Decision Analysis
Basic Terms
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Decision Alternatives (eg. Production quantities)
States of Nature (eg. Condition of economy)
Payoffs ($ outcome of a choice assuming a state of nature)
Criteria (eg. Expected Value)
What kinds of problems?
• Alternatives known
• States of Nature and their probabilities are known.
• Payoffs computable under different possible scenarios
Decision Environments
Ignorance – Probabilities of the states of nature are
unknown, hence assumed equal
Risk / Uncertainty – Probabilities of states of nature are
known
Certainty – It is known with certainty which state of nature
will occur. Trivial problem.
Example – Decisions under Ignorance
Assume the following payoffs in $ thousand for 3 alternatives –
building 100, 200, or 400 condos. The payoffs ($ million) depend
on how many are sold, which depends on the economy. Three
scenarios are considered - a Poor, Average, or Good economy at
the time the condos are completed.
Payoff Table
S1
S2
(Poor) (Avg)
S3
(Good)
A2 (200 units)
3
-1
3.5
6
4
7
A3 (400 units)
-10
-2
12
A1 (100 units)
Maximax - Risk Seeking Behavior
What would a risk seeker decide to do? Maximize payoff without
regard for risk. In other words, use the MAXIMAX criterion. Find
maximum payoff for each alternative, then the maximum of those.
A1
A2
A3
S1
S2
S3
MAXIMAX
3
-1
-10
3.5
6
-2
4
7
12
4
7
12
The best alternative under this criterion is A3, with a potential
payoff of 12.
Maximin – Risk Averse Behavior
What would a risk averse person decide to do? Make the best of the
worst case scenarios. In other words, use the MAXIMIN criterion.
Find minimum payoff for each alternative, then the maximum of those.
A1
A2
A3
S1
S2
S3
MAXIMIN
3
-1
-10
3.5
6
-2
4
7
12
3
-1
-10
The best alternative under this criterion is A1, with a worst case
scenario of 3, which is better than other worst cases.
LaPlace – the Average
What would a person somewhere in the middle of the two extremes
choose to do? Take an average of the possible payoffs. In other words,
use the LaPlace criterion (named after mathematician Pierre LaPlace).
Find the average payoff for each alternative, then the maximum of
those.
A1
A2
A3
S1
S2
S3
LaPlace
3
-1
-10
3.5
6
-2
4
7
12
3.5
4
0
The best alternative under this criterion is A2, with an average
payoff of 4, which is better than the other two averages.
Minimax Regret – Lost Opportunity
What would a person choose who wanted to minimize the worst
mistake possible? For each state of nature, find the maximum payoff,
and subtract each of the payoffs from it to compute the lost
opportunities (regrets). Then find maximum values for each
alternative, and the minimum of those.
Opportunity Loss (Regret) Table
A1
A2
A3
S1
S2
S3
Minimax
0
4
13
2.5
0
8
8
5
0
8
5
13
The best alternative under this criterion is A2, with a maximum
regret of 5, which is better than the other two maximum regrets.
Example – Decisions under Risk
Assume now that the probabilities of the states of nature are
known, as shown below.
S1
S2
(Poor) (Avg)
A1 (100 units)
A2 (200 units)
A3 (400 units)
Probabilities
S3
(Good)
3
-1
3.5
6
4
7
-10
0.30
-2
0.60
12
0.10
Expected Values
When probabilities are known, compute a weighed average of payoffs,
called the Expected Value, for each alternative and choose the
maximum value.
Payoff Table
S1
A1
A2
A3
Probabilities
3
-1
-10
0.30
S2
S3
EV
3.5
4
3.40
6
7
4.00
-2
12 -3.00
0.60 0.10
The best alternative under this criterion is A2, with a maximum EV
of 4.00, which is better than the other two EVs.
Expected Opportunity Loss (EOL)
Compute the weighted average of the opportunity losses for each
alternative to yield the EOL.
Opportunity Loss (Regret) Table
S1
A1
A2
A3
Probabilities
0
4
13
0.30
S2
S3
2.5
8
0
5
8
0
0.60 0.10
EOL
2.30
1.70
8.70
The best alternative under this criterion is A2, with a minimum
EOL of 1.70, which is better than the other two EOLs.
Note that EV + EOL is constant for each alternative! Why?
EVUPI: EV with Perfect Information
If you knew everytime with certainty which state of nature was
going to occur, you would choose the best alternative for each
state of nature every time. Thus the EV would be the weighted
average of the best value for each state. Take the best times the
probability, and add them all.
S1
S2
(Poor) (Avg)
A1 (100 units)
A2 (200 units)
A3 (400 units)
Probabilities
S3
(Good)
3
-1
3.5
6
4
7
-10
0.30
-2
0.60
12
0.10
3*0.3 = 0.90
6*0.6 = 3.60
12*0.1 = 1.20
_____________
Sum =
5.70
Thus EVUPI = 5.70
EVPI: Value of Perfect Information
If someone offered you perfect information about which state of
nature was going to occur, how much is that information worth to
you in this decision context?
Since EVUPI is 5.70, and you could have made 4.00 in the long
run (best EV without perfect information), the value of this
additional information is 5.70 – 4.00 = 1.70.
Thus,
EVPI
= EVUPI – Evmax
= EOLmin
Decision Tree
3.40
0.3
0.6
0.1
A1
0.3
0.6
A2
A2
4.00
4.00
A3
0.1
0.3
0.6
-3.00
0.1
3.00
3.50
4.00
-1.00
6.00
7.00
-10.00
-2.00
12.00
Sequential Decisions
• Would you hire a consultant (or a psychic) to get more info
about states of nature?
• How would additional info cause you to revise your
probabilities of states of nature occuring?
• Draw a new tree depicting the complete problem.
Consultant’s Track Record
Past
Fav.
Forecast Unfav.
S1
S2
S3
20
60
70
80
40
30
100
100
100
Probabilities
• P(F/S1) = 0.2
• P(F/S2) = 0.6
• P(F/S3) = 0.7
P(U/S1) = 0.8
P(U/S2) = 0.4
P(U/S3) = 0.3
• F= Favorable
U=Unfavorable
Joint Probabilities
S1
Fav.
S2
S3
Total
0.06 0.36 0.07 .49
Unfav. 0.24 0.24 0.03 .51
Prior 0.3
Probs
0.6
0.1
1.00
Posterior Probabilities
• P(S1/F) = 0.06/0.49 = 0.122
• P(S2/F) = 0.36/0.49 = 0.735
• P(S3/F) = 0.07/0.49 = 0.143
• P(S1/U) = 0.24/0.51 = 0.47
• P(S2/U) = 0.24/0.51 = 0.47
• P(S3/U) = 0.03/0.51 = 0.06
Solution
• Solve the decision tree using the posterior probabilities just
computed.