Abstract
We study the geometrical properties of the utility space (the space of expected utilities over a finite set of options), which is commonly used to model the preferences of an agent in a situation of uncertainty. We focus on the case where the model is neutral with respect to the available options, i.e. treats them, a priori, as being symmetrical from one another. Specifically, we prove that the only Riemannian metric that respects the geometrical properties and the natural symmetries of the utility space is the round metric. This canonical metric allows to define a uniform probability over the utility space and to naturally generalize the Impartial Culture to a model with expected utilities.
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- 1.
This is not always the case: for example, Gibbard [7] considers voters with expected utilities over the candidates.
- 2.
The necessary and sufficient condition is that relation \(\le \) is complete, transitive, archimedean and independent of irrelevant alternatives.
- 3.
Technically, this remark proves that \(\mathcal {U}_m\) (with its natural quotient topology) is not a \(T_1\) space [8].
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The work presented in this paper has been partially carried out at LINCS (http://www.lincs.fr).
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Durand, F., Kloeckner, B., Mathieu, F., Noirie, L. (2015). Geometry on the Utility Space. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_12
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DOI: https://doi.org/10.1007/978-3-319-23114-3_12
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