Decision & Game Theory
Chapter 4: Decision Analysis under Risk with
Additional Information
Sonia REBAI
Tunis Business School
University of Tunis
• In a risky environment, the probability distribution of the states of
nature plays an important role in the choice of the optimal decision.
• DMs have preliminary or prior probability assessments that are the
best values available at that time.
• It would then be of importance to seek additional information before
making a final decision.
• This new information can be used to revise or update the prior
probabilities so that the final decision is based on more accurate
probabilities.
• The revised probabilities are called posterior probabilities.
• Most often, additional information is obtained through experiments, a
consulting advice, or a forecast.
• Such information is usually acquired at some cost. The question is
whether the cost paid is worth.
• To answer this question we need to calculate the Expected Value of the
additional information.
Example 1
A company has the choice between producing itself or buying from a supplier
one of the electronic components that it uses in its activity. Net profit depends
on the level of demand for the product requiring this component. This
demand can be low, medium or high. The a priori distribution is estimated at
0.35; 0.35 and 0.30.
Produce
buy
S1
S2
Low Demand Medium Demand
-20
40
10
45
S3
High Demand
100
70
4
The company has the opportunity to test the demand on the market. Two
results are possible for this test favorable (F) or unfavorable (U). The
following conditional probabilities were estimated:
Favorable
S1
P(F|S1)=0.10
S2
P(F|S2)=0.40
S3
P(F|S3)=0.60
Unfavorable
P(U|S1)=0.90
P(U|S2)=0.60
P(U|S3)=0.40
Find the optimal strategy to adopt.
5
F
Pr
e
c
u
od
Buy
Sur
v
ey
e
No
Su
r
U
duc
o
r
P
Buy
ve
Produce
y
Bu
y
S1
S2
S3
S1
S2
S3
S1
S2
S3
S1
S2
S3
S1
S2
S3
S1
S2
S3
-20
40
100
10
45
70
-20
40
100
10
45
70
-20
40
100
10
45
70
Bayes’ theorem
Si: State of Nature (i = 1, …, n)
P(Si): Prior Probability
Ij: Professional Information (Experiment)( j = 1, …, n)
P(Ij | Si): Conditional Probability
P(IjÇSi) = P(SiÇIj): Joint Probability
P(Si | Ij): Posterior Probability
P(Si | Ij)
P(Si Ç I j )
P( I j | Si ) P(Si )
=
=
n
P( I j )
å P( I j | Si ) P(Si )
i =1
S1
0.35
S2
S3
0.35 0.30
F
0.1
0.4
0.6
U
0.9
0.6
0.4
The
sum
P(S1│“.”)
P(S2│“.”)
P(S3│“.”)
0.035 0.14 0.18 0.355 0.09859
0.39437
0.50704
0.315 0.21 0.12 0.645 0.48837
0.32558
0.18605
8
64.51
F (0.355)
43.90
64,51 Pro
ce
u
d
54.23
Buy
S ur
ve y
U
(0.
6
45
21.86
)
e
c
32,56 Produ
32.56
Buy
43.90
No
Su
e
rve
y
37
40.25 Produc
Bu
y
40.25
S1 (0.09859)
S2 (0.39437)
S3 (0.50704)
S1 (0.09859)
S2 (0.39437)
S3 (0.50704)
-20
40
100
10
45
70
S1 (0.48837)
S2 (0.32558)
S3 (0.18605)
S1 (0.48837)
S2 (0.32558)
S3 (0.18605)
-20
40
100
10
45
70
S1(0.35)
S2(0.35)
S3(0.30)
S1(0.35)
S2(0.35)
S3(0.30)
-20
40
100
10
45
70
• Thus, the optimal strategy is to perform the survey, if the result of
the survey is favorable then the company should produce the
product, however, if the result is unfavorable, the company should
buy the product.
• EVII = 43.90 – 40.25 = 3.65
• So if the cost paid for this information is less than 3.65, it will be
worthy to acquire. Otherwise, the information will be worthless.
• Additional information reduces risk in decision making. It can in some
extreme situations completely remove the risk and provide a certainty
environment. In General, such information is very expensive.
• How to evaluate the value of perfect information (EVPI)?
• The idea behind EVPI is that if the state of nature that will occur is
known with certainty, then the best alternative can be determined
with certainty as well.
•
The value of EPVI is just simply the expected value under certainty minus the
expected value under uncertainty.
EVPI = Expected Payoff - Expected payoff
under Certainty with no information
•
To compute the expected value under certainty simply take the best payoff
under each state of nature and multiply it by its prior probability and sum
these.
•
EVPI places an upper bound on what one would pay for additional information.
• Expected Payoff under certainty = 10*0.35+45*0.35+100*0.30 = 49.25
• EVPI = 49.25 – 40.25 = 9
• This means that we should not be willing to pay more than 9.
• The Efficiency of the imperfect information is the ratio of EVII to EVPI.
• As the EVPI provides an upper bound for the EVII, efficiency is always a
number between 0 and 1.
• The efficiency of the survey = EVII/EVPI = (3.65)/(9) = 40.556%
Example 2: Marketing a new product
Assume that a feasibility study at Getz company of a new product led to
encouraging the introduction of this product to the market. The management
of Getz is not sure whether a large plant or a small one should be built to
manufacture the product. The relevant data is presented in the table below.
1. A marketing research company requests $65000 for a perfect information.
How much does this information worth?
15
(0.5)
Favorable market
(0.5)
Unfavorable market
Construct a large plant
$200,000
-$180,000
Construct a small plant
Do nothing
$100,000
$0
-$20,000
$0
2. Getz has the possibility to conduct a survey for $10,000. The survey will result in
a positive or a negative report. Past experience shows that given a favorable
market, there is 0.7 chance of a positive report and a 0.2 chance of a positive
report under unfavorable market. What should be the optimal strategy of the
company?
3. Calculate the efficiency of the survey
16
Construct a
large plant
Construct a
small plant
Do nothing
(0.5)
Favorable market
(0.5)
Unfavorable market
EMV
$200,000
-$180,000
$10,000
$100,000
-$20,000
$40,000
$0
$0
$0
1. EVPI = Expected Value Under Certainty - Max(EMV)
= ($200,000*0.50 + 0*0.50) - $40,000
= $100,000 - $40,000 = $60,000
So Getz should not be willing to pay more than $60,000
2. Posterior distribution calculations
States
Survey
Positive
Negative
FAV
0.5
UNF
0.5
0.7
0.2
0.35
0.3
0.8
0.15
The
sum
P(FAV│ “.”)
P(UNF│ “.”)
0.10
0.45
0.78
0.22
0.40
0.55
0.27
0.73
25
$49,200
Surve
y
1
S ur
Neg . Res.
. (.5
5)
No
s
urv
ey
$106,400
2
plant
e
g
r
La
Small $63,600
plant
3
No
pla
nt
-$87,400
$2,400
$49,200
$40,000
es. )
R
.
5
r
Su s. (.4
Po
$106,400
2nd decision point
1st decision point
plant
e
g
r
a
L
Small
No plant
pla
nt
t
plan
e
Larg Sm
all
pl
an
t
No p
lant
4
$2,400
5
$10,000
6
$40,000
7
Fav. Mkt (0.78)
Unfav. Mkt (0.22)
$190,000
-$190,000
Fav. Mkt (0.78)
$90,000
Unfav. Mkt (0.22)
-$30,000
-$10,000
Fav. Mkt (0.27)
Unfav. Mkt (0.73)
Fav. Mkt (0.27)
Unfav. Mkt (0.73)
Fav. Mkt (0.5)
Unfav. Mkt (0.5)
Fav. Mkt (0.5)
Unfav. Mkt (0.5)
$190,000
-$190,000
$90,000
-$30,000
-$10,000
$200,000
-$180,000
$100,000
-$20,000
$0
Hence, if the survey results are favorable, the company should build a large
plant. However, if they are unfavorable, it should build a small plant.
The efficiency of the survey = EVII/EVPI = (19,200)/(60,000) = 32%
Example 3: Texaco vs. Pennzoil
In early 1984, Pennzoil and Getty Oil agreed to the terms of a merger. But
before any formal documents could be signed, Texaco offered Getty a
substantially better price, and Gordon Getty, who controlled most of the
Getty stock, reneged on the Pennzoil deal and sold to Texaco. Naturally,
Pennzoil felt as if it had been dealt with unfairly and immediately filed a
lawsuit against Texaco alleging that Texaco had interfered illegally in the
Pennzoil-Getty negotiations.
Pennzoil won the case; in late 1985, it was awarded $11.1 billion, the largest
judgment ever in the United States at that time. A Texas appeals court
reduced the judgment by $2 billion, but interest and penalties drove the total
back up to $10.3 billion. James Kinnear, Texaco’s chief executive officer, had
said that Texaco would file for bankruptcy if Pennzoil obtained court
permission to secure the judgment by filing liens against Texaco’s assets.
Furthermore, Kinnear had promised to fight the case all the way to the U.S.
Supreme Court if necessary, arguing in part that Pennzoil had not followed
Security and Exchange Commission regulations in its negotiations with Getty.
In April 1987, just before Pennzoil began to file the liens, Texaco offered to
pay Pennzoil $2 billion to settle the entire case. Hugh Liedtke, chairman of
Pennzoil, indicated that his advisors were telling him that a settlement
between $3 and $5 billion would be fair.
What do you think Liedtke should do? Should he accept the offer of $2
billion, or should he refuse and make a firm counteroffer? If he refuses the
sure $2 billion, he faces a risky situation. Texaco might agree to pay $5 billion,
a reasonable amount in Liedtke’s mind. If he counteroffered $5 billion as a
settlement amount, perhaps Texaco would counter with $3 billion or simply
pursue further appeals.
Below is a decision tree that shows a simplified version of Liedtke’ s
problem.
Experts expect that the Supreme Court will keep the fine with only 20%
chance, will reduce it to $5 billion with 50% chance or will eliminate it
completely with a 30% chance.
It was also believed that Texaco accepts a counter-offer of $5 billion with
1/6 chance and would place a counter-offer of $3 billion with 1/3 chance.
1. Find the optimal strategy.
2. Calculate the EVPI regarding Texaco’s reaction to a counteroffer of $5
billion? Can you explain this result intuitively?
3. The timing of information acquisition may make a difference.
a. suppose that Penzoil could obtain information about the final court
decision before making his current decision (taking the $2 billion or
counteroffer $5 billion). What would be the EVPI of this information?
b. Suppose that Penzoil knew that it would be able to obtain perfect
information only after it has made its current decision but before it
would have to respond to a potential Texaco counteroffer of $3 billion.
What would be the EVPI in this case?
4. In question 3, EVPI for (b) should be less than EVPI calculated in (a). Can
you explain why?
5. What is EVPI if Liedtke can learn both Texaco’s reaction and the final
court decision before he makes up his mind about the current $2 billion
offer? Can you explain why the interaction of the two bits of information
should have this effect?