) Introduction to Imprecise Probabilities, pp. 1–27. Wiley, Hoboken (2014)
de Cooman, G., Quaeghebeur, E.: Exchangeability and sets of desirable gambles. Int. J. Approx. Reason. 53(3), 363–395 (2012). Precisely imprecise: a collection of papers dedicated to Henry E. Kyburg, Jr
Bradley, S.: How to choose among choice functions. In: Augustin, T., Doria, S., Miranda, E., Quaeghebeur, E. (eds.) Proceedings of ISIPTA 2015, pp. 57–66. Aracne (2015)
Seidenfeld, T.: Decision without independence and without ordering: what is the difference? Econ. Philos. 4, 267–290 (1988)
Van Camp, A., de Cooman, G.: Exchangeable choice functions. In: Antonucci, A., Corani, G., Couso, I., Destercke, S. (eds.) Proceedings of ISIPTA 2017. Proceedings of Machine Learning Research, vol. 62, pp. 346–357 (2017)
Acknowledgements
The research in this paper has been supported by project TIN2014-59543-P.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Van Camp, A., Miranda, E., de Cooman, G. (2018). Natural Extension of Choice Functions. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 854. Springer, Cham. https://doi.org/10.1007/978-3-319-91476-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-91476-3_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91475-6
Online ISBN: 978-3-319-91476-3
eBook Packages: Computer ScienceComputer Science (R0)