Robust decisions under
endogenous 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
• Collaborative work with IIASA’s projects
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Energy and technology
Forestry
Global climate change and population
Integrated modeling
Land use
Risk and Vulnerability
• Case studies on catastrophic risk
management
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Earthquakes (Italy, Russia)
Floods (Hungary, Ukraine, Poland, Japan)
Livestock production and disease risks (China)
Windstorms (China)
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Overviews and further references
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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.
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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.
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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.
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A. Gritsevskii, N. Nakichenovic, 1999. Modeling uncertainties of induced
technical change, Energy policy, 28.
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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.
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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.
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Pflug (Eds.) Coping with Uncertainty: Modeling and Policy Issue, Springer
Verlag, Berlin, New York, 2006.
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
(1   )P   q , q  Q ,   0 ,   0
 Local stability solutions of differential eqs.
 Bayesian robustness
 ranges of posterior expected “losses”
 L( x, ) ( |  )d ,  
 Minimax (non-Bayesian) ranking
 endogenous random priors from

 Optimal deterministic control
 local stability of optimal trajectories
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Decision problems under uncertainty
 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.
 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.
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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 same
costs as the cost of future carbon intensive energy systems
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Technological change under increasing returns
a
b
Increasing (a)
Constant (b)
Diminishing (c)
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
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A.Gritsevskyi & N. Nakicenovic, 2000
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 d ( )
“ x  d ( ) “
 Overshooting-and-undershooting costs
cost
d
production
 Scenario analysis
d1, d2 , d3,...  x1opt  d1, x2opt  d2 , x3opt  d3, ...
Robust solution = Ed ?
 F ( x)  E (d  x) I (d  x)  E ( x  d ) I ( x  d )  E max{ (d  x),  ( x  d )}

x rob 
- quantile of d defined by slopes  ,  :
 

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(VaR)
F(xrob) = CVaR
P[ x  d ] 
 
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
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Adaptive scenario simulators: earthquakes
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Adaptive scenario simulators: floods
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100
80
60
10
9
8
7
40
3
2
6
5
4
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Monetary or natural units
Initial landscape of values
0
1
2
3
4
Locations
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6
7
8
9
10
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Scenarios of damaged values
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Initial spread of coverage:
standard feasible decisions
Insurer 1
Insurer 2
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Initial spread: high risk of bankruptcies
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Robust spread of coverage:
new feasible decisions
Insurer 1
Insurer 2
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Robust spread: reduced risk of bankruptcies
Bankruptcies
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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
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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
median(1  2  ...)  median(1)  median( 2 )  ...
Applicability of Standard Models and Methods
 Deterministic equivalent
 Expected utility models, NPV, CBA
 Intervals uncertainties
 Bellman’s equations, Pontriagin’s Principle Maximum
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and other similar decompositions schemes
Robust risk management
• Safety (chance) constraints
min Ef ( x,  )
P[ g ( x,  )  0]  1  
(Discontinuity, e.g., P[ x    0])
min[ Ef ( x,  )  NE max{ 0, g ( x,  )}]
(1996) (Convexification)
- ex-post borrowing
• CVaR measure of risk
min y
P[ g ( x,  )  y]  1  
min[ Ef ( x,  )  y  NE max{ 0, g ( x,  )  y}] (min of quantiles)
- contingent (ex-ante) credit
y
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Discontinuity: Illustrative Example
 Catastrophe model
 ( )  inf{ t : R(t )  A}
-
Potential disaster at
-
A shut down” (stopping time) decision
-
Performance function
x
ax, x  
f ( x, )  
a  b,  x
F ( x)  EfI ( x   )  EfI ( x   )
(Fast and slow components)
gradF ( x)  [ f ( x, x)  f ( x, x  0)] ( x)  Ef x ( x, )
f x ( x, )  a
If
x 
, and
f x ( x, )  0
otherwise
 Deterministic (sample mean) approximation
F N ( x)  kN1 f ( x, k ) / N
- number of jumps
 , N  
 SQG: SQG ( x,  )  [ f ( x, x)  f ( x, x  0)] ( x)  f x ( x, )
-
Fast adaptive Monte-Carlo simulators
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Parametric smoothing
 Averaged (generalized) functions: Steklov (1907),
Sobolev (1930), Kolmogorov (1934), …
- theory of distributions (Shwartz, 1966)
f ( x)  f ( x)   f ( x  z ) ( z )dz  Ef ( x   )
 (probability density) → Dirak function,   0
 Optimization (Ermoliev, Gaivoronski, Gupal, Katkovnic,
epi Lepp, Marti, Norkin, Wets, … )
f ( x)  f ( x) , strongly l.s.c.
k
k  k
- Independent of
k
1
k

f
(
x


g
)

f
(
x
)
g
gradf ( x )  k 

k
dimensionality


k
x  x k   k  k , k  0 ,  , g - Random vectors
 Parametric smoothing
F ( x)  Ef ( x,  ) f ( x,  )  u (q( x), x,  )
F ( x)  F ( x)  Eu(q( x)   , x,  )
fast estimation of functions and derivatives
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 Stochastic Processes with Stopping Time
R(t , x,  ) - a risk process
 ( x,  )  max{t  [0, T ] : R( s)  0,0  s  t , R(t )  0}
F ( x)  Ef ( , R, x,  )
H ( y)  Pr(h  y)
R( s, x,  )  R( s, x,  )  h
F ( x ) 
E Tt 0 f (t , x ,  )H  ( min R( s, x ,  )) Pr[ R(t )  0 | R( s )  0]
s  t 1
- Fast Monte-Carlo optimization
- Convergence for
 0
 Y.M. Ermoliev and V.I. Norkin, Stochastic optimization
of risk functions, in K. Marti, Y. Ermoliev, and G. Pflug (eds.):
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Dynamic stochastic optimization, Springer, 2004, pp. 225-249
 Discounting


d
V

E


t 0 t t
t  0 vt

d t  Pr[  t ]

dt  q t
Vt  Evt
For geometric discounting
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Integrated catastrophe management models
Dikes
Modification;
Failures;
Geo-Physical
Data
Rains
River
Model
Released
Water
River Module
Geo-Physical
Spatial Data;
Released Water
Inundation
Model
Standing
Waters
Vulnerability
Models
Direct
Spatial
Losses
Evaluation of
decisions with
respect to goals,
constraints
Histograms of
losses and
gains
Spatial Inundation Module
Standing waters;
Feasible Decisions;
Economic Data
Vulnerability Module
Losses of households,
farmers, producers,
water authorities,
governments,
Feasible decisions
Multi-agent accounting system
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Adaptive Monte Carlo STO Procedure
 Structural and non-structural decisions
Feasible
Policies
Monte-Carlo
Cat. Model
Gains and
Losses
Indicators
Goals
Constraints
 Optimization module: structural and non-structural decisions, premiums,
coverage, contingent credit, production allocation, …
LOCATION j

10
16
23
25
19
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17
21
13
20
18
15
10
11
4
12
19
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Cat. fund
Decisions
x
- Stopping time
W jt ( x, t ) - Property value
Rit - Risk reserve,
Ltj ( x, t ) - Scenario of loss
 tj   tj  1
 tj (x) - Premium, y - credit
- gov. compensation
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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
 Proper models and methods
- singularity (discontinuity) w.r.t. “outlyers” (rare catastrophic risks)
- importance of stochastic vs probabilistic minimax
max  L( )d i ( )  maxEli  E max li
i
i
i
- 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
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“Learning” – by simulation: Adaptive Monte Carlo procedure
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Typical “performance” of the goal function:
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