Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization

For incomplete preference relations that are represented by multiple priors and/or multiple-possibly multivariate-utility functions, we define a certainty equivalent as well as the utility indifference price bounds as set-valued functions of the claim. Furthermore, we motivate and introduce the notion of a weak and a strong certainty equivalent. We will show that our definitions contain as special cases some definitions found in the literature so far on complete or special incomplete preferences. We prove monotonicity and convexity properties of utility buy and sell prices that hold in total analogy to the properties of the scalar indifference prices for complete preferences. We show how the (weak and strong) set-valued certainty equivalent as well as the indifference price bounds can be computed or approximated by solving convex vector optimization problems. Numerical examples and their economic interpretations are given for the univariate as well as for the multivariate case.

Automated summary

This study focuses on incomplete preference relations represented by multiple priors and/or multiple-possibly multivariate-utility functions. It defines a certainty equivalent and utility indifference price bounds as set-valued functions of the claim, and introduces weak and strong certainty equivalents. The definitions are shown to encompass some found in the literature on complete or special incomplete preferences, with monotonicity and convexity properties demonstrated for utility buy and sell prices in analogy to scalar indifference prices. Computational methods for the (weak and strong) set-valued certainty equivalent and indifference price bounds are discussed, with numerical examples and economic interpretations provided for both univariate and multivariate cases.