CSM/KSS'2005Knowledge Creation and Integration for Solving Complex Problems

August 29-31, 2005, IIASA, Laxenburg, Austria ______________________________________

A Multi-Attribute, Non-Expected Utility Approachto Complex Problems of Optimal Decision Making under Risk

Gebhard Geiger

Fakultät für WirtschaftswissenschaftenTechnische Universität München

Stiftung Wissenschaft und Politik(German Institute for International and Security Affairs)

Berlin

Germany

 

Overview

Application context of multi-attribute utility theory (MAUT)

Model assumptions

Utility model of optimal choice under risk

MAUT (expected utility, or EU)

Non-EU MAUT Non-EU MAUT: example with two-attributes Properties of present account of Non-EU MAUT Applications

Application context of MAUT: complex problems of optimal choice under risk

• Risk assessment in complex, large-scale (industrial, environmental, ...) systems

• Risk management must be effective and cost-efficient → demand for quantitative structuring multi-attribute risk preferences

• Risk management: optimisation of several objectivessimultaneously

Trade-offs between attributes of decision consequences. . .

Model assumptions

• Mathematical model building leading toincreased precision, logical coherence

• Statistics, probability theory

• Instrumentalist view of model building in the domain of rational human behaviour

• Engineering optimal choice under risk

Utility model of optimal choice under risk

Action under risk

• Uncertainty about the (quantitative, monetary, ...) consequences of actor´s choices

• Characterised as performance of random experiments (or “lotteries“)

• Uncertain outcome (or “risk“) is random variable X with probability distribution p (discrete), or density function f (continuous), and real value x

• Decision alternatives constitute set C = {p, q, ...} of lotteries associated with real random variables X, Y, ... (discrete case)• Risk assessment in terms of “utility functional“ U: C ℝ so that

U(p) U(q) actor prefers p to q, or is indifferent between p and q

Special cases

Expected utility (EU) with “utility“ function u(x) (von Neumann & Morgenstern, 1944)

U(p) = Eu(p) = in u(xi)p(xi), in p(xi) = 1, p(xi) 0

Non-expected utility with probability-dependent utility function u(p, x) (Becker & Sarin, 1987; Schmidt, 2001; Geiger, 2002)

U(p) = in u(p, xi)p(xi) dx

MAUT (EU)

• m 1 measureable attributes X1, ..., Xm of outcomes of risky decisions• Joint (discrete) probability function p(x1, ..., xm) with marginals pi(xi) • Utility function u(x1, ..., xm) in the EU case so that

Eu(p) = i ,j ,... u(xi1, ..., xj

m)p(xi1, ..., xj

m)

Important special cases (Keeney & Raiffa, 1976)

Utility independence of each attribute Xi of all the other attributes

u(x1, ..., xm) = i kiui(xi) + i j>i kijui(xi) uj(xj) + ...

+ k12...m u1(x1) ... um(xm)

Mutual utility independence of Xi and Xj, i j

u(x1, ..., xm) = i kiui(xi) + k2 i,j>i kikjui(xi) uj(xj) + ...

+ km-1k1k2 ... km u1(x1) ... um(xm)

Additivity

u(x1, ..., xm) = i kiui(xi)

Non-EU MAUT

Problem: MAUT (EU) extremely difficult to operationalise

• Single-attribute utility functions ui(xi) hard to specify

• “Consistent scaling“ of the ui‘s cumbersome (specification of

many constants k, ki, kij, ..., k12...m)

• Consistent scaling of non-expected (e. g., probability-dependent), multi-attribute utility generally much more cumbersome than in EU case

Simplification

Single-attribute utility functions ui(pi, xi)

• ui´s completely specified in the present approach

• ui´s cconsistently scaled by construction

Non-EU MAUT: example with two attributes (m=2)

X1 utility independent of X2

u(p, x1, x2) = u2(p2, x2) + u1(p1, x1) (u2(p2, x2) – cu´2(p´ 2, x´ 2) + 1)

x´ 2 = x2 –1, p´ 2 (x´ 2) = p2(x2 – 1)c = –1 – u2(δa, a)

δa = 1 if x2 = a, and δa = 0 otherwise(–1, –1) and (0, a) are indifferent (riskless case!)

Mutual utility independence of X1 and X2

Scaling of u(p, x1, x2)u(p, –1, x2) = cu2(p2, x2) – 1u(p, x1, –1) = u1(p1, x1) – 1

u(p, x1, x2) = u2(p2, x2) + u1(p1, x1)(u2(p2, x2) – cu2(p2, x2) + 1)

Additivity of X1 and X2 : c = 1

u(p, x1, x2) = u1(p1, x1) + u2(p2, x2)

Properties of present account of Non-EU MAUT

• Appropriate representation of the multi-attribute risks inherent in the management of complex systems

• Exploits the applicability and reality properties and computational simplicity of underlying single-attribute Non-EU theory

• Admits straightforward decomposition of multi-attribute utility functions into single-attribute utility components

• Aviods notorious problems of consistent scaling of single-attribute components in MAU assessments of risk

VALUE

LOSSES GAINS

Kahneman & Tversky, 1979 (observed)Utility function (theoretical),

after Geiger, 2002

ApplicationsSocial costs of fatality risk in complex industrial systems (m=2)

-20 -10 0 10 20-10

-8

-6

-4

-2

0

2

4

6

8

u(p, x)

x

Social costs of fatality risk in complex industrial systems (m=2)Case of maximally acceptable involuntary social fatality risk

Non-EU MAUT: Equivalent social costs as a concave function of loss (fatality)(after Geiger, 2005)

x ~ y1/3 (Starr)

x ~ y0.56 (Otway & Cohen)

0

x =

u X

-1(p

, - y

)

-y = u X (p, x)

x < 0

-y < 0 x > 0

-y > 0

Convex (single-attribute case) and convcave (two-attribute case) utility function

-20 -10 0 10 20-10

-8

-6

-4

-2

0

2

4

6

8

u(p, x)

x

Selected literature

Keeney, R. L. and Raiffa, H. (1976) Decisions with Multiple Objectives, New York: John Wiley.

Kahneman, D. and Tversky, A.: (1979), Prospect Theory: An Analysis of Decision under Risk, Econometrica 47, pp. 763-791.

Machina, M. J.: (1987), Decision-Making in the Presence of Risk, Science 236, pp. 537-543.

Beaudouing, F., Munier, B. and Serquin, Y.: (1999). Multi-Attribute Decisison Making and Generalized Expected Utility Theory in Nuclear Power Plant Maintenance. In: Machina, M. J. and Munier, B. (eds), Preferences, Beliefs, and Attributes in Decision Making. Dordrecht: Kluwer.

Geiger, G.: (2002), On the Statistical Foundations of Non-Linear Utility Theory: The Case of Status Quo-Dependent Preferences, European Journal of Operational Research 136: 459-465.

Geiger, G.: (2004), Risk Acceptance from Non-Linear Utility Theory, Journal of Risk Research 7: 225-252.

Example of catastrophic risk

After US-Canada Power System Outage Task Force, Interim Report: Causes of the August 14th Blackout in the United States and Canada, Washington DC and Ottawa, November 2003, p. 67.

acceptable

unacceptable

n = 2

n = 1

unacceptable