Robust decisions under endogenous uncertainties and risks Y. Ermoliev,

1. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) Robust Decisions, December 10-12 2007,  IIASA, Laxenburg, Austria Robust decisions under endogenous uncertainties and risks Y. Ermoliev, T. Ermolieva, L. Hordijk, M. Makowski

2. Collaborative work with IIASA’s projects • Energy and technology • Forestry • Global climate change and population • Integrated modeling • Land use • Risk and Vulnerability • Case studies on catastrophic risk management • Earthquakes (Italy, Russia) • Floods (Hungary, Ukraine, Poland, Japan) • Livestock production and disease risks (China) • Windstorms (China)

3. Overviews and further references • Y. Ermoliev, V. Norkin, 2004. Stochastic Optimization of Risk Functions via Parametric Smoothing. In K. Marti, Y. Ermoliev, G. Pflug (Eds.) Dynamic Stochastic Optimization, Springer Verlag, Berlin, New York. • T. Ermolieva, Y. Ermoliev, 2005. Catastrophic risk management: flood and seismic risk case studies. In Wallace, S.W. and Ziemba, W.T., Applications of Stochastic Programming, SIAM, MPS. • Fischer, G., Ermolieva, T., Ermoliev, Y., and van Velthuizen, H. (2006). Sequential downscaling methods for Estimation from Aggregate Data” In K. Marti, Y. Ermoliev, M. Makovskii, G. Pflug (Eds.) Coping with Uncertainty: Modeling and Policy Issue, Springer Verlag, Berlin, New York. • A. Gritsevskii, N. Nakichenovic, 1999.Modeling uncertainties of induced technical change, Energy policy, 28. • Y. Ermoliev, L. Hordjik, 2006. Global changes: Facets of robust decisions. In K. Marti, Y. Ermoliev, M. Makovskii, G. Pflug (Eds.) Coping with Uncertainty: Modeling and Policy Issue, Springer Verlag, Berlin, New York, 2006. • B. O’ Neill, Y. Ermoliev, and T. Ermolieva, 2006. Endogenous Risks and Learning in Climate Change Decision Analysis. In K. Marti, Y. Ermoliev, M. Makovskii, G. Pflug (Eds.) Coping with Uncertainty: Modeling and Policy Issue, Springer Verlag, Berlin, New York, 2006.

4. Concepts of robustness • Term ‘robust’ was coined in statistics, Box, 1953 • true sampling model of uncertainty P • insensitivity of estimates to assumptions on P • Robust statistics, Huber, 1964 • continuity w.r.t outlyers: uniform convergence of estimates • for small perturbation of P • Local stability solutions of differential eqs. • Bayesian robustness • ranges of posterior expected “losses” • Minimax (non-Bayesian) ranking • endogenous random priors from • Optimal deterministic control • local stability of optimal trajectories

5. Statistical decision theory deals with situations in which the model of uncertainty and the optimal solution are defined by a sampling model P with an unknown vector of true parameters Vector defines the desirable optimal solution, its performance can be observed from the sampling model and the problem is to recover from these data. Decision problems under uncertainty • The general problems of decision making under uncertainty deal with • fundamentally different situations. The model of uncertainty, feasible, • solutions, and performance of the optimal solution are not given and all • of these have to be characterized from the context of the decision • making situation, e.g., socio-economic, technological, environmental, • and risk considerations. As there is no information on true optimal • performance, robustness cannot be also characterized by a distance • from observable true optimal performance. Therefore, the general • decision problems may have rather different facets of robustness.

6. Global changes (including global climate changes) pose new methodological challenges • affect large territories, communities, and activities • require proper integrated modeling of socio-economic and environmental processes (spatio-temporal, multi-agent, technological, etc.) • a key issue: inherent uncertainty and potential “unknown” endogenous catastrophic risks, discontinuities • Path-dependencies, increasing returns require forward-looking policies • exact evaluation vs robust policies • Integrated climate assessment models: A. Manne and R. Richels (1992), W. Nordhaus (1994). Typical conclusions: • damage/losses are not severe enough • adaptive “wait-and-see” solutions

7. Standard modeling approaches (a new bumper to the old car) • aggregate indicators (production and utility functions, GDP) • spatial heterogeneity ? • average global temperature vs extreme events • exogenous TC, convexity (incremental market adaptive adjustments) • discounting • standard exogenous risks • IIASA’s studies (A. Gritsevski, N. Nakicenovic, A. Grubler and Y. Ermoliev, 1994-1998) • technological perspectives, interdependencies, interlinkages • endogenous TC, uncertainties and risks (VaR and CVaR –type) • increasing returns (non-convexity) of new technologies • Conclusions: earlier investments lead to CO2 stabilization at the same • costs as the cost of future carbon intensive energy systems

8. a b c Technological change under increasing returns Increasing (a) Constant (b) Diminishing (c) • J. Schumpeter (1942): Technological changes occur due to local search • of firms for improvements and imitations of practices of other firms • B. Arthur, Y. Ermoliev, Y. Kaniovski (1983, Cybernetics). Outcomes of natural • myopic evolutionary rules are uncertain. The convergence takes place, • but where it settles depends completely on earlier (even small) random movements. • Results may be dramatic without strong policy guidance. • A. Gritsevskyi, N. Nakicenovic, A. Gruebler, Y. Ermoliev (1994-1998) • The design of proper robust policy is a challenging STO problem • Critical importance of uncertainty • Non-convexities (markets ?) • Proper random time horizons • Bottom-up modeling • Conclusions:earlier investments have the greatest impact vs wait-and-see

9. A.Gritsevskyi & N. Nakicenovic, 2000

10. Projected surface of risk-adjusted cost function

11. Implementation • Cray T3E-900 at National Energy Research Scientific Computer Center, US • 640-processor machine with a peak CPU performance of 900 MFlops per processor • C/C++ with MPI 2.0

12. Intuition. Simple models. Does it work? cost b - x ) ( d - a x d ( ) d production - quantile of d defined by slopes , : (VaR) F(xrob) = CVaR • Production (emission reduction) x = demand “ “ • Overshooting-and-undershooting costs • Scenario analysis Robust solution = Ed ?

13. Ignorance of potential catastrophic risks • Methodological reasons K. Arrow: Catastrophes Don’t Exist in standard economic models • Decisions makers, Politicians demand simple answers A “magic” number, scenario • Scenario thinking • Extreme events are simply characterized by (expected) intervals 1000 year flood, 500 year wind storm, 107 year nuclear disaster, which are viewed as improbable events during a human life

14. Adaptive scenario simulators: earthquakes

15. Adaptive scenario simulators: floods

16. 100 80 Monetary or natural units 60 40 10 9 8 20 7 6 5 4 3 0 2 1 2 3 4 5 6 7 8 9 10 Locations Initial landscape of values

17. Scenarios of damaged values

18. Insurer 2 Insurer 1 Initial spread of coverage: standard feasible decisions

19. Initial spread: high risk of bankruptcies

20. Insurer 1 Insurer 2 Robust spread of coverage: new feasible decisions

21. Bankruptcies Robust spread: reduced risk of bankruptcies

22. Discontinuities, stopping time Typical random scenarios of growth (decline) under shocks Deterministic (average) scenarios are linear (red) functions They ignore a vital variability (discontinuity, ruin) and can not be used for designing robust strategies

23. Can we use average values • Expected costs, average incomes • Need for median and other quantiles • Non-additive characteristics, collapse of separability and linearity Applicability of Standard Models and Methods • Deterministic equivalent • Expected utility models, NPV, CBA • Intervals uncertainties • Bellman’s equations, Pontriagin’s Principle Maximum and other similar decompositions schemes

24. (Discontinuity, e.g., ) - contingent (ex-ante) credit Robust risk management • Safety (chance) constraints (Convexification) (1996) - ex-post borrowing • CVaR measure of risk (min of quantiles)

25. Discontinuity: Illustrative Example If , and otherwise , - number of jumps • Catastrophe model • Potential disaster at • A shut down” (stopping time) decision • Performance function (Fast and slow components) • Deterministic (sample mean) approximation • SQG: - Fast adaptive Monte-Carlo simulators

26. (probability density) → Dirak function, , strongly l.s.c. - Random vectors k Parametric smoothing • Averaged (generalized) functions: Steklov (1907), Sobolev (1930), Kolmogorov (1934), … - theory of distributions (Shwartz, 1966) • Optimization (Ermoliev, Gaivoronski, Gupal, Katkovnic, Lepp, Marti, Norkin, Wets, … ) - Independent of dimensionality , , • Parametric smoothing fast estimation of functions and derivatives

27. - a risk process • Fast Monte-Carlo optimization • Convergence for • Stochastic Processes with Stopping Time • Y.M. Ermoliev and V.I. Norkin, Stochastic optimization of risk functions, in K. Marti, Y. Ermoliev, and G. Pflug (eds.): Dynamic stochastic optimization, Springer, 2004, pp. 225-249

28. For geometric discounting • Discounting

29. Integrated catastrophe management models River Module Spatial Inundation Module Vulnerability Module Losses of households,farmers, producers, water authorities, governments, Feasible decisions Evaluation of decisions with respect to goals, constraints Histograms of losses and gains Multi-agent accounting system

30. Optimization module: structural and non-structural decisions, premiums, coverage, contingent credit, production allocation, … Decisions x - Stopping time - Property value - Scenario of loss - Premium, - credit - Risk reserve, • gov. compensation • Ins. contract Adaptive Monte Carlo STO Procedure • Structural and non-structural decisions

31. Robust strategies • Proper treatment of “uncertainties – decisions – risks” interactions • there is no true model of uncertainty • decisions (in contrast to estimates) affect uncertainty, and risks e.g., • CO2 emissions • Proper models and methods • singularity (discontinuity) w.r.t. “outlyers” (rare catastrophic risks) • importance of stochastic vs probabilistic minimax • standard extreme events theory deals with i.i.d.r.v. • spatial and temporal distributional heterogeneity (growth, wealth, incomes, risks) • discontinuinity, stopping time, spatio-temporal risk measures, multi agent aspects • system’s risk, discounting • Proper concept of solutions • risks modify feasible sets of solutions • flexibility: anticipation-and-adaptation, ex-ante - and - ex-post, risk • averse – and – risk taking • - ex-post options require ex-ante decisions

32. Typical “performance” of the goal function: “Learning” – by simulation: Adaptive Monte Carlo procedure