Comparison of linear interpolation and arctan approximation of one-dimensional monotonic utility functions based on experimental data

Abstract

Elicitation of utilities is among the most time consumingtasks in decision analysis. We search for ways to shortenthis phase without compromising the quality of results. We use theresults from an empirical experiment with 104 participants. Theyelicited 9 inner nodes from their one-dimensional utility functionover monetary gains and losses using three elicitation techniques.A specific feature of the results is their interval character, as theelicitators are fuzzy rational individuals. The data is used to constructarctan-approximated and linearly interpolated utilities andto compare the results. We form partial samples with 3, 4 and 5nodes for each participant and each elicitation method, and againinterpolate/approximate the utilities. We introduce goodness-of-fitand deterioration measures to analyze the decrease in quality ofthe utility function due to reduced data nodes. The analysis, usingpaired-sample tests, leads to the following conclusions: 1) arctanapproximationis more adequate than linear interpolation over thewhole samples; 2) 5 inner nodes are sufficient to construct a satisfactoryarctan-approximation; 3) arctan-approximation and linearinterpolation are almost equal in quality over the partial samples,but the local risk aversion of the linearly interpolated utility functionis of poor quality unlike that of the arctan-approximated utilityfunction.

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