Decision Analysis under
Uncertainty
Christopher Grigoriou
Executive MBA/HEC Lausanne
2007-2008
Introduction
Examples of decision making under uncertainty in the business
world;
=> Trade-off between bidding low to win the bid and bidding high to
make a larger profit
=> Introducing a new product into the market (customers reaction),
test market?
=> Insurance market
All the problems have three common elements
1- The set of strategies available to the decision maker
2- The set of possible outcomes and the probabilities of these
outcomes
3- A value model that prescribes monetary values for the various
decision/outcome combinations
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Executive MBA- HEC Lausanne- 2007/2008
Case of an Insurance Market
Uncertain prospect: during the next year probability p to have
an accident
Your wealth is w1 at the beginning of the year
At the end of the year
=> wealth still w1 if no accident
=> wealth now w0<w1 if accident
(w1-w0 = cost of repairing your car etc.)
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The expected value of your wealth, w*, is the weighted average
of the two possible outcomes:
w* = p.w0 + (1-p).w1
Figure 1: Expected Wealth
w0
w*
w1
wealth
The figure suggests a relatively high likelihood of accident (a low
probability would place w* very close to w1)
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Executive MBA- HEC Lausanne- 2007/2008
If indifferent between
- w* with certainty
- the prospect of w0 with probability p and w1 with
probability (1-p)
=> risk neutral
Willing to pay an insurance premium: w1-w* for a
policy that paid off w1-w0 in the event of an accident so
that you get w* (w1-your premium) whatever happens
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Executive MBA- HEC Lausanne- 2007/2008
Example:
A 20’000$ car / probability of a wreck is 1%
=> w1= 20’000 ; w0= 0 ; w*= 19800
If the insurance company insures a large number of drivers, it can believe that its $20’000 payouts will
be made close to one in one hundred of its policy holders and it receives 200$ from each policy holder.
Real insurance companies
Costs other than claims payouts
⇒the premiums they charge exceed the amount they pay out
⇒The expected wealth of the consumer who does not insure exceeds the certain
wealth of an ensured person
⇒Given a choice between an uncertain prospect with expected value of w* and
alternatively w* with certainty, consumers prefer the certain alternative
⇒Consumers get more utility from the certain prospect
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Executive MBA- HEC Lausanne- 2007/2008
Expected utility with certain levels of utility
1- The following figure graphs consumer utility for the two points w0 and w1:
=> u(w0) is the utility you receive if you have suffered a clamity
=> u(w1) is the utility you receive if you do not have to worry about that clamity.
=> These u(w0) and u(w1) are the utility of having a certain level of wealth with
certainty.
2- We can represent the different levels of utility depending on p and (1-p)
Utility
u(w1)
p.u(w0)+(1-p).u(w1)
u(w0)
w0
w’
w* w1
Wealth
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Executive MBA- HEC Lausanne- 2007/2008
Expected utility and Insurance Premiums
utility
u(w1)
u(w*)
p.u(w0)+(1-p).u(w1)
u(w0)
w0
w’ w*
w1
Wealth
You exactly know, i.e. with certainty, the utility you would get with w0 or w1.
Between w0 and w1, the expected utility not to be insured is less than the
utility associated with w* (the case where we are insured)
=> the utility function for certain prospects that passes through (w0,
u(w0)) and (w1, u(w1)) must lie above the line connecting the two points
If the red line represents your utility, how much are you willing to pay for an
insurance policy?
=> the consumer is indifferent between the uncertain prospect with expected
value w* and w’ with certainty
=> the consumer starting with w1 would be willing to pay up to w1-w’ to avoid
the uncertain prospect.
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What do we know? What have we learnt? (1)
Risk aversion = the reluctance of a person to accept a bargain with an
uncertain payoff rather than another bargain with a more certain but
possibly lower expected payoff.
Choice between a bet of
=> either receiving 100$ (50%) or nothing (50%)
=> or receiving some amount with certainty.
Risk neutral = indifferent between the bet and a certain 50$ payment
Risk averse = accept a payoff of less than 50$ (e.g 40$) with
probability 100%
Risk loving = Required that the payment be more than 50$ (e.g 60$)
to induce him to take the certain option over the bet.
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Executive MBA- HEC Lausanne- 2007/2008
What do we know? What have we learnt? (2)
The average payoff of the bet is called the « Expected
(Monetary) Value » (50$)
The certain amount accepted instead of the bet is called the
« certainty equivalent », the difference between it and the
expected value is called the risk premium
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Definition and example:
Imagine a decision maker must choose amongst three decisions (d1, d2 and d3) with
three possible outcomes (O1, O2 and O3)
Payoff tables = listing of payoffs for all decision-outcome pairs,
positive values = gains/ negative = losses
Outcome
Decision
⇒
⇒
⇒
O1
O2
O3
D1
10
10
10
D2
-10
20
40
D3
-30
30
70
Safe decision = chosing D1
D3 the riskier; greater possible gains and losses
Decision makers must make rational decisions based on the information they have
when the decisions must be made
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Executive MBA- HEC Lausanne- 2007/2008
Possible decision criteria
The maximin criterion: choose the decision that maximizes the worst payoff (for a
pessimistic decision maker) => D1 with payoff 10
Avoid large losses but also fails to consider large rewards => not commonly used
The maximax: choose the decision that maximizes the best payoff (optimistic
decision maker or risk taker)
Focuses on large gains but ignores possibles losses => seldom used
Maximin and maximax criteria make no reference to how likely each outcome is
(decision makers typically have at least some idea of these likelihoods and ought to
use this information in the decision making process).
=> if outcome O1 is very unlikely, then the maximin users are overly conservative
=> the same if O3 is quite unlikely, the maximax users take an unnecessary risk
• Expected Monetary value (EMV)
=> The EMV approach assesses probabilities for each outcome of each decision and then
calculates the expected payoff from each decision
⇒For any decision, the EMV is the weighted average of the possible payoffs for this
decision, weighted by the probabilities of the outcomes.
⇒We choose the decision with the largest EMV.
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Executive MBA- HEC Lausanne- 2007/2008
Expected Monetary Value
The decision maker assesses the probabilities of the three outcomes
(O1, O2, O3) as 0.4, 0.4 and 0.2
For each decision:
EMV for D1: 10x0.4 + 10x0.4 + 10x0.2 = 10
EMV for D2: -10x0.4 + 20x0.4 + 40x0.2 = 12
EMV for D3: -30x0.4+30x0.4+70x0.2 = 14
⇒ The optimal decision is then to choose D3 since it has the largest
EMV.
⇒
=> We’ll never get 14$ (either -30 or +30 or +70) but on average, if
running that decision many times we will make a gain of about 14$
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Executive MBA- HEC Lausanne- 2007/2008
Sensitivity Analysis
Changing slightly the inputs, how are the outputs
(EMVs and the best decision) modified?
Modify either the outcomes or the probabilities and
see how the final decision change
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Executive MBA- HEC Lausanne- 2007/2008
Decision trees
=>Used to analyse complex problems with a
sequence of events (decisions and
outcomes)
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Methodology
Decompose problem into chronological sequence of decisions and
events
Decisions: - You decide
Events:
- ‘Others’ decide
Determine all possible scenarios (sequences of decisions and events)
For each scenario: Outcome? Likelihood?
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Executive MBA- HEC Lausanne- 2007/2008
Decision nodes and Chance nodes
Decision node:
You choose which way to go
Chance node:
‘Chance’ decides which way you go
Decision 1
Event 1 Probability 1
Decision 2
Event 2 Probability 2
Decision 3
Event 3 Probability 3
Probabilities sum to 1
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Executive MBA- HEC Lausanne- 2007/2008
Example
0.5
Failure
Marketing effort
0.5
Success
0.75
Failure
...
No Marketing effort
0.25
Success
Posterior probabilities = Conditional probabilities:
Depend on decisions and events preceding this event
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Executive MBA- HEC Lausanne- 2007/2008
Using Precision Tree or Treeplan
Summarize the data (costs, revenues, probabilities)
Structure the tree: Chronological sequence of decisions and
events
Insert the costs, revenues and probabilities for each branch
Indicate whether you are minimizing or maximizing EMV
Interpret the solution: - Expected Monetary Value
- Risk Profile
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Executive MBA- HEC Lausanne- 2007/2008
The value of information
How much is the information worth? Should we
purchase it?
=> the answers to these questions are embedded in the
decision tree itself
Expected Value of the Perfect Information
=> the most we would be willing to pay for the sample
information
Price of Information =
EMV with perfect information - EMV without information
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Executive MBA- HEC Lausanne- 2007/2008
Example 1
Assume you have to ship a gift, but there is a probability 0.4
that the shipment fails. In this case you have a loss of 80. Your
total wealth is 100$. You have the opportunity to insure the
gift and the insurance premium costs 30.
How much are you willing to pay in order to know what is
going on before choosing?
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Executive MBA- HEC Lausanne- 2007/2008
Example 2
2 risky investment plans.
Plan A:
prob. high market 0.8 payoff associated 100;
prob. low market 0.2 payoff associated 20.
Plan B:
prob. bad event 0.2 payoff 10;
prob. nothing change event 0.5 payoff 60;
prob. good event 0.3 payoff 100.
1- Which one would you chose?
2- How much is your willingness to pay (two cases) ?
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Executive MBA- HEC Lausanne- 2007/2008