Decision-network polynomials and the sensitivity of decision-support models

Highlights

• We study the sensitivity of decision-support models.

• We introduce the notion of decision-network polynomials.

• The absence of differentiability at indifference points is investigated in detail.

• We utilize finite change sensitivity indices for providing managerial insights.

• The representation of certainty equivalents is discussed.

Abstract

Decision makers benefit from the utilization of decision-support models in several applications. Obtaining managerial insights is essential to better inform the decision-process. This work offers an in-depth investigation into the structural properties of decision-support models. We show that the input–output mapping in influence diagrams, decision trees and decision networks is piecewise multilinear. The conditions under which sensitivity information cannot be extracted through differentiation are examined in detail. By complementing high-order derivatives with finite change sensitivity indices, we obtain a systematic approach that allows analysts to gain a wide range of managerial insights. A well-known case study in the medical sector illustrates the findings.

Introduction

Decision trees, influence diagrams, Bayesian networks and event trees support the solution of decision analysis problems in several applications. Their use is nowadays facilitated by a number of software programs (see Bielza et al., 2011, Jensen and Nielsen, 2007).¹ Computer implementation enables analysts to develop sophisticated codes that incorporate a variety of aspects of the problems under investigation (Dillon, Paté-Cornell, & Guikema, 2003). However, model complexity exposes analysts and decision-makers to the risk of a partial understanding of the model input–output response. Then, deriving insights about the structure of the model becomes essential to make robust conclusions and inferences.

Researchers have developed methods for exploring the informational content of decision-support models. For Bayesian networks several of the most recent findings rest on the fundamental result that the input-mapping in Bayesian networks is a multilinear polynomial (Castillo et al., 1996, Castillo et al., 1997). For instance, multilinearity is crucial for arithmetic and decision circuits (Bhattacharjya and Shachter, 2012, Darwiche, 2003). However, there are unique challenges in sensitivity analysis for influence diagrams due to the non-linearities created by maximization operations for making decisions (Bhattacharjya & Shachter, 2010, p. 1). Indeed, this issue is transversal to all decision-support models that include a maximization (or minimization) operator. The maximization operation induces piecewise-definiteness and impairs differentiation. Thus, properties of Bayesian networks cannot be transferred directly to models such as influence diagrams, decision trees and decision circuits. The difficulties associated with differentiation then raise broader issues concerning the methodology for deriving managerial insights from decision-support models.

In this work, we conduct a systematic investigation into the mathematical properties of decision-support models. We show that the decision-theoretical principles underlying their construction (Savage, 1954) lead to a piecewise defined input–output mapping. Each piece corresponds to an available strategy and is multilinear in probabilities and utilities. We call the mapping a decision-network polynomial.

Non-differentiability occurs at those values of the model inputs (probabilities/utilities) for which the decision-maker is indifferent among alternative strategies. Using the terminology of Howard (1968) (see also Bhattacharjya & Shachter, 2010),² this result suggests that differentiation finds its natural application in an open-loop analysis, i.e., when the preferred strategy (or any other strategy) is under scrutiny. In a closed-loop analysis differentiation might not be possible. We then employ sensitivity measures based on an orthogonal decomposition via finite difference operators (Borgonovo, 2010), that do not require differentiability. These finite change sensitivity indices allow us to obtain a clear understanding of the model response, by apportioning the change in expected utility to the individual effects and interactions among the model inputs.

The next step is the analysis of the model structure when consequences are monetary and certainty equivalents are the output of interest. Findings show that the input–output mapping remains piecewise-defined but each piece is composite multilinear. We link high order derivatives of certainty equivalents polynomials analytically to the derivatives of the corresponding decision-network polynomials. Finite change sensitivity indices acquire a direct interpretation as the monetary gain (loss) associated with the model input variations.

The well known case-study of Felli and Hazen (2004) helps us illustrating how the approach can be used for: (i) understanding direction of change, (ii) quantifying the relevance of interactions; and (iii) identifying the model inputs on which to focus managerial attention during implementation (Eschenbach, 1992, pp. 40-41).

The remainder of the paper is organized as follows. Section 2 offers a literature review. Section 3 presents decision-network polynomials. Section 4 investigates the link between indifference and differentiation. Section 5 discusses the presence of interactions and specializes finite change sensitivity indices to the case of decision-network polynomials. Section 6 discusses results for certainty equivalents. Section 7 illustrates the derivation of decision-making insights through a case study in the medical sector. Section 8 offers conclusions. All proofs are in Appendix A.