Robust decisions underendogenous uncertainties and risks

Y. Ermoliev,

T. Ermolieva, L. Hordijk, M. Makowski

IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) Robust Decisions, December 10-12 2007,  IIASA, Laxenburg, Austria

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• 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)

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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.

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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

(1 )P q , q Q , 0 , 0

Bayesian robustness

( , ) ( | )L x d ,

Minimax (non-Bayesian) ranking

Optimal deterministic control

continuity w.r.t outlyers: uniform convergence of estimatesfor small perturbation of P

Local stability solutions of differential eqs.

ranges of posterior expected “losses”

endogenous random priors from

local stability of optimal trajectories

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Decision problems under uncertainty

The general problems of decision making under uncertainty deal withfundamentally different situations. The model of uncertainty, feasible, solutions, and performance of the optimal solution are not given and

allof 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 generaldecision problems may have rather different facets of robustness.

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.

* *

*

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• 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

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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 samecosts as the cost of future carbon intensive energy systems

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Technological change under increasing returns

Increasing (a)Constant (b)Diminishing (c)

J. Schumpeter (1942): Technological changes occur due to local searchof firms for improvements and imitations of practices of other firms

B. Arthur, Y. Ermoliev, Y. Kaniovski (1983, Cybernetics). Outcomes of naturalmyopic 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

a

b

c

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A.Gritsevskyi & N. Nakicenovic, 2000

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Projected surface of risk-adjusted cost function

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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

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Intuition. Simple models. Does it work? Production (emission reduction) x = demand

Overshooting-and-undershooting costs)(dx

d production

cost

)

(

dx

)

(x

d

Scenario analysis

... dx dx dx ddd optoptopt ,,,,...,, 332211321 Robust solution = Ed ?

- quantile of d defined by slopes , :

robx

][ dxP (VaR) F(xrob) = CVaR

)(d

“ “

)()()()()( dxIdxExdIxdExF )}(),(max{ dxxdE

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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

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

Scenario thinking

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Adaptive scenario simulators: earthquakes

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Adaptive scenario simulators: floods

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Initial landscape of values

0

20

40

60

80

100

Mo

net

ary

or

nat

ura

l u

nit

s

23456

78910

1

Locations

2 3 4 5 6 7 8 9 10

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Scenarios of damaged values

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Insurer 2Insurer 1

Initial spread of coverage: standard feasible decisions

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Initial spread: high risk of bankruptcies

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Insurer 2Insurer 1

Robust spread of coverage: new feasible decisions

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Robust spread: reduced risk of bankruptcies

Bankruptcies

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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

Discontinuities, stopping time

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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

...)()(...)( 2121 medianmedianmedian

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Robust risk management

1]0),([ xgP

)}],(,0max{),(min[ xgNExEf

• Safety (chance) constraints

- ex-post borrowing

(1996)

ymin

1]),([ yxgP

}]),(,0max{),(min[ yxgNEyxEf

• CVaR measure of risk

y - contingent (ex-ante) credit

(Convexification)

(min of quantiles)

(Discontinuity, e.g., )]0[ xP

),(min xEf

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- Potential disaster at

- A shut down” (stopping time) decision

- Performance function

xba

xaxxf

,

,),(

x

N- number of jumps ,

- Fast adaptive Monte-Carlo simulators

NxfxF Nk k

N /),()( 1

),()()]0,(),([),( xfxxxfxxfxSQG x

})(:inf{)( AtRt

Discontinuity: Illustrative Example Catastrophe model

)()()( xEfIxEfIxF

),()()]0,(),([)( xEfxxxfxxfxgradF x

axfx ),( x 0),( xfx If , and otherwise

Deterministic (sample mean) approximation

SQG:

(Fast and slow components)

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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, … )

Parametric smoothing

fast estimation of functions and derivatives

)()()()()( xEfdzzzxfxfxf

)()( xfxfepi , strongly l.s.c.

kkkk

kk gxfgxf

)()(1)( kxgradf - Independent of

dimensionality

(probability density) → Dirak function, 0

),,)(()()( xxqEuxFxF

),()( xEfxF ),),((),( xxquxf

kk

kkxx 0 - Random vectorsg,

k, ,

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}0)(,0,0)(:],0[max{),( tRtssRTtx

hxsRxsR ),,(),,(

]0)(|0)(Pr[)),,(min(),,(

)(

10

sRtRxsRHxtfE

xF

ts

Tt

),,,()( xREfxF )Pr()( yhyH

Stochastic Processes with Stopping Time

),,( xtR - a risk process

0

- Fast Monte-Carlo optimization

- Convergence for

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

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Discounting

0 0t t ttt vEVd

tt EvV

]Pr[ td t

tt qd For geometric discounting

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Geo-PhysicalSpatial Data;

Released Water

InundationModel

StandingWaters

Geo-PhysicalSpatial Data;

Released Water

InundationModel

StandingWaters

StandingWaters

Spatial Inundation Module

Integrated catastrophe management models

ReleasedWater

River Model

DikesModification;

Failures;Geo-Physical

Data

Rains

ReleasedWater

River Model

DikesModification;

Failures;Geo-Physical

Data

RainsRains

River Module

Standing waters;Feasible Decisions;

Economic Data

VulnerabilityModels

DirectSpatialLosses

Standing waters;Feasible Decisions;

Economic Data

Standing waters;Feasible Decisions;

Economic Data

VulnerabilityModels

DirectSpatialLosses

Vulnerability Module

Histograms oflosses and

gains

Multi-agent accounting system

Losses of households, farmers, producers,water authorities,

governments,Feasible decisions

Evaluation of decisions with

respect to goals,constraints

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Adaptive Monte Carlo STO Procedure

IndicatorsGoals

Constraints

Gains andLosses

FeasiblePolicies

Monte-CarloCat. Model

IndicatorsGoals

Constraints

Gains andLosses

FeasiblePolicies

Monte-CarloCat. Model

Structural and non-structural decisions

Optimization module: structural and non-structural decisions, premiums, coverage, contingent credit, production allocation, …

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18 4 12 19 34

16 23 25 19 32

21 13 20 18 15 10

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11

1211

LOCATION j Cat. fund

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18 4 12 19 34

16 23 25 19 32

21 13 20 18 15 10

17

11

1211

LOCATION j Cat. fund

),( ttj xW - Property value

- Stopping time

),( ttj xL - Scenario of loss

tiR - Risk reserve,

1 tj

tj - gov. compensation

- Ins. contract

)(xtj - Premium, y - credit

Decisions x

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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

max ( ) ( ) max maxi i i

i i iL d El E l

- 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 models and methods

- singularity (discontinuity) w.r.t. “outlyers” (rare catastrophic risks)- importance of stochastic vs probabilistic minimax

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

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“Learning” – by simulation: Adaptive Monte Carlo procedure

Typical “performance” of the goal function: