Constructing Preference Relations from Utilities and Vice Versa

In decision making, the suitability of decision options may be specified by utility values for each option or by preference values for each pair of options. This paper introduces a new approach to construct a matrix of pairwise preference values from a vector of utility values, which is called the U2P transformation. The U2P transformation satisfies reciprocity, triangle condition, weak transitivity, restricted max–min transitivity, and restricted max–max transitivity, but it violates (unrestricted) max–min transitivity, (unrestricted) max–max transitivity, multiplicative transitivity, and additive transitivity. Inversion of the U2P transformation yields the P2U transformation which can be used to construct a vector of utility values from a matrix of pairwise preference values. Numerical experiments with movie ratings illustrate the practical use of the U2P and P2U transformations.