Decision Analysis

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Decision Analysis
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Basic Terms

• 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)

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What kinds of problems?

• Alternatives known
• States of Nature and their probabilities are
  known.
• Payoffs computable under different possible
  scenarios

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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.

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Example Decisions under Ignorance
Assume the following payoffs in thousand for 3
alternatives building 10, 20, or 40 condos. The
payoffs 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 Payoff Table Payoff Table Payoff Table
S1 (Poor) S2 (Avg) S3 (Good)
A1 (10 units) 300 350 400
A2 (20 units) -100 600 700
A3 (40 units) -1000 -200 1200
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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.
S1 S2 S3 MAXIMAX
A1 300 350 400 400
A2 -100 600 700 700
A3 -1000 -200 1200 1200
The best alternative under this criterion is A3,
with a potential payoff of 1200.
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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.

S1 S2 S3 MAXIMIN
A1 300 350 400 300
A2 -100 600 700 -100
A3 -1000 -200 1200 -1000
The best alternative under this criterion is A1,
with a worst case scenario of 300, which is
better than other worst cases.
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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.

S1 S2 S3 LaPlace
A1 300 350 400 350
A2 -100 600 700 400
A3 -1000 -200 1200 0
The best alternative under this criterion is A2,
with an average payoff of 400, which is better
than the other two averages.
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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 Opportunity Loss (Regret) Table Opportunity Loss (Regret) Table Opportunity Loss (Regret) Table Opportunity Loss (Regret) Table
S1 S2 S3 Minimax
A1 0 250 800 800
A2 400 0 500 500
A3 1300 800 0 1300
The best alternative under this criterion is A2,
with a maximum regret of 500, which is better
than the other two maximum regrets.
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Example Decisions under Risk
Assume now that the probabilities of the states
of nature are known, as shown below.
S1 (Poor) S2 (Avg) S3 (Good)
A1 (100 units) 300 350 400
A2 (200 units) -100 600 700
A3 (400 units) -1000 -200 1200
Probabilities 0.30 0.60 0.10
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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 Payoff Table Payoff Table Payoff Table Payoff Table
S1 S2 S3 EV
A1 300 350 400 340
A2 -100 600 700 400
A3 -1000 -200 1200 -300
Probabilities 0.30 0.60 0.10
The best alternative under this criterion is A2,
with a maximum EV of 400, which is better than
the other two EVs.
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Expected Opportunity Loss (EOL)

• Compute the weighted average of the opportunity
  losses for each alternative to yield the EOL.

Opportunity Loss (Regret) Table Opportunity Loss (Regret) Table Opportunity Loss (Regret) Table Opportunity Loss (Regret) Table Opportunity Loss (Regret) Table
S1 S2 S3 EOL
A1 0 250 800 230
A2 400 0 500 170
A3 1300 800 0 870
Probabilities 0.30 0.60 0.10
The best alternative under this criterion is A2,
with a minimum EOL of 170, which is better than
the other two EOLs. Note that EV EOL is
constant for each alternative! Why?
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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.
3000.3 90 6000.6 360 12000.1
120 _____________ Sum 570 Thus EVUPI
570
S1 (Poor) S2 (Avg) S3 (Good)
A1 (100 units) 300 350 400
A2 (200 units) -100 600 700
A3 (400 units) -1000 -200 1200
Probabilities 0.30 0.60 0.10
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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 570, and you could have made 400
  in the long run (best EV without perfect
  information), the value of this additional
  information is 570 - 400 170.
• Thus, EVPI EVUPI Evmax
• EOLmin

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Decision Tree
300
0.3
340
0.6
350
0.1
400
A1
-100
0.3
0.6
A2
600
0.1
400
A2 400
700
-1000
0.3
A3
0.6
-200
-300
0.1
1200
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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.

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Consultants Track Record
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Probabilities

• P(F/S1) 0.2 P(U/S1) 0.8
• P(F/S2) 0.6 P(U/S2) 0.4
• P(F/S3) 0.7 P(U/S3) 0.3
• F Favorable UUnfavorable

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Joint Probabilities
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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

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Solution

• Solve the decision tree using the posterior
  probabilities just computed.