1. CSM/KSS'2005 Knowledge Creation and Integration for Solving Complex Problems August 29-31, 2005, IIASA, Laxenburg, Austria ______________________________________ A Multi-Attribute, Non-Expected Utility Approach to Complex Problems of Optimal Decision Making under Risk Gebhard Geiger Fakultät für Wirtschaftswissenschaften Technische Universität München Stiftung Wissenschaft und Politik (German Institute for International and Security Affairs) Berlin Germany

2. 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

3. 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 objectives simultaneously  Trade-offs between attributes of decision consequences...

4. Model assumptions Mathematical model building leading to increased precision, logical coherence Statistics, probability theory Instrumentalist view of model building in the domain of rational human behaviour Engineering optimal choice under risk

5. 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) = E u (p) =  i  n u(x i )p(x i ),  i  n p(x i ) = 1, p(x i )  0 Non-expected utility with probability-dependent utility function u(p, x) (Becker & Sarin, 1987; Schmidt, 2001; Geiger, 2002) U(p) =  i  n u(p, x i )p(x i ) dx

6. MAUT (EU) m  1 measureable attributes X 1,..., X m of outcomes of risky decisions Joint (discrete) probability function p(x 1,..., x m ) with marginals p i (x i ) Utility function u(x 1,..., x m ) in the EU case so that E u (p) =  i, j,... u(x i 1,..., x j m )p(x i 1,..., x j m ) Important special cases (Keeney & Raiffa, 1976) Utility independence of each attribute X i of all the other attributes u(x 1,..., x m ) =  i k i u i (x i ) +  i  j>i k ij u i (x i ) u j (x j ) +... + k 12...m u 1 (x 1 )... u m (x m ) Mutual utility independence of X i and X j, i  j u(x 1,..., x m ) =  i k i u i (x i ) + k 2  i, j>i k i k j u i (x i ) u j (x j ) +... + k m-1 k 1 k 2... k m u 1 (x 1 )... u m (x m ) Additivity u(x 1,..., x m ) =  i k i u i (x i )

7. Non-EU MAUT Problem: MAUT (EU) extremely difficult to operationalise Single-attribute utility functions u i (x i ) hard to specify “Consistent scaling“ of the u i ‘s cumbersome (specification of many constants k, k i, k ij,..., k 12...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 u i (p i, x i ) u i ´s completely specified in the present approach u i ´s cconsistently scaled by construction

8. Non-EU MAUT: example with two attributes (m=2) X 1 utility independent of X 2 u(p, x 1, x 2 ) = u 2 (p 2, x 2 ) + u 1 (p 1, x 1 ) (u 2 (p 2, x 2 ) – cu´ 2 (p´ 2, x´ 2 ) + 1) x´ 2 = x 2 –1, p´ 2 (x´ 2 ) = p 2 (x 2 – 1) c = –1 – u 2 (δ a, a) δ a = 1 if x 2 = a, and δ a = 0 otherwise (–1, –1) and (0, a) are indifferent (riskless case!) Mutual utility independence of X 1 and X 2 Scaling of u(p, x 1, x 2 ) u(p, –1, x 2 ) = cu 2 (p 2, x 2 ) – 1 u(p, x 1, –1) = u 1 (p 1, x 1 ) – 1 u(p, x 1, x 2 ) = u 2 (p 2, x 2 ) + u 1 (p 1, x 1 )(u 2 (p 2, x 2 ) – cu 2 (p 2, x 2 ) + 1) Additivity of X 1 and X 2 : c = 1 u(p, x 1, x 2 ) = u 1 (p 1, x 1 ) + u 2 (p 2, x 2 )

9. 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

10. Kahneman & Tversky, 1979 (observed) Utility function (theoretical), after Geiger, 2002 Applications Social costs of fatality risk in complex industrial systems (m=2)

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

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

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

14. 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.

15. 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