Lecture 10: Decision Trees and Expected
Utility
A cynic is a man who knows the price of everything and the value of nothing
– Oscar Wilde (Lady Windermere’s Farm)
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Introduction
In the previous lecture, we discussed the basics of probability theory. In this
lecture, we want to apply probability theory to analyze decision making. We
will focus on two concepts: the value of information and the value of control
over a decision. Both of these are covered in great detail by Lave and March.
So, if you do not understand these examples, do some of the problems in
Lave and March.
After the decision tree analysis, we will consider the subject of risk. In
doing so we will differentiate between trees with monetary payoffs and trees
with utility payoffs.
Our discussion of decision trees seems a bit removed from our core motivations of predict, explain, design, and think. Though trees prove helpful for
all four, they might be better thought of as a tool for helping people make
better decisions. The key insight is this: if we assume that people make good
decisions (using trees or not), then trees will be helpful for explaining why
people did what they did.
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Decision Trees And Expected Value
We begin with some simple examples to show how we can use trees to get at
expected value. In the first example, we use a tree to compute a probability.
Example 1: There is a 40% chance that you will get a call back from a firm.
If called back there is a 50% chance that you get a job offer. To compute the
probability that you get the job, you draw a tree beginning with 100 chances,
40 times you will get a call back. And of those 40 call backs, you will get the
job 50% of the time or 20 times.
In the second example, we compute an expected value.
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Example 2: You lose the extension cord for your laptop during a time when
you are contemplating buying a new laptop for $1500. You know that there
is a 50% chance that the same company will introduce a new model of the
laptop. If so, there is a 60% chance it will be $200 cheaper and a 40% chance
it will cost the same amount. You can buy a new extension cord for $50. Do
you do it? Out of 100 chances, 50 times the company will introduce a new
computer. Of those 50 times, 30 times (60%) the new model will cost less.
Therefore, 30% of the time, you save $200. That means that if you wait, you
save $60.
In our last example, we are not given probabilities. Instead, we back out
the appropriate probabilities for each choice.
Example 3: As President, you want to get rid of a dictator who has destabilized a region. If you attack without cause, you anticipate a payoff of 5.
(Ignore the units of the payoffs.) However, you think that this person might
have violated a treaty. You can detect violations 80% of the time. If you find
violations and then attack, your payoff is 9. If you do not find violations,
then you cannot attack and your payoff is 0. How sure do you have to be of
violations in order to want to investigate?
To determine this probability, we draw the tree with 100 chances on top.
Let p be the probability that they are cheating. This means that of the
100 chances, that 100p times they will be cheating and you will detect those
violations 80p times. Therefore, the probability of finding violations is 0.8p
and the payoff is 7.2p. If you do not inspect the payoff is five. You inspect
when 7.2p > 5, or p > 5/7.2 = 69.44%.
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The Value of Information and Control
The simple tree models are easy. Computing the value of information and
the value of control are more difficult. We first construct an example in which
we compute the value of having information.
Example 4: Suppose that I am forced into an agreement with someone
whom I don’t know. I have two options: To trade in good faith or to renege.
Suppose that I have reason to think that there is a 75% chance this person
is honest. If the person is honest and I trade I make $100. If the person is
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dishonest and I trade I lose $200. In either case, if I bail out I lose $100.
How much would I pay for information whether the person is honest?
To answer questions like this we draw two trees. The first tree we use to
compute the expected value without the information. In the second tree,
we compute the expected value with the information. First, we compute my
expected value. If I do not trade, I lose $100. If I do, I gain $100 75% of the
time and lose $200 25% of the time for a net gain of $25. Therefore, I should
trade. If I had perfect information, then I would trade when he is honest and
get $100 75% of the time. And, when he is dishonest, I would not trade and
I would only lose $100. This happens 25% of the time. So, my net would be
$50. Therefore, I would pay $25 for the information.
Here is another example
Example 5: You can buy either a Dell or a Mac computer. You are indifferent between the two but the Dell is $50 cheaper. You read in the paper
that Apple is going to send free IPODs to 20% of their customers. You were
just about to spend $300 on an IPOD. A friend tells you that she has learned
that Apple is using a simple random number generator applied to your social
security number to determine who gets the IPODs. She says that for $20,
she will find out if you would get the IPOD or not. Should you take her up
on her offer?
Example 6: Party A is considering moving its convention until after party
B’s. The reason for this is that they would like to have second mover advantage in the selection of their vice president. Party B will choose a governor
from either California or New York. Party A will choose a senator from one
of three states: California, Virginia, or Ohio. The payoffs to party A for each
pair of choices are as follows:
Virginia
Ohio
California
California New York
5
7
6
2
2
8
Suppose that the probability that part B picks the governor from New
York is 20%. The expected payoff from choosing Virginia is 4 + 1.4 = 5.4.
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The expected payoff from choosing Ohio is 4.8+0.4 = 5.2, and from choosing
California equals 1.6+1.6 = 3.2. Therefore, if A goes first, they pick Virginia
and get a payoff of 5.4. If A picks second and B picks California, A picks
Ohio and gets 6. This happens 80% of the time. If B picks New York, then A
chooses California and gets 8. This happens 20% of the time. The expected
payoff equals 4.8 + 1.6 = 6.4. The gain from full information equals 1.
Note, that the senator from Virginia hates this plan. More information
can mean that a choice that was previously made is now never made. Lave
and March also discuss the value of control. The value of control is computed in the same way as the value of information. The difference between
the two concepts is that the value of control is not buying information but
implementing a plan so that there is no longer any uncertainty.
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What’s it all about?
This is a class about building models. Why did we just study these trees
in such detail when these trees seem to be prescriptive: here’s how to make
good decisions. To repeat what I said at the beginning, we typically assume
that if people do something often enough than they probably don’t do it
too badly. If we’re modeling it from the outside, we can use these trees to
boil a decision down to its essence. Look for example at Lave and March’s
discussion of voting.
To see the power of trees, let’s consider an example that looks at the effect
of unpredictability. Congress has passed a law that forces states to pay for
some environmental damage. Everyone agrees that the law is constitutional
under current law. The states can challenge this law at a cost C. The only
way that the states can win in the courts is if the courts forge new law. The
probability of this is p. The loss to the states of the new law is L. If we draw
out the tree we see that the states challenge only if C + (1 − p) ∗ L ≤ L,
which reduces to C ≤ p ∗ L. A whole bunch of obvious and cool implications
pop out of this.
1. the lower C, the more likely the states challenge (obv)
2. the higher L, the more likely the states challenge (obv)
3. the higher p, the more likely the states challenge (obv)
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4. congress wants L to be big but not too big
5. the court wants p to be large enough that laws get challenged
6. which gets set first L or p makes a huge difference
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Expected Utility
We’ve been using monetary terms and utility terms interchangeably so far.
There is a problem with using money if we’re talking about individuals and
not firms or organizations: risk. Most people would rather get $50 for sure
than $100 half of the time and nothing the other half of the time. (Three
caveats: we like small risks, hence all of this gambling. People tend to be risk
loving over losses. And, for really big risks we hate risk.) Therefore, when
we cut branches off the trees and compute expected values, we are lopping
of risk as well. That’s a mistake.
This begs the question. What about utilities? Is the utility of 5 the same
as half of the time getting a utility of 10 and half of the time getting a utility
of 0? The answer is, it can be if you want it to provided that two assumptions
hold.
Substitution Axiom If x is preferred to y, then a lottery that gives x with
probability p and z with probability (1 − p) is preferred to lottery that gives y
with probability p and z with probability (1 − p).
Archimedean Axiom If x is preferred to y is preferred to z, then there exist
probabilities p and q in (0, 1) such that a lottery that gives x with probability
p and z with probability (1 − p) is preferred to y which is in turn preferred to
lottery that gives x with probability q and z with probability (1 − q).
The first assumption tells us that substituting something better always
makes us happier.. The second says that we can take combinations of something better than y and something worse than y to give us something both
better and worse than y. This assumption rules out extreme risk aversion or
risk loving behavior. I couldn’t have a y that I preferred to any lottery of
something better and worse than it.
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5.1
Does this make sense
As in the case of creating utility functions, we again have these innocuous
looking axioms that give a powerful result. This happens because axioms
build upon themselves quickly. As you might expect, whether people satisfy
expected utility theory is complicated. It is easy to construct examples where
people violate it.
Example 6: Lottery 1: would you rather have $4 million for sure or a 90%
chance at $6 million? Lottery 2: would you rather have a ten in a million
chance at $4 million or a nine in a million chance at $6 million?
If you choose the former and the latter in this example, you violated the
substitution axiom. The only difference between the two lotteries is that the
second includes a big chance at getting nothing.
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Practice Question
You have two options: law school or a government internship. However, your
government contacts are all republican and if the Republicans lose the House,
then your internship won’t be as much fun. Your utility from the internship
if they maintain the House majority equals 8. But your utility if they lose is
only 2. You figure that there is a 75% chance that they will maintain it. Your
utility from going to Law School is 6. You have to decide whether to accept
at the law school by September 1 but the internship deadline is December
1. Suppose that the internship deadline were to be moved to September 1.
Suppose that on September 1 that you figure that you will know either that
the republicans have a 100% chance of keeping the House or that they have
a 50% chance. If each utility unit is worth about $10,000, how much would
you pay to get an extension to September 1st for the internship deadline?
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