# Certainty Equivalent and Utility Indifference Pricing for Incomplete Preferences via Convex Vector Optimization

## TL;DR

This paper introduces a framework for defining and computing certainty equivalents and indifference prices for incomplete preferences using convex vector optimization, extending existing models to multivariate and multi-prior cases.

## Contribution

It develops set-valued certainty equivalents and indifference prices for incomplete preferences, generalizing previous models and providing computational methods.

## Key findings

- Properties of utility buy and sell prices mirror those in complete preferences.
- Set-valued certainty equivalents can be computed via convex vector optimization.
- Numerical examples illustrate economic interpretations for univariate and multivariate cases.

## Abstract

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 buy and sell prices and 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.