[Paper Review] Least worst regret analysis for decision making under uncertainty, with applications to future energy scenarios

Stan Zachary|arXiv (Cornell University)|Aug 2, 2016

Risk and Portfolio Optimization4 references3 citations

TL;DR

This paper demonstrates that least worst regret (LWR) and minimax decision analyses are dominated by two 'extreme' scenarios whose cost functions determine the outcome, regardless of other scenarios. For convex cost functions, LWR results are highly sensitive to these two scenarios and can be replicated by Bayesian analysis with specific probability weights, showing robustness in the UK electricity capacity planning case study.

ABSTRACT

Least worst regret (and sometimes minimax) analysis are often used for decision making whenever it is difficult, or inappropriate, to attach probabilities to possible future scenarios. We show that, for each of these two approaches and subject only to the convexity of the cost functions involved, it is always the case that there exist two "extreme" scenarios whose costs determine the outcome of the analysis in the sense we make clear. The results of either analysis are therefore particularly sensitive to the cost functions associated with these two scenarios, while being largely unaffected by those associated with the remainder. Great care is therefore required in applications to identify these scenarios and to consider their reasonableness. We also consider the relationship between the outcome of a least worst regret and a Bayesian analysis, particularly in the case where the regret functions associated with the scenarios largely differ from each other by shifts in their arguments, as is the case in many applications. We study in detail the problem of determining an appropriate level of electricity capacity procurement in Great Britain, where decisions must be made several years in advance, in spite of considerable uncertainty as to which of a number of future scenarios may occur, and where least worst regret analysis is currently used as the basis of decision making.

Motivation & Objective

• To investigate the structural sensitivity of least worst regret (LWR) and minimax decision-making under uncertainty.
• To identify the conditions under which LWR results are dominated by only two scenarios, regardless of the full set of possible future scenarios.
• To explore the relationship between LWR and Bayesian decision-making, particularly when regret functions differ by shifts in arguments.
• To assess the robustness of LWR outcomes in real-world applications, specifically in UK electricity capacity planning.
• To provide guidance on scenario selection and reasonableness assessment in LWR applications.

Proposed method

• Theoretical analysis of LWR and minimax criteria under convex cost functions, proving that only two extreme scenarios determine the outcome.
• Use of regret functions derived by subtracting the minimum cost over the decision set from each scenario’s cost function.
• Identification of 'extreme' scenarios as those with the most divergent regret function arguments, particularly when regret functions differ by shifts.
• Numerical experimentation to compare LWR results with Bayesian outcomes under various probability assignments.
• Application of the method to the 2015 UK Electricity Capacity Report, using real data from National Grid plc.
• Derivation of probability weights that reproduce LWR results under Bayesian analysis, using the condition that regret functions are approximately shifted versions of each other.

Experimental results

Research questions

• RQ1 Which two scenarios, among a set of possible future scenarios, are primarily responsible for determining the outcome of a least worst regret (LWR) analysis?
• RQ2 Under what conditions does the LWR result become insensitive to all but two scenarios, regardless of the number of scenarios considered?
• RQ3 Can a Bayesian analysis be constructed such that its optimal decision matches the LWR result, and if so, under what probability assignments?
• RQ4 How does the structural form of regret functions—particularly when they differ by shifts in arguments—affect the sensitivity of LWR outcomes?
• RQ5 To what extent is the LWR result for UK electricity capacity procurement robust to different probability assignments in a Bayesian framework?

Key findings

• For convex cost functions, the outcome of both LWR and minimax analysis is determined solely by two 'extreme' scenarios, regardless of the number of scenarios considered.
• The LWR result for Great Britain’s electricity capacity procurement is 47.8 GW, which matches Bayesian results when probabilities of 0.68 and 0.32 are assigned to the two dominant scenarios.
• Assigning uniform probabilities to the five major scenarios in the 2015 Electricity Capacity Report yields a Bayesian result of 47.7 GW, very close to the LWR result.
• Excluding the DECC scenario increases the Bayesian result to 48.0 GW, while including all 19 scenarios also yields 48.0 GW, indicating robustness across probability assignments.
• The LWR result is robust because the regret functions are approximately shifted versions of each other, and the scenarios are relatively evenly spaced between their extremes.
• A Bayesian analysis with probabilities assigned to only the two extreme scenarios (0.68 and 0.32) replicates the LWR result exactly, confirming the theoretical equivalence under these conditions.