Linear Transformation of One-Dimensional Utility Functions: Empirical Study on the Impact on the Final Ranking of Alternatives in Personal Decisions

Determining one-dimensional utility functions for each objective in multiattribute utility theory takes time and effort from decision makers. They must consider including a decreasing or increasing marginal utility and/or their relative risk attitude, resulting in a nonlinear shape. This assessment is prone to errors and distortions. We analyze to what extent a linear transformation of one-dimensional utility functions compromises the quality of the decision. Therefore, we examine the impact of one-dimensional utility functions on the final ranking of alternatives in practice, focusing on three aspects: the use of (non)linear utility functions, their impact on the ranking of alternatives, and the stability of best alternatives concerning utility differences of alternatives assuming linear transformation. We examine 2,536 carefully modeled personal decisions analyzed by students with the decision support tool Entscheidungsnavi. Our results show that 95.9% of the participants used at least one nonlinear utility function in their decision, and 76.4% of all objectives were evaluated with nonlinear utility functions. Simplifying preference-accurate utility functions with linearization led to a rank reversal of the best alternative in 15.5% of the decisions. The top-three set of alternatives changed in 14% of the decisions. In 98.8% of the decisions, the best alternative could be found in the top three alternatives ranked under linearity. Based on our results, we recommend determining the utility functions preference-accurately using (non)linearity to model the decision as precisely as possible, especially for important decisions. However, no rank reversal for the best alternative was detected in our data set from an absolute utility difference greater than 0.27 between the best and second best alternatives under linearity. In these cases, assuming linear utility functions is useful if decision makers want to save time and effort.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/deca.2024.0317.