Abstract
This paper gives a way of analyzing decisions in the case of unknown utility function, or more precisely, when we know only a linear order on an income space. It is shown that in this situation, decisions and corresponding probability measures are partially ordered, and this order is identical to the inclusion relation of comonotone fuzzy sets. It enables us to use inclusion indices of fuzzy sets to analyze the comparability of decisions. To do this, we introduce an inclusion index having properties, which are close to ones of the classical expected utility.
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Notes
Actually, this order \(\preceq \) on probability measures coincides with the first order stochastic dominance if \(R\) is a real line.
One can find the proof of this proposition in (Bronevich and Karkishchenko (2002), Theorem 5, p. 192).
We can define the inclusion index for general case as \(\psi (F_1 \subseteq F_2 ) = \int \nolimits _0^1 {w(p)\psi _p (F_1 \subseteq F_2 )dp} \), where \(w\) is a non-negative weight function with \(\int \nolimits _0^1 {w(p)dp} = 1\). Here we assume that \(w(p) = 2p\). This assumption leads to the properties of inclusion index, which are close to ones of the classical expected utility functional.
This theorem follows directly from Property 3 given in the next section.
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Bronevich, A.G., Rozenberg, I.N. Ranking probability measures by inclusion indices in the case of unknown utility function. Fuzzy Optim Decis Making 13, 49–71 (2014). https://doi.org/10.1007/s10700-013-9169-6
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DOI: https://doi.org/10.1007/s10700-013-9169-6