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Analysis of a Decision Tree
Q: What question are we trying to answer?
A: Whether or not we should buy some market research data that has been offered to us for $5,000.
D: We have actually answered this question three times, with EVPI, EVSI, and by looking at the expected values
of each major portion of the tree.
Q: Why did we answer it three times?
A: A question of speed.
D: EVPI is a quick, but rough estimate, while EVSI takes longer to set up. Having set up the EVSI tree, we used
the expected values for the third answer.
Q: What did EVPI recommend?
A: To buy the information.
D: We would pay $90K for perfect information, they were asking for only $5K, and our boss said the research
company was pretty good, though certainly not perfect.
Q: What did EVSI recommend?
A: To buy the information.
D: We would pay $8,240 for information of this accuracy, and they are asking only $5,000.
Q: What did we decide when looking at the expected values of each major portion of the tree?
A: To not buy the information.
D: Whether we looked at the decision alternative to “Not Buy” or either of the “Buy – Favorable” or “Buy –
Unfavorable” branches, the decision with the highest expected value was to build. This meant that if we were
going to rely on expected values (EVPI and EVSI), then there was no point in buying the information because it
did not matter what the information said. Whatever it said, we would make the same decision – to build.
Q: So, since we can’t trust the expected values to make our decision for us, how do we analyze the tree?
A: Figure 1 is an outline of the steps we will follow to analyze Figure 2, shown on the next page:
Step
1
2
3
4
5
6
7
8
Instructions
Start on the right-hand side of the tree
Analyze the squares, not the circles
For each decision alternative, compare the probabilities
and payoffs for one state-of-nature
Identify risks and rewards, tradeoff, and decide
Repeat for each remaining state-of-nature
Reconcile any conflicts to make an overall decision
Move down to the next square on this level.
When done with a level, use the decisions you have
made to analyze the next set of squares to the left
Page 1 of 12
684085936, Page 2 of 12
Figure 1: Steps to Analyze a Decision Tree
High (.4)
1200
Expand
Mod (.4)
660
500
Low (.2)
-100
High (.4)
2500
Not Buy
1380
Build
Mod (.4)
1380
1200
Low (.2)
-500
High (.4)
1500
Second
Mod (.4)
910
800
Low (.2)
-50
High (.50)
1195
Expand
Mod (.43)
803
495
Low (.07)
-105
High (.50)
2495
Favorable
1383
1726
Build
Mod (.43)
1726
1195
Low (.07)
.56
-505
High (.50)
1495
Mod (.43)
Subcont
1086
Buy
795
Low (.07)
-55
1383
High (.28)
1195
Expand
Mod (.36)
475
495
Low (.36)
-105
High (.28)
2495
.44
Unfavorable
947
Build
Mod (.36)
947
1195
Low (.36)
-505
High (.28)
1495
Mod (.36)
Subcont
685
795
Low (.36)
-55
684085936, Page 3 of 12
Figure 2: Decision Tree
Q: Why do we start on the right-hand side?
A: To make our job easier.
D: You may think the tree in Figure 2 is too complex to easily analyze, but it is rather simple if we take it small
pieces. By starting on the right, we don’t have to look at the whole tree, just one small piece at a time.
Q: Why do we analyze the squares, but not the circles?
A: The squares represent decision alternatives, which we control, while the circles show states-of-nature, which we
don’t control.
D: Since the purpose of analysis is to lead to a decision, you wouldn’t want to waste time analyzing something
where you can’t make a decision (that is, circles). So, we concentrate on the squares.
Q: Why do we look at the states-of-nature one at a time?
A: They are mutually exclusive and, again, it makes our job easier.
D: As only one state-of-nature will eventually turn out to be the true one, it doesn’t make any sense to look at all of
them together. Besides, there is too much data to do that. Comparing all of the alternatives for one state-ofnature is similar to the EVPI approach: if we knew the future were going to be that state-of-nature, which
alternative would we pick?
Q: What happens when we repeat the analysis for the other states-of-nature?
A: We get an answer for each of the other states-of-nature.
Q: Why do we have to reconcile anything?
A: You may (most likely, will have) choose different alternatives for different states-of-nature.
D: As you must make a single, overall decision, you must look at the various alternatives you have preferred under
different conditions and, in effect, ask yourself where you are most willing to be wrong.
Before beginning the analysis, I want to point out this is a very simple tree, mostly because the probabilities are
not affected by the decision alternatives.
Q: How can the decision alternatives affect the probabilities?
A: Quite easily.
D: Suppose you are making plans for Spring Break. Your alternatives are that you could go out west to the
Rockies to go skiing, or you could go to the Caribbean on a cruise. You might use weather conditions as your
states-of-nature, something like “hot” and “cold.” In this setting, the probability of hot weather is rather more
likely in the Caribbean than it is in the Rockies, and the reverse is true for cold weather.
It is much more typical to have the probabilities changing within a tree than it is to have them stay the same. I
chose this example because it is easy, and because it allows me to illustrate domination in a decision tree.
Q: How do you look for domination in a decision tree?
A: By comparing the payoffs and the probabilities for each alternative
684085936, Page 4 of 12
D: In a payoff table, you did not need to worry about the probabilities changing, but that is a concern for decision
trees.
Q: Should we check the probabilities or the payoffs first?
A: The probabilities.
D: If the probabilities of the states-of-nature change from one alternative to another, then you cannot have
domination, no matter what the payoffs show
Q: If the probabilities do not change, then should we check for domination?
A: Yes.
D: For our convenience, I have copied one square (and the subsequent states-of-nature) into Figure 3, below.
Look at that figure and compare the decision alternatives “Expand” and “Subcontract.” First make sure the
probabilities are the same on the states-of-nature (they are, 0.40, 0.40, and 0.20), and then compare the payoffs
for each state-of-nature. Under “High,” subcontracting wins (1500 to 1200), under “Moderate,” subcontracting
wins (800 to 500) and under “Low,” subcontracting wins (-50 to -100). This means that subcontracting
dominates expansion, and “Expand” can be ignored while analyzing the tree (which makes the tree even easier
to analyze).
One more caveat: this analysis is based on my preferences. You may not agree with all of my choices, because
you may be more or less risk-prone than I am. I will try, though, to explain why I am making those choices as I
go, so you can see the types of explanations you should be providing.
Q: How do we get started?
A: Concentrate on the first square on the right-hand side of the tree.
D: Being an excessively orderly person, I chose to start at the top-right, so the decision node (square) that I am
analyzing is the one for the initial decision to “Not Buy.”
Q: Should we always start at the top-right?
A: That’s up to you.
High (.4)
1200
Expand
Mod (.4)
660
500
Low (.2)
-100
High (.4)
2500
Not Buy
1380
Build
Mod (.4)
1380
1200
Low (.2)
-500
High (.4)
1500
Mod (.4)
Subcont
910
800
Low (.2)
-50
684085936, Page 5 of 12
Figure 3: First Analysis Block
D: A good reason for starting at the top right is that if you approach the tree in an organized way, you are less
likely to make a mistake, such as forgetting to analyze one part of the tree. There is no rule, though, that says
you have to do it that way.
Q: Is there anything that can help us sort through all the data that is still there?
A: Yes, something you have already used: the payoff table
D: You may recall that we drew this portion of the tree by starting with a payoff table. At that time, I mentioned
that any table could be drawn as a tree, and that tables were better for focusing on a single decision. We are not
going to do a full-blown payoff table analysis. Most people, however, see data in a table better than they do in a
tree, so we will present it that way for our own benefit.
Q: How do we use a payoff table to analyze a decision tree?
A: Begin by drawing the table.
D: Just as you did when setting a payoff table, begin by listing the decision alternatives (remember that “Expand”
is dominated, so it is not shown) down the left-hand side of the table, as shown in Table 1:
Alternatives
Build
Subcontract
Table 1: Decision Alternatives in Table Form
Now, follow your instructions and compare the alternatives for the first state-of-nature, high growth.
Q: What do we do with the payoff table when comparing the alternatives for the first state-of-nature?
A: Copy in whatever data you need.
D: Table 2 shows the payoffs for the “High” state-of-nature.
(0.40)
High
Build
2500
Subcontract
1500
Table 2: Payoff Table with First State-of-Nature
Alternatives
Q: Why didn’t you show the probabilities?
A: I did, above the word “High.”
D: I don’t need to show the probabilities for the decision alternatives, because the probabilities are the same. Since
the probabilities are the same for the two alternatives, they do not affect our decision. That is what makes
analyzing this tree so easy. All we have to do is see that “Build” has a higher payoff that “Subcontract” and we
know that “Build” is the better alternative for the “High” state-of-nature.
Q: As we are finished with the first state-of-nature, what comes next?
A: Move on to the second state-of-nature.
684085936, Page 6 of 12
D: Table 3 (next page) adds the second state-of-nature, “Moderate,” to the table. Again the probability of moderate
growth is the same (40%) for each alternative. Comparing the two on payoffs, “Build” wins (1200 versus 800).
(0.40)
(0.40)
High Moderate
Build
2500
1200
Subcontract
1500
800
Table 3: Payoff Table with Two States-of-Nature
Alternatives
Q: Can we say that “Build” is dominating “Subcontract” so far?
A: Yes, but I wouldn’t.
D: It is a rather pointless statement, because we are about to compare the two alternatives for the final state-ofnature, and then we will know whether or not domination is present. So why raise an issue that may not exist?
Q: Do we get the same result for the third state-of-nature?
A: We use the same process, but get a different conclusion.
D: Table 4 shows all three states-of-nature. Looking at the payoffs for “Low” growth, this time “Subcontract”
wins over “Build,” -50 to -500.
(0.40)
(0.40)
(0.20)
High
Moderate
Low
Build
-500
2500
1200
Subcontract
1500
800
-50
Table 4: Payoff Table for the Decision to “Not Buy”
Alternatives
Q: Do we have our decision yet?
A: No, because we made different choices for different states-of-nature.
D: Now we have to perform Step 6, reconcile the differences. To do this, summarize the risks and rewards for the
two alternatives. One of the nice things about trees is that the risks of one alternative tend to be the rewards of
the other.
Q: What are the rewards of the “Build” alternative?
A: There is an 80% chance of a payoff that is at least $400K better.
Q: Where did that come from?
A: Combining the “High” and “Moderate” states-of-nature where “Build” won and looking at the difference in the
payoffs.
D: “High” and “Moderate” both had 40% probabilities, so that makes an overall 80% chance that “Build” is the
right choice. Under “High” that payoff for “Build” is $1,000K better than “Subcontract” (2500 – 1500) and
under “Moderate” the payoff for “Build” is $400K better (1200 – 800).
Q: What are the risks of the “Build” alternative?
A: There is a 20% chance of losing an extra $450K.
D: That comes from the “Low” state-of-nature.
684085936, Page 7 of 12
Q: Are the rewards and risks for “Subcontract” just the reverse of “Build?”
A: Yes.
D: The reward for “Subcontract” is that there is 20% chance you will have avoided losing an extra $450K, but the
risk is that there is an 80% chance of giving up the opportunity to make at least an extra $400K, possibly an
extra $1,000K.
Q: So how do you decide?
A: Trade-off the risks and rewards.
D: There are three issues here: the additional profit (made or given up), the additional loss (incurred or avoided)
and the probability of each.
Q: Does the amount of the avoidable loss ($450K) make you think twice about the additional profits ($400K and
$1,000K)?
A: Almost certainly.
Q: Does the chance of losing the money affect your decision?
A: It should.
Q: Are you willing to accept a 20% chance of losing an extra $450K (that you could have avoided) to get an 80%
chance of gaining an extra $400K to $1,000K?
A: If you say, “Yes,” then you choose “Build.” If you say, “No,” then you choose “Subcontract.”
D: That’s really all there is to it. Which reward is more attractive to you: avoiding the loss or making the extra
money? Which risk do you want to avoid: losing the extra money, or giving up the chance to make more
money? I choose “Subcontract” because, to me, the 20% probability of losing almost half a million dollars that
I didn’t have to lose is just too much. I think payoffs of $1,500K or $800K are good enough, so I am not all that
attracted by the extra payoffs. Remember: you don’t have to agree with me.
Q: Are we finished with the analysis of the first part of the tree?
A: Yes.
Q: What next?
A: Step 7 says to move on to the next square on the same level.
D: Looking at Figure 2, above, and staying to the right, the next square follows the decision to “Buy” the
information and have it turn out to be “Favorable.” Figure 4 (top of the next page) shows that portion of the
tree.
Q: Do we go through the same process?
A: Yes, but this time we look for changes from the numbers of the first analysis.
Q: Are the payoffs the same?
A: No, they have decreased by $5K, to pay for the survey data.
684085936, Page 8 of 12
Q: Are the probabilities the same?
A: No, they have changed based on the accuracy of the survey data, as calculated via Bayes’ Rule.
High (.50)
1195
Expand
Mod (.43)
803
495
Low (.07)
-105
High (.50)
2495
Favorable
1726
Mod (.43)
Build
1726
1195
Low (.07)
.56
-505
High (.50)
1495
Subcont
Mod (.43)
1086
795
Low (.07)
-55
Figure 4: Decision Tree Following a Favorable Survey Result
Q: Is “Expand” still dominated?
A: Yes.
Q: Do we setup another payoff table?
A: Yes.
D: Table 5 shows the same decision alternatives, but the new probability and payoffs for the “High” state-ofnature. “Build” still wins, and by the same margin, $1,000K.
(0.50)
High
Build
2495
Subcontract
1495
Table 5: Payoff Table with One States-of-Nature
Alternatives
Q: How can the margin be the same?
A: The same amount, $5K, was subtracted from each payoff.
D: All that happened was each number went down by the same amount, so the difference is unchanged, and so is
our preference.
Q: What about the other two states-of-nature?
A: The preferences are the same there as well, as shown in Table 6:
Alternatives
Build
Subcontract
(0.50)
High
2495
1495
(0.43)
Moderate
1195
795
(0.07)
Low
-505
-55
684085936, Page 9 of 12
Table 6: Payoff Table Following a Favorable Survey Result
Q: Does that mean our overall decision is the same?
A: That depends on what decision you made the first time and how you feel about the new probabilities.
Q: How is the analysis the same?
A: The payoffs are effectively unchanged.
D: While every payoff has gone down slightly ($5K), that is at worst 10% of the payoff amount, and usually less
than 1%. The differences between the payoffs are the same, so that part of the analysis is unchanged.
Q: How has the analysis changed?
A: The probabilities have changed dramatically.
D: Previously, we had an 80/20 preference for “Build.” Now, with a favorable survey result, we have 93/7
preference for “Build.”
Q: What are the risk/reward tradeoffs?
A: Are you willing to accept a 7% chance of losing an extra $450K (that you could have avoided) to get a 93%
chance of gaining an extra $400K to $1,000K? Conversely, are you willing to accept a 93% chance of giving
up the chance to make an extra $400K or $1,000K, to avoid a 7% chance of losing an extra $450K?
Q: Isn’t that a little obvious?
A: Maybe not.
D: I’ll admit I am willing to accept that 7% chance of an extra loss, but quite often I will have students that still see
the amount of the loss as too high, and still prefer to subcontract. If you chose to build in the first analysis, then
you would have to choose to build here (otherwise your decision making would be hopelessly inconsistent). If
you chose to subcontract initially, then you may of may not choose to build here. In any case, make your choice
and move on.
Do you notice that the second analysis went a lot faster than the first one? That is because we learned what to
look for in the first analysis and applied that to the second. Also, we could refer back to the reasoning of the
first analysis, which made the explanations a lot shorter. That progression will usually be true for analysis of
decision trees.
Q: What do we analyze next?
A: We stay to the right, move down the tree, and analyze the alternative to “Buy” the information when the results
turn out to be “Unfavorable.” This portion of the tree is shown as Figure 5 at the top of the next page.
Q: How have the payoffs changed?
A: They haven’t, as compared to the second (most recent) analysis.
Q: Have the probabilities changed?
A: Yes, significantly.
Q: Do we need to set up a payoff table for this analysis?
684085936, Page 10 of 12
A: Maybe not, but we will.
D: We have learned a lot about the risks and rewards for these alternatives, so it is tempting to simply make the
decision, but it is better to go slowly. Table 7 follows the tree in Figure 5, on the next page.
High (.28)
1195
Expand
Mod (.36)
475
495
Low (.36)
-105
High (.28)
2495
.44
Unfavorable
947
Build
Mod (.36)
947
1195
Low (.36)
-505
High (.28)
1495
Mod (.36)
Subcont
685
795
Low (.36)
-55
Figure 5: Decision Tree Following an Unfavorable Survey Result
(0.28)
(0.36)
(0.36)
High
Moderate
Low
Build
-505
2495
1195
Subcontract
1495
795
-55
Table 6: Payoff Table Following an Unfavorable Survey Result
Alternatives
Q: What are the risk/reward tradeoffs now?
A: Are you willing to accept a 36% chance of losing an extra $450K (that you could have avoided) to get a 64%
chance of gaining an extra $400K to $1,000K? Conversely, are you willing to accept a 64% chance of giving
up the chance to make an extra $400K or $1,000K, to avoid a 36% chance of losing an extra $450K?
D: Now the chance of losing the extra money is little higher than 1-in-3. This increases the risk significantly. If
you chose to subcontract on the first analysis, with only a 20% chance, then you would certainly choose to
subcontract here, regardless of what you chose in the second analysis. If you chose to build in the first two
analyses, then you have a decision to make. If you think the probability of the loss has gotten too high, then
you would choose to subcontract here. If you are still more attracted by the extra gains (after all, there is still
almost a 2/3’s chance of getting the extra money), then you accept the risk and choose to build. You have to
decide. As implied above, here I would choose to subcontract, because that was my initial choice.
Q: Ok, we’ve finished all the squares to the right-hand side of the tree. What comes next?
A: Back up to the square on the left-hand side of the tree.
D: We are back at the start of the tree, but look at what we have accomplished. By making the decisions on the
right-hand side first, we have eliminated 2/3’s of the branches that need to be considered now.
Q: Do we repeat the process of comparing the decision alternatives for each state-of-nature?
A: Yes.
684085936, Page 11 of 12
D: More precisely, we try to, but quickly realize that the tree is not perfectly symmetric. The “Not Buy” branch
doesn’t have any states-of-nature. All we can do here is set up the payoff table by treating the two “Buy”
outcomes as separate alternatives, as shown in Table 7, on the next page:
Alternative
Not Buy
Favorable
Buy
Unfavorable
Table 7: Payoff Table for Buy/Not Buy Decision
Please note that “Buy – Favorable” and “Buy – Unfavorable” refer to how the survey turns out. They do not
imply you can select what kind of survey information you see.
Q: What data do we put into this table?
A: Whatever you chose during the earlier analyses.
D: This time, the probabilities are changing, so we have to show a little more. Even though we have already made
our decisions for what follows each alternative shown in Table 7, we still have to consider the
High/Moderate/Low growth states-of-nature, so we copy in all of the data, as shown in Table 8:
Alternatives
Decision
Outcomes
High:
40%
Not Buy
Subcontract Moderate: 40%
Low:
20%
High:
50%
Favorable
Build
Moderate: 43%
Low:
7%
Buy
High:
28%
Unfavorable Subcontract Moderate: 36%
Low:
36%
Table 7: Payoff Table for Buy/Not Buy Decision
1500
800
-50
2495
1195
-505
1495
795
-55
These are my decisions. I’ll show the other possibilities later on, but I want to make a point first.
Q: What is the question we are trying to answer?
A: Whether or not to buy the survey information.
D: At the end of the EVSI lecture, I used the expected values of the major branches to show that we did not need to
buy the information. The logic was that since we were going to chose “Build” no matter what the information
told us, there was no benefit to buying it.
We can turn that logic around and ask a different question:
Q: Does the result of the survey information change the decision I make?
A: Yes.
D: If the survey says “Favorable,” then I choose “Build.” If the survey says “Unfavorable,” then I choose
“Subcontract.” Clearly, the survey results matter to me.
Q: What is the change in payoff, moving from a decision to “Build” to a decision to “Subcontract?”
684085936, Page 12 of 12
A: It varies based on the state-of-nature, but either $1,000K, $400K, or $450K.
Q: Would you pay $5K to gain information that could make at least a $400K change in your decision?
A: I would.
D: For me, for the set of decisions I made, the information is worth $5K and a whole lot more. I would be willing
to pay up to $400K for really good survey data. Given that, I might want to tell my boss to commission a
survey, one that will give us more precise information. That may or may not happen (there may not be time, or
your boss might be satisfied with the available information), but it is something to think about.
Notice that in this case, I did not have to really look at the payoffs or probabilities. It was enough to know that
the information would make me change my mind.
Q: What if the information didn’t make you change your mind?
A: Then you wouldn’t buy it.
D: Table 8 shows all of the reasonable decision patterns for this analysis.
Decision Sets
Alternatives
1
2
3
Not Buy
Build
Build
Subcontract
Favorable
Build
Build
Build
Buy
Unfavorable
Build
Subcontract
Subcontract
Table 7: Payoff Table for Buy/Not Buy Decision
4
Subcontract
Subcontract
Subcontract
Find the one that matches your decisions from the analysis. If you are in Set 1 or Set 4, then you wouldn’t buy
the information, it doesn’t affect your decision making. If you are in Set 2 or Set 3, then you just got a bargain,
because that information is worth a lot more than just $5K.
As I said, this was an easy to tree to analyze, because it is a very simple tree. The percentages stayed the same
within a decision node (square), one alternative was dominated, and the final decision did not rest on the
payoffs or probabilities. To show you how we analyze a more complex tree, a have taken an old mid-term
problem and written out an analysis of it. That is the next lecture.
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