Preference and choice

Introduction

In this chapter, we begin our study of decision-making theory.

individuals when considering it in a completely abstract framework. The remaining chapters of
Part 1 develops the analysis in the context of explicitly economic decisions.

The starting point for any individual decision problem is a set of possible
alternatives (mutually exclusive) from which the individual must choose. In the
discussion that follows, we denote this set of alternatives abstractly by X .
For the moment, this set can be anything. For example, when an individual is
faces a decision on which career to pursue, the alternatives in X could be: {Go to school
of law, go to graduate school and study economics, go to business school, ...
to become a rock star}. In chapters 2 and 3, when we consider the problem
of consumer decision, the elements of the set X these are the possible options of
consumption.

There are two different approaches to modeling individual choice behavior.

First, which we present in section 1.B, addresses the preferences of the decision-maker.
summarized in relation to preference, as the primitive characteristic of the individual.
theory is developed by first imposing axioms of rationality on preferences of
decision maker and then analyzing the consequences of these preferences for their
choice behavior (that is, in the decisions made). This approach based on the
preferences is the more traditional of the two, and it is the one we emphasize throughout the book.

The second approach, which we developed in Section 1.C, addresses choice behavior.
of the individual as the primitive characteristic and proceeds to make assumptions directly
about this behavior. A central assumption in this approach, the weak axiom of the
revealed preference imposes an element of consistency in choice behavior,
in a parallel sense to the assumptions of the redundancy of the approach based on the
preferences. This choice-based approach has several attractive features. Leaves
space, in principle, for more general forms of individual behavior than what is
possible with the preference-based approach. It also makes assumptions about
objects that are directly observable (choice behavior), rather than things
which are not (preferences). Perhaps most importantly, it makes clear that the theory of decision making
individual decisions do not need to be based on a process of introspection, but rather
A purely behavioral basis can be established.
The understanding of the relationship between these two different approaches to modeling the
individual behavior is of considerable interest. In section 1.D this is investigated
question examining first the implications of the preference-based approach for the
choice behavior and then the conditions under which choice behavior
it is compatible with the existence of an underlying preference (a matter that also appears
in chapters 2 and 3 for the most restricted configuration of demand for the
consumers).

For an in-depth advancement, the treatment of the materials in this chapter, see Richter.
(1971).

1.B Preference relationships

In the preference-based approach, the goals of the decision maker are summarized.
in a relationship of preferences, which we denote by ≽ .Technically, ≽ it is a relationship
binary in the set of alternatives X, allowing for the comparison of pairs of alternatives
yx,x is greater Xor equal
ϵ than We read
to y like x is at least as good as y of>
we can derive two other important relationships in X :
(I) The strict preference relationship, ¿ defined by
x> y is equivalent to x, therefore y is greater than or equal to x
and it reads " x it is preferable toy".¹

(Ii) The relationship of indifference.∼ , defined by

x i s s i⇔
m i l axr t≽o yyyy is greater than or equal to x
and it readsxis indifferent toy

In much of microeconomic theory, it is assumed that individual preferences are

rational. The hypothesis of rationality is embodied in two basic assumptions about
from the relationship of preference: complete study transit visibility.²

Definition 1.B.1: the preference relation≽ it is rational if it possesses both properties

next:
I) Integrity: For everything x , y ϵ Xwe have to x ≽ yoy is greater than or equal to x(o
both).
II) Transitivity: For everythingx , y, z a n d X , yesx is greater than or equal to yyy is greater than or equal to z, thenx is greater than or equal to z.
The assumption that ≽ It is complete, it says that the individual has a good preference.
defined between two possible alternatives. The solidity of the assumption should not be underestimated.
integrity. Introspection quickly reveals how difficult it is to evaluate alternatives that are
far from the realm of common experience. Serious work and reflection are needed to discover its
own preferences. The axiom of completeness states that this task has taken place: our
decision-makers only make media choices.

Transitivity is also a strong assumption, and it goes to the heart of the concepts of

1. The symbol <> is read as 'if and only if'. Literature sometimes refers to x> as 'x is
weakly preferred to y" and x > y as "x is strictly preferred to y". We will adhere to the
previously presented terminology.

Note that there is no unified terminology in the literature; weak order and complete pre-order.
they are common alternatives to the term rational preference relation. Furthermore, in some
presentations, the assumption that it is reflexive (defined x x for all x in X) is added to the
completed and the assumptions of transitivity. This property is, in fact, implied by the
integrity and therefore it is redundant.
Transitivity implies that it is impossible to address the decision together with a sequence of
choice pairs in which their preferences seem to align with the cycle: for example, to feel that a
an apple is less good than a banana and that a banana is less good than a
orange, but then also prefers an orange over an apple. Like the property of
Completed, transitiveness can be difficult to satisfy when evaluating alternatives far from the
common experience. However, compared to the property of completeness, it is also
more fundamental in the sense that substantial parts of economic theory do not
they would survive if economic agents could not assume that they had transitive preferences.

The assumption that the preference relation ≥ is complete and transitive has implications.
for the strict preference relationships and indifference relationships > and ~. These are
recounted in Proposition 1.B.1, whose test is waived (after completing this
section, try to establish these properties yourself in exercises 1.B.1 and A.B.2)

Proposition 1.B.1: If ≥ is rational then:

i) it is both irreflexive (x > x never happens) and transitive (if x > y and y > z then
x > z).
ii) ~ is irreflexive (x ~ x for all x), transitive (if x ~ y and y ~ z, then x ~ z), and
symmetric (if x ~ y, then y ~ x)
iii) If x > y ≥ z, then x > z.

The irreflexivity of > and the irreflexivity of the symmetry of ~ are sensitive properties for
strict preferences and indifference relations. An even more important point in the Proposition
1.B.1 is that the rationality of ≥ implies that both > and ~ are transitive. In addition, a
The transitive property is also valid for > when it is combined with a property.
The transitive property also holds when combining a relationship at least as
Good how, ≥.

Some individual preferences may fail to satisfy the property of transitivity due to
a number of reasons. One difficulty arises due to the problem of the barely existing differences
perceptible. For example, if we ask a person to choose between two shades of gray very
similar to paint your room, you may be unable to distinguish colors and, therefore,
it will be indifferent. Let's suppose now that we offer an option between the clearer of the two
gray paintings and a slightly more subdued tone. She may again be unable to distinguish.
we continue like this, letting the colors of the paint become progressively lighter
With each successive choice experiment, she can express indifference at each step.
Still, if offered between the original (darker) shade of gray and the final (almost white) color,
she could be able to distinguish between colors and is likely to prefer one of them.
However, this would violate transitivity.

Another potential problem arises when the way in which the alternatives are presented is
matter of choice. This is known as the framing problem. Let's consider the
Next example, paraphrased from Kahneman and Tversky (1984):

Imagine that you are about to buy a stereo for 125 dollars and a calculator of
15 dollars. That seller tells him that the calculator is on sale for 5 dollars less.
another store of latenda, located 20 minutes away. The stereo is the same price there.
Could you make the trip to the other tent?
It turns out that a part of the respondents answered that they would travel to the other store for the
a discount of 5 dollars is much greater than the fraction they say they would travel when the
The question changes so that the saving of 5 dollars is in the stereo. This is so that even
it was thought that the final savings obtained by incurring the inconvenience of traveling is the same in
both cases. In fact, we expect the response to the following questions to be indifferent:

Due to a shortage that must travel to the other side to obtain the two elements, but
You will receive a $5 discount on any of the items as compensation. Will you
Which item is this 5 dollar discount applied to?

If so, however, the individual violates transitivity. To see this, denote

Travel to the other store and get a 5 dollar discount on the calculator.

Travel to the other store and get a 5-dollar discount on the stereo.

z= Buy both items in the first store

The first two elections say that x > zy y > z, but the last election reveals x ~ y. Many
framing problems arise when individuals face choices between
alternatives yield uncertain results (the topic of chapter 6). Kahneman and Tversky
(1984) provides a series of other interesting examples.

At the same time, it is often the case that the seemingly intransitive behavior
it can be explained fruitfully as the result of the interaction of several
more primitive rational preferences (and therefore, transitive). Consider the following two
examples.

(i) A family consisting of mom (M), dad (d), and child (C) makes decisions for
majority voting. The alternatives for Friday night entertainment
to attend an opera (O), a rock concert (R) or a skating show
(I). The three members of the household have rational individual preferences:
O> MF R> MFI ,I> DFO> DF R, R> CF I > CF O , where ¿ M, ¿ D, ¿C they are relationships of
strict transitive individual preferences. Now imagine three majority votes:
O versus R, R versus I and I versus O. The result of these votes (O will win the first,
R the second and I the third) will make the household preferences ≥ have the form
intransitive: O> R> I> O. (The intransitivity illustrated in this example is known
like the Condorcet paradox, and it is a central difficulty for the current of
group decision making. For more information, see Chapter 21.)

(ii) Decisions of intrusiveness can also be seen at some point as a

manifestation of a change in tastes. For example, a potential smoker can
prefer to smoke one cigarette a day rather than not smoke and may prefer not to smoke over smoking
strongly. But once she is smoking a cigarette a day, her tastes
they can change, and she may want to increase the amount that smoke.
Formally, let y be abstinent, x be a smoker of one cigarette a day, and z
a habitual smoker, his initial situation is, and his preferences in which the
Initial situations are x > y > z. But once x is chosen over y and z, and there is a
change from the individual's current situation from y to x, their tastes change to z > x > y.
Therefore, we have an intransitivity z > x > z. This change of model of the
Gustostene has an important theoretical significance in the analysis of behavior.
 addiction. It also raises interesting questions related to commitment.
in decision making [see Schelling (1979)]. A decision maker
rational will anticipate the induced change in tastes and therefore will try to tie its
hand to his initial decision (Ulysses had tied the mast to approaching the island of the
mermaids
It often happens that this point of view on changing tastes gives us a way
well-structured way of thinking about non-rational decisions. See Elster (1979) for
philosophical discussions on this and other similar points.

Utility functions

In economics, we often describe preference relationships through a

utility function. A utility function u ( x ) assign a numerical value for
each element of X, classification of elements of X in accordance with the
preferences of individuals. This is more accurately stated in the definition
1.B.2

3. Kahneman and Tversky are credited with finding that individuals maintain
"mental calculations" in which the savings are compared to the price of the item in the
what is received.

elements of X according to individual preferences. This is stated more precisely

in definition 1.B.2.

Definition 1.B.2: A function u : X → R it is a utility function that represents the

preference relationship ≳ yes, for everything ∈
yx,X
x is preferred
( yto) y, and y is greater than or equal to u
Please note that a utility function represents a preference relationship. ≳
it is not unique.

For any strictly increasing function f:R→R, v ( x )=f (u ( x) ) it's a new one
utility function that represents the same preferences as u ( ∙ ) See the exercise
1.B.3. The properties of utility functions that are invariant for any
strictly increasing transformations are called ordinal. The cardinal properties are
those that are not preserved under all these transformations. Therefore, the relationship between
Preference associated with a utility function is an ordinal property. On the other hand, the
numerical values associated with the alternatives in X, and therefore the magnitude of any
difference in Utility between alternatives, they are cardinal properties.

The ability to represent preferences by a utility function is closely

linked to the assumption of rationality. In particular, we have the result shown in the
proposition 1.B.2.
Proposition 1.B.2: A preference relation ≳ can be represented by a function of
utility only if it is rational.

Demostration: To demonstrate this proposition, we show that if there is a utility function

what represents preferences ≳ , then ≳ It must be complete and transitive.

Completed. Because u ( ∙ ) it is a real-valued function defined on X, it must be that

for anyonex , y ∈ X , u ( x ) ≥ u ( y ) y o (uy ) ≥u ( x ) But because of u ( ∙ ) it is a
utility function that represents ≳ , this implies both x is at least as
how preferred
y is
as greater
y than or equal to x
remember
Definition 1.B.2). Therefore, ≳ it must be complete.
Transitivity. Let's suppose thatx i s a p p r oyx i m
y isagreater
t e l ythan
g roreequal r ztDue
a t e to ( ∙y) represents≳ ,
h a unto
we must have u ( x ) ≳ u( y ) y u ( y ) ≳u( z ) Therefore, u ( x ) ≳ u( z ) Because u ( ∙ )
represents ≳ , this implies x i s g. rThus,
e a t ewe
r thave
h a ndemonstrated x i m axt eilsy geryqe ua atyelisrgreater
o r a p p r othat t ot hza than
n oorr equal
a p ptorzo x i m a t e l y e q u a l t o y
they imply x i s gand
r e thus
a t e rthet htransitivity
a n o r aispestablished.
proximately equal to z
At the same time, to ask oneself, can any relationship of preference ≳ rational being
described by some utility function? It turns out that, in general, the answer is no. A
An example in which it is not possible to do so will be discussed in section 3.G. A case in which
We can always represent a rational preference relationship with a utility function.
it arises when X is finite (see exercise 1.B.5) The results of utility representation
more interesting (e.g., for sets of alternatives that are not finite) They will be presented in
subsequent chapters.

1.C election rules

In the second approach of decision-making theory, choice behavior

it is taken as the primary object of the theory. Formally, the behavior of
election is represented by a choice structure. A choice structure
( B,C (∙ ) ) It consists of two ingredients:
(I) B it is a family (a set) of nominal subsets of X; That is, each element
of B it's a set B i s a Bys u banalogy
s e t o fwith
X consumer theory that is
it will be developed in chapters 2 and 3, we call the elements B ∈ B sets
budget. The set of budgets in B it should be considered as a list
exhaust from all choice experiments that the social institutional, physical or
In another restricted way, it can conceivably pose to the decision-maker. It is not
necessary, however, to include all possible subsets of X. In fact, in the case of the
consumer demand studied in later chapters, it will not.

(ii) C ( ∙ ) it is a rule of choice (technically, it is a correspondence) that analyzes a

non-empty set of chosen elements C ( B) ⊂B for each budget set
B ∈ B . When C ( B) contains only one element, that element is the choice of
individuals among the alternatives in B. The set C ( B) it can, however, contain more
of an element. When it does, the elements of C ( B) they are the alternatives in B that the
decision maker could choose; that is, they are their acceptable alternatives in B. In this
In that case, one can think that the set ww contains those alternatives that really
we would see chosen if the decision maker faced the problem repeatedly of
choose an alternative from set B.
 Example 1.C.1: suppose that X = {zyx, } y B= { {x , y}, { x , y , z}} . A possible
the election structure is ( B , C1 (∙ )) , where the rule of choice C1 ( ∙ )
C1 ( { x, y }) = { x }
y C1 ( { x })= { x } In this case, we see x chosen regardless of the
Budget that the decision-maker faces.

Another possible choice structure is ( B,C 2 (∙ )) , where the rule of choice C2 ( ∙ )

C2 ( { x, y }) = { x } y C2 ( { x , y , z})= { x } In this case, we see x chosen every time the
decision maker faces { xy, } budget, but we can see x or y chosen
when facing budget {zyx, } .
When choice structures are used to model individual behavior, it is
possible that you wish to impose some "reasonable" restrictions regarding the
choice behavior of an individual. An important assumption, the weak axiom of the
revealed preference [first suggested by Samuelson; See chapter 5 of
Samuelson (1947) reflects the expectation that the choices observed by individuals
they will show certain consistency. For example, if an individual chooses x alternative (and only that)
when faced with a choice between x and y, we would be surprised to see her choose y when she
face a decision between x, y, and a third alternative z. The idea that the choice of x
when faced with the alternatives { xy, } reveals a propensity to choose x over y that
We should expect to see reflected in the behavior of individuals in response to the alternatives.
{zyx, } 4

The weak axiom formally stated in definition 1.C.1.

Definition 1.C.1: the choice structure ( B,C (∙ ) ) satsface the weak axiom of the
revealed preference if the following property is fulfilled:
If for some B ∈ B withx , y ∈ B we havex i s i n C ( B ) , then for any
B nx
with i , syi ∈' B B ' y n )C , we must also havex b e l o n g s t(oB 'C)
y i s (iB'

In words, the weak axiom says that x is always chosen when y is available, then
There cannot be a budget set that contains both alternatives for which it is.
chosen and x is not.
This proclivity could reflect some underlying 'preference' for x over y, but it could also arise in other ways; for example, it could be the result

of some evolutionary process.

Observe how the assumption that choice behavior satisfies the weak axiom
capture the idea of consistency: if c ({x, y}) = {x}, then the weak axiom says that we cannot
having C ({x, y,z}) = {y }.

A slightly simpler assertion of the weak axiom can be obtained by defining a relationship of
revealed preference ≽ of the choice behavior observed in C (.)

Definition 1.c.2: given an election structure (B, C (.)) The revealed preference relation
≽ is defeated by.

X ≽ * y ↔ there exists some B in Β such that x, y are in B and x is in C (B).

We read ≽ * as 'x is revealed to be at least as good as y'. Note that the

revealed preference relation ≽ * does not need to be either complete or transitive. In particular,
for any pair of comparable alternatives x, y, it is necessary that, for some B in B,
Let x, y be in B, x is in C (B), or y is in C (B), or both.

We could also informally say that 'x is revealed preferred to y' if there is some B in B such that
that x, y are in B, x is in C (B) and y is in C (B), that is, if x is always chosen over y when both are
factbles.

With this terminology, we can reaffirm the weak axiom as follows: 'if x is revealed by the
less than good as and, then and cannot be revealed preferred to x.

Example 1.C.2: Do the two choice structures considered in example 1.C.1 satisfy the
weak axiom? Consider the structure of choice ( B , C1 .) With this structure of
election, we have x ≽ *y y x≽ z, but there is no revealed preference relationship
that can be deduced between y and z. This choice structure satisfies the weak axiom because y
y z are never chosen.

Now let's consider the choice structure (B, C2 .)). Because C2 (x, y, z) = {x,
y}, we have y ≽ *x (just like x ≽ *y, x ≽ *z, y y ≽ *z). But because of the fact that C2
(x, y) = {x}, x is revealed preferred to y. Therefore, the choice structure (B, C2 )
it violates the weak axiom.

We must point out that the weak axiom is not the only hypothesis about behavior of
choice that we can impose in a particular context. For example, in the context of the
consumer demand discussed in chapter 2, we impose additional conditions that
they arise naturally in that context. The weak axiom recovers the behavior of choice
in a way that resembles the use of the assumption of rationality for relationships of
preference. This raises a question: what is the precise relationship between the two approaches? In
In section 1.D, we explore this matter.

1. The relationship between preference relations and choice rules

Now we address two fundamental questions about the relationship between the two approaches.
discussed so far.

5. In fact, it says more: we must have C ({x, y, z}) = {x}, = {z}, or = {x, y}. You will be asked to show
this in exercise 1.C.1. See also exercise 1.C.2.
(i) if a decision-maker has a rational preference order ≥, do they make their decisions
when facing budget set elections in B they necessarily generate a
election structure that satisfies the weak axiom?

(ii) if the behavior of an individual for a family of budget sets B

captured by a choice structure (B, C (.)) that satisfies the weak axiom, is there
necessarily a rational preference relationship that is consistent with these choices?

As we will see, the answers to these two questions are, respectively, 'yes' and 'maybe'.
answer the first question, let's assume that an individual has a relationship of
rational preference ≥ in x. If this person faces a non-empty subset of
alternatives B ⊏ X, its preference maximization behavior is to choose
any of the elements of the set:

C¿ (B, ≽ = {x ∈ B: x ≽ and for each y in B

The elements of the set C (B, ≽ they are the most preferred alternatives of the taker
¿

decisions in B. In principle, we could have C¿ (B,≽ ) = φ for some B; But if X is

finished, or if the right conditions (continuity) are maintained, then C¿ (B, ≽ )
it will not be empty. From now on, we will only consider preferences ≽ and the families
of budget sets B such that C (B, ≽ ) the sea is not empty for all B єB
¿

we say that the relationship of rational preference ≽ generate the election structure B
¿
, C (., ≽ )) .

The result of proposition 1.D.1 tells us that preferences necessarily satisfy the
weak axiom.

Proposition 1: D.1: assumes that ≽ it is a relationship of rational preference. Then the

election structure generated by≽ . ( B , C¿ (., ≽ )), Satsface the weak axiom.
Test: let us assume that for some Bis Bwe have x, y in B and x in C¿ (B,
≽ ). By the definition of C (B, ≽ ), this implies x ≽ and to verify if the axiom
¿

weak holds on, let's suppose that for some B ' there are B we have x,y are weB'
C≽ ), this implies that and≽ for all z that belong to the set
we have and є (B, ¿
B ' But we already know
what x≽ and by transitivity, x≽ for all z belongs to B ' , and thus x is inC ( B ' , ≽ ) This
¿

It is precisely the conclusion required by the weak axiom.

Proposition 1.D.1 constitutes the affirmative answer to our first question. That is, if
behavior is generated by rational preferences then satisfies the requirements of
consistency incorporated in the weak axiom. In the other direction (from choice to the
preferences), the relationship is more subtle. To answer this second question, it is useful
start with a definition.

Definition 1.D.1: given a choice structure (( B we say that the relationship of

rational preference ≽ rationalize C (.) with respect to ( B yes.

C(B)= C ( B ' , ≽ )
¿

For all B in B that is, if ≽ generate the structure of choice (( B , C (.)).

In words, the relationship of rational preference ≽ rationalize the choice rule C (.) In
B if the optimal elections generated by ≽ (captures taken by C¿ (., ≽ ),
They coincide with C (.) For
6. exercise 1.D.2 asks you to establish the absence of C (B ≽ ), For the case when X
¿

It is finished. For the general results, see sections M.F of the mathematical appendix and the
section 3.C for a specific application

All the budget established in B. In a certain sense, preferences explain the

behavior; We can interpret the decisions of the decision-maker as if it were
a preference maximizer. Please note that in general, there may be more than one
preference relation of rationalization≿ For a given choice structure (B, C(∙) )
(See exercise 1.D.1).
Proposition 1.D.1 implies that the weak axiom must be satisfied if there is a relationship of
rationalizing preference. In particular, since C*(∙ , ≿ ) Well, satisfy the weak axiom
for anyone ≿ only a choice rule that satisfies the weak axiom can be
rationalized. However, it turns out that the weak is not sufficient to ensure the existence of
a relationship of rationalizing preference.

Example 1.D.1 suppose thatX = { zx y, } , B¿ {{ x , y} , { y,z } , {x , z}} , C ( {x , y }) = { x } ,

C ( { y, z })= { y } , andC ( {x,z }) = { z } This choice structure satisfies the weak axiom.
(you must verify this).
However, we cannot have rationalizing preferences. To see this, what for
rationalize the options { x } and { y,z } it may be necessary for us to havex ≻y,
y y ≻z. But, due to transitivity, we would thenx ≻ have
z. That contradicts the
choice behavior { xz, }
Therefore, there cannot exist a relationship of rationalizing preference.

Understand example 1.D.1 Please note that the budget sets are more in B.
The weaker the axiom restricts choice behavior, the more there is simply.
opportunities for the decision maker's decisions to contradict each other. In example 1.D.1
the set
1.D.3) how
{
x, y, z} it is not an element of B. As this occurs, it is crucial (see example
now we show in proposition 1.D.2.
If the budget family establishes B includes enough subsets of X, and if (B, C(∙)
If the weak axiom is satisfied, then there exists a rational preference relationship C.(∙) what
Rationalizes relative to B. This was first shown by the arrow (1959).

Proposition 1.D.2. If (B, C(∙) ) is a choice structure that

(i) The weak axiom is satisfied
(ii) B Includes all subsets of X with up to three elements

Then there is a relationship of rational preference≿ what rationalizes C(∙) in relation to B

that is to say, C ( B)=C * ( B , ≿ ) for everyone ϵ B Additionally, this relationship of
Rational preference is the only preference relation that does it.

Test: The natural candidate for a rationalizing preference relationship is the relationship of
revealed preference ≿ To test the result, we must first show two things: (i)
what ≿ * it is a rational preference relationship, and (ii) that ≿ * rationalize C(∙) about
B. Then we argue, as point (iii), that ≿ it is the unique preference relationship
that makes it.

(i) First we check that≿ It is rational (that is, it satisfies completeness

and transitivity.
Completeness of course (ii) { xy, } ϵ Because x or y must be an element ofC ( {x, y })
we must have x≿ *y , o y≿ *x or both. Therefore ≿ It is complete.

Let's go transit x≿ *y
y y≿ Consider the established budget
{zyx, }ϵ i s i n C{zyx,(,
B. It is enough to prove thatx }) , since this implies by the definition of
≿ that x≿ *z Because C ( {x , y , z})=∅ , at least one of the alternatives x, y or z
it must be an element ofC ( {x , y , z}) Assume thaty is an element ( {zyx,of C }) Sincex ≿
*y, then the weak axiom yields x i s ( a{x,n y,e lze}m
) e, nast we
of C wish. Let's suppose in
change thatz is in C ( {x }) since y≿ *z, the weak axiom yields y is in C ( {zyx, }) y
we are in the previous case.

B,≿ ϵ B; That is, the

(ii) Now we show thatC ( B)=C * for everybody
¿
revealed preference relation≿ * deduced from C(∙) it actually generates C
(∙) Intuitively, this seems reasonable. Formally, we show this in
two steps. First, let's assume thatx i s a n e l e(m B)e. n t So
ox≿
f Cand for everything
y is in B
so we have x i s a nCe l*e mBe,≿n t oThis
f means that
¿
C ( B) ⊂ C * B,≿ Next, let's assume that x i s a nCe* l e m e n t o f
¿
B,≿ x≿ and for everything y is in B And so for each y is in B
This implies that
¿
There must exist some set By ϵ B such that xy∈ , Byy x ϵ C (By).
Because C ( B) ≠∅ , the weak axiom then implies that x i s ( iBn) CFor the
B≿
, *) ⊂ C ( B) . Together, these inclusion relationships
so much C *
¿
imply thatC ( B)=C* B,≿
*).
¿
(iii) To establish uniqueness, just note that because B
includes all the subsets of two elements of X, the behavior of
election in C(∙) completely determines the preference relationships in
pairs about X of any preference for rationalization.
This completes the test

We can conclude from proposition 1.D.2 that for the special case in which the election is defined
for all subsets of X, a theory based on choice that satisfies the axiom
weak is completely equivalent to the decision-making theory based on
rational preferences. Unfortunately, this special case is too special for the
economics. For many situations of economic interest, such as demand theory of
consumer, the choice is defined only for special types of budget sets. In
In these contexts, the weak axiom does not exhaust the implications of preference choice.
rational. We will see in section 3.J, however, that the strengthening of the weak axiom
(which imposes more restrictions on choice behavior) provides a condition
necessary and sufficient for the behavior to be rationalized by the
preferences.

Definition 1.D.1 defines a rationalizing preference as a C ( B)=C * ( B , ≿ ) .

An alternative notion of a rationalizing preference that appears in the literature requires
just that C ( B) ⊃ C * ( B , ≿ ) ; that is,≿ It is said that it rationalizes C.(∙) in B
yes C ( B) it is a subset of the most preferred options generated by ≿,C *
(B , ≿ ) for each budgetB ϵ B.
There are two reasons for the possible use of this alternative notion. The first is, in a sense
philosophical. It is possible that we want to allow the decision-maker to resolve
your indifference in some specific way. We may want to allow the person
that makes the decision resolve its indifference in a specific way, instead of insisting
in which indifference means that anything can be chosen.
The vision embodied in definition 1.D.1 (and implicitly in the weak axiom as well) is
that if she chooses in a specific way then she is, de facto, not indifferent. The second
the reason is empirical. If we are trying to determine based on the data whether the choice of a
individual is compatible with the maximization of rational preference, we will have in the
practice only a finite number of observations on the choices made of any
set budget B determined.
If C ( B) It represents the set of choices made with this limited set of
observations, then due to these limited observations may not reveal all
the options for maximizing the preferences of decision-makers is the requirement
natural C ( B) ⊃ C* (B , ≿ ) to impose a preferential relationship for
rationalize the observed choice data.

Two points are worthless about the effects of using this alternative notion. First, it is a
weakest requirement. As long as we can find a preference relationship that
Rationalizes the choice in the sense of definition 1.D.1, we have found one that does so.
also in its other sense. Secondly, in the abstract context studied here,
finding a rationalizing preference relationship in this last sentence is actually
trivial: the preferences that the individual has among all the elements of X will rationalize
any choice behavior in this sense. When this alternative notion is
it is used in economic literature, there is always an insistence that the preference relationship
Rationalizing must satisfy some additional properties that are constraints.
natural to the specific economic context being studied.
REFERENCE

Arrow, K. (1959). Functions of rational choice and arrangements. Econometrics 26: 121-27.

Ulises and the Sirens

Kahneman, D. and A. Tversky. (1984). Choices, values, and frames. American psychologist 39: 341-
50.

Plotl. C. R. (1973). Path to Independence. Chapter 2 in Preferences, Utility, and Demand

edited by J. Chipman, L. Hurwiez, and H. Sonnenschein. New York: Harcourt Brace Jovanovich.

Samuelson.P.(1947). Fundamentals of Economic Analysis. Cambridge, MA: Harvard University

Press.

Schelling.T.(1979). Micromotives and Macrobehavior. New York: Norton.

Thurstone, I. I. (1927). A comparative right of judgment. Psychological Review 34: 275-86.

EXERCISE

1.B.1Bdemonstrate property (iii) of Proposition 1.B.1.

1.B.2Ademonstrate properties (i) and (ii) of proposition 1.B.1.

1.B.3Bthey show that if f: R → R it is a strictly increasing function and u : X → R is

a utility function that represents the preference relationship ≽ then the function
v:X→Rdefined byv ( x )=f (u ( x ) ) it is also a utility function that represents
the preference relationship ≽ .

1.B.4Aconsider a rational preference relation≽ Demonstrate that if u ( x )=u ( y )

impliesx i s s i m i l a r t ou ( x
y )and
>u ( yif) impliesx> ythen u(.) it is a function of
utility that represents ≽ .

1.B.5Bthey show that if X is finite and ≽ it is a relationship of rational preference in X, then

there is a function u : X → R of usefulness that represents ≽ Suggestion: Let's consider
first of all, the case in which the individual's ranking between any two elements of
X is strict (that is, there is never indifference), and to establish a utility function that
represent these preferences; then extend your argument in the general case.

{x } , {x , y, z }
1.C.1Bconsider the choice of structure ( B,C ( . ) ) With y
B=¿
C ( {x, y }) = { x } Prove that ifB,C ( ( .) ) satsface the weak axiom, we must have
C ( {x , y , z})= {x } , {z } , o ¿ { zx, } .

1.C.2Bthey show that the weak axiom (definition 1.C.1) is equivalent to the following property
celebration
Suppose thatB, B' ∈ B, yBxni, erya yb ,ex l o n g t o B ' So, if x i s (iBn ) C y
y i s i n C (we '
B )must have { xy, } ⊏C ( B) y { x }⊏ '
C( B ) .

1.C.3CLet's assume that the election structure ( B , C ( . ) ) weak axiom of choice. Consider
the following two possible revealed preferred relationships, ≽∗¿ y ≽∗¿ :
 x is greater than or equal to∗⇔ There are someB ∈ B such thatx , y( Bi) n yB ,yxi si nn( BC
o) t i n C

x is greater than or equal to∗¿But x≽∗yy≽∗x

⇔ no
Where ¿∗¿ the revelation is at least as good as defined in definition 1.C.2.

(a) Prove that ¿∗¿ y ¿∗¿ and the same relationship throughout X; that is to say,
for anyonex , y ∈ X , x > ¿ y≠x >¿∗yIs this still true if
( B,C ( . )) Does the weak axiom not hold?

(b) ¿∗¿ Should it be transitive?

(c) Show that if B include the three elements of subsets of X, then

¿∗¿ it is transitive.

1.D.1BGive an example of a choice structure that can be rationalized by several

preference relationships. Please note that if the budget family B includes two
elements of all the subsets of X, then there may be more than one rationalization
of the preference relationship.

1.D.2Athey show that if X is finite, then any rational preference relationship generates
a non-empty choice rule; That is, C(B) ≠ ø for any B⊂ X with B is not equal to empty.

1.D.3BLet X = (x, y, z), and consider the choice structure (B, c (.)) with

B = {{x,y},{y,z},{x,z},{x,y,z}}

Y C ({x, y}) = {x}, C ({y,z}) = {y} and C ({x,z}) = {z}, as in Example 1.D.1. They show that (B, c (.))
It must violate the weak axiom.

1.D.4Bthey show that a choice structure (B, C (.)) for which there is a relation of
preference for rationalization > satsface the property of trajectory invariance: For each
by B 1 , B 2 ∈ B such that B 1 ᴗ B 2 ∈ B and CB 1( ) ᴗ C ( B 2 ) ∈ B, we have C (
B1 ᴗ B 2 ) = C (C ( B 1 B 2 )) means that the decision problem can
to subdivide safely. See plot(1973) for the second discussion.

1.D.5CLet X = {x, y, z} and B = {{x, y}, {y, z}, {z, x}}. Let's assume the choice is now
it is used in the sense that for each B∈ B, C (B) is a frequency distribution over
the alternatives in B. For example, if B = {x, y}, we write C (B) = ( C x B) Cy (B), where
C xCy(B)
C x and (B) are non-negative numbers with (B) + Cy (B) = 1. We say that
The stochastic choice function C (.) can be rationalized by preferences if we can
find a probability distribution Pr over the six possible preference relations
(strict) in X such that for each B∈ B, C (B) is precisely the election frequency
induced by Pr. For example, if B = {x, y}, then C x (B) = Pr ({>: x> y}). This concept is
originates in Thurstone (1927), and is of considerable econometric interest (in fact, it provides
a theory for the term of error in observable choice.
1
(a) show that the stochastic selection function C ({x, y}) = C ({y, z}) = C ({z, x}) = ( ,
2
1
It can be rationalized by preferences.
2
 1
(b) shows that the stochastic choice function C ({x, y}) = C ({y, z}) = C ({z, x}) = ( ,
4
3
it is not rationalizable by preferences.
4
Determine the 0 < α <1 in which C ({x, y}) = C ({y, z}) = C ({z, x}) = (α,1-α) changes from
rationalizable to non-rationalizable.