The Hurwicz Decision Rule’s Relationship to Decision Making with the Triangle and Beta Distributions and Exponential Utility

Abstract

Non-probabilistic approaches to decision making have been proposed for situations in which an individual does not have enough information to assess probabilities over an uncertainty. One non-probabilistic method is to use intervals in which an uncertainty has a minimum and maximum but nothing is assumed about the relative likelihood of any value within the interval. The Hurwicz decision rule in which a parameter trades off between pessimism and optimism generalizes the current rules for making decisions with intervals. This article analyzes the relationship between intervals based on the Hurwicz rule and traditional decision analysis using a few probability distributions and an exponential utility functions. This article shows that the Hurwicz decision rule for an interval is logically equivalent to: (i) an expected value decision with a triangle distribution over the interval; (ii) an expected value decision with a beta distribution; and (iii) an expected utility decision with constant absolute risk aversion with a uniform distribution. These probability distributions are not exhaustive. There are likely other distributions and utility functions for which equivalence with the Hurwicz decision rule can also be established. Since a frequent reason for the use intervals is that intervals assume less information than a probability distribution, the results in this article call into question whether decision making based on intervals really assumes less information than subjective expected utility decision making.Abstract Non-probabilistic approaches to decision making have been proposed for situations in which an individual does not have enough information to assess probabilities over an uncertainty. One nonprobabilistic method is to use intervals in which an uncertainty has a minimum and maximum but nothing is assumed about the relative likelihood of any value within the interval. The Hurwicz decision rule in which a parameter trades off between pessimism and optimism generalizes the current rules for making decisions with intervals. This article analyzes the relationship between intervals based on the Hurwicz rule and traditional decision analysis using a few probability distributions and an exponential utility functions.This article shows that the Hurwicz decision rule for an interval is logically equivalent to: (i) an expected value decision with a triangle distribution over the interval; (ii) an expected value decision with a beta distribution; and (iii) an expected utility decision with constant absolute risk aversion with a uniform distribution. These probability distributions are not exhaustive. There are likely other distributions and utility functions for which equivalence with the Hurwicz decision rule can also be established. Since a frequent reason for the use intervals is that intervals assume less information than a probability distribution, the results in this article call into question whether decision making based on intervals really assumes less information than subjective expected utility decisio...

From the Editors: Decision Analysis Focus and Trends

“…Papers providing a balanced comparison of several decision methodologies would also be welcome. A recent example of a paper that provides a decision-analytic perspective on a nondecision analytic method is Sivaprasad and MacKenzie (2018), showing the conditions under which a wellknown nonprobabilistic decision rule yields results equivalent to those of expected-value or expectedutility decision theory.…”