Minimax decision rules for planning under uncertainty

TL;DR

This paper analyzes minimax decision rules, especially minimax regret, in planning under uncertainty with finite scenarios, highlighting their advantages and sensitivity issues, and examining their behavior in different decision spaces.

Contribution

It provides a detailed analysis of minimax rules' behavior with finite scenarios, including their sensitivity and potential for manipulation, in both convex and finite decision spaces.

Findings

Minimax rules avoid probability assessments but are sensitive to scenario choices.

Minimax regret rules can violate independence of irrelevant alternatives.

Analysis covers decision variables in convex sets and finite choices.

Abstract

It is common to use minimax rules to make decisions for planning when there is great uncertainty on what will happen in the future. Minimax regret is one popular version of this. We give an analysis of the behaviour of minimax rules in the case with a finite set of possible future scenarios. The use of minimax rules avoids the need to determine probabilities for each scenario, which is an attractive feature in many public sector settings. However, minimax rules will have sensitivity to the choice of scenarios. In many cases using a minimax approach will mean the requirement for what may be regarded as arbitrary probabilities on scenarios is replaced by a similarly arbitrary choice of a very small number of specific scenarios. We investigate this phenomenon. When regret-based rules are used there are also problems arising since the independence of irrelevant alternatives property fails, which can lead to opportunities to game the process. Our analysis of these phenomena considers cases where the decision variables are chosen from a convex set in $R^n$, as well as cases with a finite set of decision choices.