1. Decision making under uncertainty
Stochastic methods in Engineering
Risto Lahdelma
Professor, Energy for communities

2. Content What is uncertainty Stochastic dominance Utility theory
Risk attitude
Multi-criteria decision analysis
Multicriteria utility function

3. ”I used to be uncertain, but now I am not sure”
What is uncertainty
”I used to be uncertain, but now I am not sure”
Uncertainty can mean different things
Inaccurate: his height is 190 cm
Imprecise: his height is in range [185,195]
Fuzzy (vague): he is tall
Classical probability: tossing heads on a fair coin has 50% probability
Subjective probability: I believe there is 50% chance for rain tomorrow
Statistical probability: I win my friend in Badminton 6 times out of 10

4. Representing uncertainty
Many ways
Standard uncertainty, confidence margins, fuzzy sets, rough sets, …
Different kinds of uncertainty can be represented using probability distributions
In most cases this is an approximation of reality
Almost nothing is certain – not even the probability distribution to represent uncertainty
Still we must represent uncertainty the best we can and consider it in decision making

5. Biases in estimating uncertainty
People cannot assess probabilities well
After tossing 6 heads on a coin, what is more likely next, heads of tails?
When tossing a coin, which sequence is more likely, HHHTTT or HTHTHT ?
You see a good looking woman in a bar. Is she more likely a model or nurse?
Is it more likely that an English word starts with K or that K is the third letter?
Is the percentage of African countries in UN greater or less than xx%? How large?

6. Biases in estimating uncertainty
Which is more likely, getting 7 figures right on Lotto or dying in car accident while submitting Lotto coupon?
If the probability of rain is 50% on Saturday and 50% on Sunday, what is the probability that it will rain during weekend?
Which number series has greater variance: A: B:
Which is more likely:
A: That Williams loses the first set in Tennis.
B: That Williams loses the first set but wins the match.
Estimate 90% confidence margin for the distance to the second brightest star Canopus (in light years).

7. Decision under uncertainty
Many different approaches to compare uncertain alternatives
Single criterion decision making
Typically, money is the criterion
Different aspects of the decision alternatives are expressed in monetary units
Multiple criterion decision making
Multiple non-commensurate criteria (objectives) must be considered simultaneously

8. Stochastic dominance
You have to choose between a1,a2,a3 with outcomes at probabilities 1, 2, 3
Such dominating alternative does not often exist

9. Maximin / pessimistic criterion
Choose alternative with best worst case
Often this is too extreme, in particular if the probability for the worst case is very small

10. Maximax / optimistic criterion
Choose alternative with best best case
Also this is may be unwise

11. Expected value Choose best expected value Commonly used
People do not often behave like this

12. Gamble situation Which alternative do you prefer? 10 000 € for sure
€ for p=0.6 but 0 for p=0.4
Expected value of B is

13. Gamble situation Which alternative do you prefer? Many people choose A
€ for sure
€ for p=0.6 but 0 for p=0.4
Expected value of B is
Many people choose A
This behaviour is called risk aversion
A certain outcome is preferred to a risky choice with a little higher expected value

14. Utility theory
Preferences in the precence of risk can be modelled using a utility function u(x)
The u(x) is a monotonic function that maps the worst outcome to 0 and the best outcome to 1
The decision maker (DM) maximises the expected utility E(u(x))

15. E(u(x)) = x f (x)u(x)dx
Expected utility
For discrete case x = <p1,x1 ; ... pn,xn >
E(u(x)) = i piu(xi)
For continuous case x with density f(x)
E(u(x)) = x f (x)u(x)dx

16. Risk averse DM
Risk averse DM prefers certain outcome to gamble with (little) higher expected value
A rational DM is normally assumed to be risk averse or risk neutral
Opposite of risk averse is risk-seeking, who prefers a gamble

17. Concave utility function
Risk averse behaviour corresponds to utility function with decreasing marginal utility, i.e. concave shape
Concavity
p1u(x1)+(1-p1)u(x2) <= u(p1x1+(1-p1)x2)

18. Shape of utility function
Linear=Risk-neutral
Concave=Risk-averse
Convex = Risk-seeking

19. Example To drill or not drill for oil Don’t drill.
Cost of drilling is 20, chance of finding is 25%, profit in this case is 80
Don’t drill, no cost, no profit
Utility function u(x) = (20+x)0.5/10
u(A) = (0.25* *0)/10 = 0.25
u(B) = (20)/10  0.447
Don’t drill.
(using expected values, E(A)=5, E(B)=0)

20. Multicriteria decision making
Choosing most preferred alternative considering multiple non-commensurate criteria
If all criteria can be made commensurate, expressed e.g. in money, then the problem reduces into single criterion
Sometimes this is not possible or desired to do that
Some criteria are really difficult to combine

21. Non-commensurate criteria
Examples
Cost, quality, reliability
Economy, environment, human health
Effects on different interest groups
Energy efficiency, emissions, service level

22. Multicriteria utility function
Utility is a function with multiple arguments u(x1,x2,…,xn)
Cost, quality, reliability
Economy, environment, human health
Effects on different interest groups
Energy efficiency, emissions, service level

23. Multicriteria utility function – 2-criterion example
The City is considering four sites (A,B,C and D) for a new power plant
The objectives of the city are to
minimise the cost of building the station;
minimise the area of land damaged by building it.
Factors influencing the objectives include
the land type at the different sites,
the architect and construction company hired,
cost of material and machines used, the weather, etc.

24. Multicriteria utility function – 2-criterion example
Criteria evaluation
Costs, are estimated to fall between 15M€ and 60M€
between 200 and 600 hectars of land will be damaged
The possible consequences of the decision alternatives are captured by two criteria.
We need to determine a two-attribute utility function:
u(Cost, Area) or u(C,A)

25. Multicriteria utility function assessment
Let x1, ,xn, n ≥ 2, be a set of criteria associated with the consequences of a decision problem.
The utility of a consequence (x1, , xn) can be determined from direct assessment:
estimate the overall utility u(x1, , xn) over the given values of all n attributes;
Decomposed assessment:
estimate n partial utilities ui(xi) for given values of the n criteria;
Define u(x1, , xn) as combination of ui(xi) for all criteria:
u(x1,…, xn) = u(u1(x1),…, un(xn))

26. Decomposed assessment
Decomposed assessment is commonly used especially when there are many criteria
The additive form is often used
u(x1,…,xn) = u(u1(x1),…, un(xn)) = w1u1(x1)+…+wnun(xn)
Here w1,…wn are importance weights given for the criteria
Weights are subjective information that should be obtained from the DMs
If the partial utility functions are linear, then the overall utility function is linear (risk neutral behaviour)
Concave partial utility functions means risk averse behaviour

27. Decomposed assessment
The City models the utility as additive function.
The partial utility functions uC and uA are assessed
uC(15) = 1; uC(30) = 0.5; uC(50) = 0.2; uC(60) = 0
uA(200) = 1; uA(300) = 0.8; uA(400) = 0.5; uA(600) = 0
Cost is considered three times as important as Area.
Utilities for various Cost,Area pairs are then computed:
u(50, 300) = 3 · uC(50) + 1 · uA(300) = 3 · · 0.8 = 1.4
u(30, 400) = = 2.0
u(60, 200) = = 1.0
u(15, 600) = = 3.0
u(15, 200) = = 4.0

28. Excercise Choose (Cost,Area) values for the four alternatives A,B,C,D
For Cost, let each alternative have two possible values with different probabilities
Compute the expected utility for the alternatives