On estimating the determinate equivalent of a lottery (Q1284336)

Suppose a so called utility function \(u(x)\) is given on the interval \([-v, 0]\). Let us denote by \[ u(\widetilde{x}) = \int_{-v}^{0} u(x) d\varphi (x) \tag{1} \] an expected utility of the lottery (of a random quantity \(\widetilde{x}\)) which has a distribution function \( \varphi (x)\) on the interval \([- v , 0]\). Usually the utility function is not known but it is possible to reconstruct its values in some points of the interval. A quantity \(\widehat{x}( u) = u^{-1} ( u(\widetilde{x}))\) is called a determinate equivalent of the lottery \(x\). A problem of searching of the value \[ \alpha = \inf_{u\in U} \widehat{x} (u), \tag{2} \] where \(U\) are sets of all utility functions \( u( x):u ( -x_{0}) = u_{0}\), \(-x_{0} \in ( - v, 0)\), is considered in the article. The value \(\alpha\) is a lower boundary of the determinate equivalent of the lottery and the problem of its evaluation (estimation) is equivalent to the problem of finding roots of an equation. Among the other applications of the estimation method proposed may be some problems of insurance companies.