Binary Preferences
Zhaochen He
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OR
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is bad at teaching
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that is great at teaching.
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OR
Time travel 200 years
into the past
Time travel 200 years
into the future
The Big Picture
• We will be spending the next few lectures
discussing the most fundamental model in
microeconomic theory: the theory of
consumer choice.
• Consumer choice theory is a mathematical
description of how people might make
purchasing decisions, but can be generalized
to much broader situations.
Meet Mark’s Dilemma
?
The Big Picture
• The mathematics of consumer choice theory
can make a prediction about choice a person
will make, but it needs two pieces of “given”
information.
– A description of the person’s preferences, usually
in the form of a utility function
– A description of the person’s financial situation
(the money he has available, and how expensive
his various options are); usually called a budget
constraint.
The Big Picture
A description of a person’s preferences usually
comes in the form of a utility function.
By the end of this lecture, we’ll begin to talk about
utility functions. But utility functions themselves
are based off of a even more fundamental way to
represent preferences. It all begins with binary
preferences.
• A binary preference is a preference between
two distinct options.
– This is in some sense the simplest form of
preference we could consider.
– When faced with a binary preference A vs B, an
agent could prefer A to B, B to A, or be indifferent
between the two.
– From now on, we’ll write these possibilities as:
• ApB
• B pA
• AiB
• Of course, we often have more than two
options when we make a choice.
• However, we could reduce your preferences
over multiple items to a series of binary
comparisons.
1
2
3
vs
• A good way to represent this set of binary preferences is with
a table.
vs
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vs
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This collection of all binary preferences over a
group of items is called a preference relation
over those items.
vs
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1. Reflexivity – Any good is
indifferent with itself
vs
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2. Symmetry - The table is symmetric
across the diagonal of indifference
vs
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3. Transitivity: If A p B, and B p C,
then A p C
vs
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3. Transitivity: If A p B, and B p C,
then A p C
vs
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3. Transitivity: If A p B, and B p C,
then A p C
vs
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i
3. Transitivity: If A p B, and B p C,
then A p C
vs
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3. Transitivity: If A p B, and B p C,
then A p C
vs
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A B C D
A
i
A B C D
A C D
A
i
C B
B A
i
C
C C C
i
C
D D B C
I
D D D C
I
B A
C C C
Alice
i
A C D
i
Bill
C D
A B C D
A
i
B A
A
A C D
i
C C C
C D
i
C
D D D C
I
With transitive preferences,
we can reduce all of the above
to a simple list, or ranking.
C
B
D
1.C
2.D
3.A
4.B
A
B
C
D
E
A
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B
C
D
E
B
B
i
B
B
E
C
C
B
i
C
E
D
D
B
C
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E
E
E
E
E
E
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1.E
2.B
3.C
4.D
5.A
Option
Utility
E
?
B
?
C
?
D
?
A
?
Utility Functions
• A utility function simply assigns a numerical
value to each option. The SIZE of these
numerical value fully represent the
consumer’s binary preferences over all
choices.
• For example, if he prefers A to B, then the
utility of A will be higher than the utility of B.
Utility Functions
• IMPORTANT: The magnitudes given by a utility
function are not unique – that is, many
different utility functions could describe the
same set of binary preferences.
• Another way of saying this: A utility of 10 isn’t
necessarily “twice as good” as a utility of 5.
– Utility functions are ordinal, not cardinal.
Towards Mark’s Dilemma
• So far, we’ve looked at multiple goods, but
with a quantity of one.
• We could also look at only one good, but
allow any quantity.
• Or, we could look at multiple goods, and allow
any quantity.
One good, any quantity
1
2
3
4
5
1
i
2
3
1
1
2
2
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3
4
5
3
3
3
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4
5
4
4
4
4
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5
5
5
5
5
5
i
1. 1
2. 2
3. 3
4. 4
5. 5
6. 5
7. Etc…
One good, any quantity
1. 4
2. 3, 5
3. 2, 6
4. 1, 7
5. 0, 8
6. 9
7. 10
8. 11
9. Etc.