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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, …
10
18 4 12 19 34
16 23 25 19 32
21 13 20 18 15 10
17
11
1211
LOCATION j Cat. fund
10
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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6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101
106
111
116
121
126
131
136
141
146
151
156
161
166
171
176
181
186
191
1966 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101
106
111
116
121
126
131
136
141
146
151
156
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166
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“Learning” – by simulation: Adaptive Monte Carlo procedure
Typical “performance” of the goal function: