Appendix: Some Extensions
Representability
houthakker−1961−e p.709 saw the aim of the revealed preference approach to be “to for-
mulate equivalent systems of axioms on preferences and on demand functions.” The first
issue raised by such a project is representability.
Given a weak (i.e., reflexive) binary relation R, define the choice sets CS(·, R) to be
the set of “R-maximal” elements, so that for any menu A ⊆ X 1
CS(A, R) = {x ∈ A | x R x0 ∀x0 ∈ A}
We say a binary relation R represents (or rationalizes) a choice function C(A) iff C(A) =
CS(A, R) ∀A ∈ D. When R represents C(·), we say choice is representable (or rationalizable) by R. We want to know when it is possible to find a binary relation R such that
CS(A, R) = C(A) ∀A ∈ D (i.e., such that the binary relation rationalizes the choice
function C(·)).
The Base Relation: x RB y ⇔ x ∈ C({x, y})
We will call a choice function binary if it is rationalized by some binary relation R.
That is, if there exists R such that C(A) = {x ∈ A | x R x0 ∀x0 ∈ A} ∀A ∈ [X].
Following Sen (1986), we will call a choice function basic binary if it is rationalized by
the base relation. In this case, C(A) = {x ∈ A | x RB x0 ∀x0 ∈ A} ∀A ∈ [X].
Assumption 1 (Basic Domain)
x, y ∈ A ∈ D → {x, y} ∈ D
(1)
Note that we allow x = y, so the basic domain assumption is that all pairs and singletons
underlying D are in the domain.
1
Equivalently, defining Uw (x) = {y ∈ X | y R x}, we can set CS(A, R) = A ∩ [∩x∈A Uw (x)].
1
Looking ahead:
With finite X, C(·) is basic binary (and therefore binary) whenever C(·) satisfies Houthakker’s
axiom of revealed preference.
Theorem 1
C(A) = CS(A, RB ) ∀A ∈ [X]
x, y ∈ A, B
if (Sufficiency!!)
& x ∈ C(A) & y ∈ C(B) =⇒ x ∈ C(B)
or equivalently
C(A) ∩ B 6= ∅ & C(B) ∩ A 6= ∅ =⇒ C(A) ∩ B = C(B) ∩ A
Proof: postponed until page 15.
Let’s wait a bit for the proof, and first think a bit more about CS(A, R). In particular,
let’s begin by looking for necessary conditions for CS(A, R) to define a choice function
on [X]. [For necessary and sufficient conditions, see Mukherji (1977).]
We have found necessary and sufficient conditions for CS(A,R) to define a choice
function on finite X. Thus we know that if C(A) can be represented by a binary relation
R, then R will satisfy completeness and acyclicity. These are necessary conditions for R
to represent C(A).
We know a further necessary condition for R. Recall that a binary relation R represents
a choice function C(A) iff
{x ∈ A | x R x0
∀x0 ∈ A} = C(A)
∀A ∈ D
Thus any binary relation that represents C(A) will be completely determined by the
choices over pairs and singletons! To show this
2
Recall the definition of the base relation
RB = {(x, x0 ) | x ∈ C({x, x0 })}
RB characterizes the consumer’s choices over pairs and singletons.
This is known as the base relation (Herzberger, 1973) Note: x and x0 are not necessarily
distinct.
So ∀x, x0 ∈ x we will write x RB x0 ⇔ x ∈ C({x, x0 }).
This makes it clear that the question of whether C(A) can be represented by a binary
relation is just the question of whether RB represents C(A). That is, the base relation is
the unique rationalization of any representable choice function.
Lemma: If R represents C(A), then R = RB . (Sufficiency!)
Proof: Assume R represents C(A). Since R rationalizes C(·), we know R is reflexive.
Thus
a R a0 ⇔ a ∈ {x ∈ {a, a0 } | x R x0 ∀x0 ∈ {a, a0 }} ⇔ a ∈ C({a, a0 }) ⇔ a RB a0
(2)
=⇒ reflexivity, def. rep., def. base relation
⇐ obvious ” ”
We will call this relation “weak preference” since, based on the consumer’s choices, it
appears that x is weakly preferred to (or at least as good as) x0 whenever x RB x0 .
(Unlike Kreps, we focus on the relation of weak preference. However, we may also find
it interesting to consider RA defined by xR̂C x0 iff ∃A s.t. x, x0 ∈ A and x ∈ C(A). How
does this relate to Kreps’s (1990, p.29) c defined as x c x0 iff ∃A s.t. x ∈ C(A) and
x0 6∈ C(A)?)
3
Preference Relations
This binary relation, defined on (consumption) set X, can therefore be interpreted as a
subset RB ⊂ X ⊗ X such that (x, x0 ) ∈ RB iff x is as least as good as x0 . Note however
that “at least as good as” refers here only to pairwise comparisons. Note also that if
we try to treat choices as primitive, “at least as good as” must be seen as an imposed
psychological interpretation. For example, consider X = <1 , the case of 1 good, and a
consumer who always chooses the maximum amount available.
Of course our earlier exploration of the necessary conditions for a rationalization tell
us that RB must be complete and AAS if it is to rationalize a choice function.
It turns out that RB satisfies completeness trivially.
Since the choice function C(A) is by definition non-empty for A 6= ∅, the binary
relation RB is
1. Reflexive: x RB x ∀x ∈ X [o/w ∃x ∈ X s.t. C({x}) is empty]
2. Weakly connected: x 6= x0 =⇒ x RB x0 or x0 RB x [o/w ∃x, x0 ∈ X s.t. C({x, x0 })
is empty.]
3. Complete (or connected): x RB x0 or x0 RB x ∀x, x0 ∈ X [i.e., reflexive and weakly
connected]
When X has only two elements, say x, x0 , completeness is all we need to be assured of
equivalence between C(A) and CS(A, RB ).
4
Example 1
(X = {x, x0 }):
A
C(A)
RB
CS(A, RB )
x, x0
x
x R x0 ,¬x0 R x
x
x
x
xRx
x
x0
x0
x0 R x 0
x0
Recall that we have already seen that any choice function generates a weak preference relation that is complete. So RB generated from a rational choice function will be
complete, since RB generated from any choice function will be complete.
Whenever X contains 3 or more distinct elements, completeness does not ensure that
we can recover our choice function from weak preference. If we are going to be able to
do that, we need to know at least that CS(A, RB ) is a choice function (i.e., is non-empty
∀A ∈ [X]). This just repeats our first example, except our binary relation is now derived
from the choice function.
A
C(A)
x, y, z
x (or whatever)
x, y
x
Example 2 (No AAS)
y, z
y
RB
CS(A, RB )
∅
x R y,¬y R x
x
y R z,¬z R y
y
x, z
z
z R x,¬x R z
z
x
x
xRx
x
y
y
yRy
y
z
z
zRz
z
In example 2, CS(A, RB ) is not even a choice function! If we want to rule out this possibility, we will have to rule out choice functions that generate weak preference which does
not always imply choice. Our approach will be to insist that the choice function conform
5
to certain minimum restrictions, which are often considered standards of rationality.
Note that RB can be decomposed into a symmetric subrelation IB ⊆ RB and an asymmetric subrelation PB ⊆ RB (which might be thought of as corresponding to indifference
and strict preference).
Define indifference: xIB x0 ⇔ x RB x0
& x0 RB x. Note that xIB x0 =⇒ x0 IB x,
indifference is symmetric.
Define strict preference: x PB x0 ⇔ x RB x0
& ¬x0 RB x. Note that x PB x0 =⇒
¬x0 PB x, strict preference is asymmetric.
Is acyclicity of RB a requirement of rational choice? [Remember, this will restrict
the admissible choice functions C(·).] Unless we agree that all rational choice functions
generate this acyclicity, rationality alone will not suffice for C(A) = CS(A, RB ). If the
answer to this question is not obvious a priori, we might ask if we accept axioms that
imply this acyclicity.
Recall (from the definition of the base relation) that we have found that C(·) is representable iff it is representable by the base relation. Thus representability implies
∀A ∈ [X],
C(A) = {a ∈ A | a ∈ C({a, a0 }) ∀a0 ∈ A}
This suggests the key to representability is a kind of irreversibility of the choices over
pairs.2
The most interesting and very well known characterization of the necessary and sufficient conditions for choice to be representable is that of sen71restud.3
2
See page 7.
3
schwartz76jet offers a compressed characterization of the necessary and sufficient conditions:
C(A) ∩ C(B) = C(A ∪ B) ∩ A ∩ B
∀A, B ∈ [X]
HW: Show the equivalence between the Sen and Schwartz conditions.
Bandyopadhyay and Sengupta’s (1991) more recently proposed condition—that every decision situation
has an element to which no revealed inferior element is ever revealed preferred—does not offer much insight
into the problem of representability, whatever its other virtues.
6
Sen’s α and γ:4 ∀A, B ∈ [X]
α : C(A ∪ B) ⊆ [C(A) ∪ C(B)]
γ : [C(A) ∩ C(B)] ⊆ C(A ∪ B)
The first half of Sen’s conjunction, “Sen’s α,” is often rephrased as
A ⊆ B =⇒ [A ∩ C(B) ⊆ C(A)]
which makes it clearer that it just says that choices persist as options shrink. Thus is it
known as “contraction consistency.” Here is a final formulation that may be even more
transparent
x∈A⊆B
& x ∈ C(B) =⇒ x ∈ C(A)
(3)
With all pairs in the domain, this implies that rejection by the base relation is never
reversed in larger sets. HW: show that.
The second half of Sen’s conjunction, “Sen’s γ,” is often rephrased as5
x ∈ C(A) & x ∈ C(B)
=⇒ x ∈ C(A ∪ B)
(4)
This makes it clear that Sen’s γ is a kind of expansion consistency: an element continues
to be judged best in a larger set as long as it has no new competitors. With all pairs
in the domain, this implies that acceptance of an element in all pairwise comparisons in
never reversed in a larger set.
HW: show that.
4
5
This is the formulation for finite X.
More generally (covering infinite sets), x ∈ C(Sj ) ∀j ∈ I
7
=⇒ x ∈ C(∪j∈I Sj ).
Representability and Irreversibility
Since C(A) can be represented by a binary relation iff it can be represented by RB ,
the necessity of these irreversibilities is obvious and not surprising: its violation would
imply the existence of some decision situation A such that either [a ∈ C(A) & a0 ∈
A & ¬aRB a0 ] or [a ∈ A/C(A) & aRB a0 ∀a0 ∈ A]. Sufficiency is just as transparent,
since to say it is representable by RB
C(A) = {a ∈ A | a RB a0 ∀a0 ∈ A} ∀A ∈ [X]
is certainly to say it is representable.
Still, it may be instructive to work through the implications of irreversibility in these
two parts: an implication of irreversibility of rejection, and an implication of the irreversibility of acceptance. Irreversibility of rejection, as noted by herzberger73e in his
Proposition 10, implies C(A) ⊆ {x ∈ A | x RB x0 ∀x0 ∈ A}:
a ∈ A\{x ∈ A | x RB x0 ∀x0 ∈ A} =⇒ ∃a0 ∈ A s.t. a 6∈ C({a, a0 }) =⇒ a ∈ A\C(A)
The first implication follows from the definition of the base relation, and the second follows
from irreversibility.
To this we can add that irreversibility of acceptance implies {x ∈ A | x RB x0 ∀x0 ∈
A} ⊆ C(A):
a ∈ {x ∈ A | x RB x0 ∀x0 ∈ A} =⇒ a ∈ C({a, a0 }) ∀a0 ∈ A =⇒ a ∈ C(A)
Again, the first implication follows from the definition of the base relation, and the second
follows from irreversibility. We can therefore easily see why irreversibility (i.e., Sen’s α
and γ) is necessary and sufficient for representability.
8
Chernoff Condition
Of the axioms above, the one which has had the broadest appeal is the Chernoff Condition
(Chernoff 1954), also known as Independence of Irrelevant Alternatives, or Sen’s condition
α.6
The Chernoff Condition (α) is one type of contraction consistency: choice in a contracted set reflects the choices in a larger set. (For a weaker version that is also satisfactory
for much of our project, see Baigent (1990).)
We adopt the notation that S ⊆ S + . This gives a somewhat compressed statement of
the Chernoff Condition:
α: x ∈ S, x ∈ C(S + ) =⇒ x ∈ C(S) or C(S + ) ∩ S ⊆ C(S)
Theorem 2
α implies that PB is acyclical. (Sufficiency!)
Proof: Recall that ∀A ∈ [X], C(A) 6= ∅. Note that ∀{x1 , . . . , xn },
α
xi ∈ C({x1 , . . . , xn }) =⇒ xi ∈ C({xi , xj })
def RB
=⇒ xi RB xj
∀j = 1, . . . , n
∀j = 1, . . . , n
def PB
=⇒ ¬∃j s.t. xj PB xi
Therefore ¬∃{x1 , . . . , xn } s.t. xn PB xn−1 , . . . , x1 PB xn .
Comment: This is a sufficient condition for the acyclicity of PB . We can still have
a choice function C(A) for which the derived strict preference relation is acyclical but
where C(A) does not satisfy α.
6
The earliest statement of this condition that I know of is in Nash (1950), where it is introduced
as an axiom for bargaining outcomes. Note that our choice functions can readily represent bargaining
outcomes, although this may change the set of axioms that we would introduce to characterize rational
choice.
9
A
C(A)
x, y, z
x, y, z
RB
CS(A, RB )
z
x, y
x, y
x R y,y R x
x, y
Example 3 (AAS of RB 6 =⇒ α)
y, z
z
z R y,¬y R z
z
x, z
z
z R x,¬x R z
z
x
x
xRx
x
y
y
yRy
y
z
z
zRz
z
Here C(A) violates α (although not Expansion Consistency (β)), but note weak preference is acyclical (AAS). This is of course enough to imply that CS(·, RB ) is a choice
function, since RB is always complete given our domain assumption. In fact, in this
example, RB is complete and transitive! (So, CS satisfies WARP, although C does not!)
Comment: CS(·, R) always satisfies α, even if it is not a choice function, so naturally
any representable C(·) will have to as well.
Theorem 3
α implies RB = RC .
Proof:
First note that x RB y =⇒ x RC y by definition. Then note that if x RC y we can shrink
the set in which this preference was revealed down to {x, y}, where by α we must have
x RB y.
HW: under what conditions is RC acyclical?
Sen’s β
Although Sen’s γ is intuitively interpreted as the irreversibility of acceptance, economists
have been more interested in a different kind of expansion consistency. A choice function
10
satisfies Sen’s β, a kind of expansion consistency (β), iff:
x, x0 ∈ C(S) =⇒ [x ∈ C(S + ) =⇒ x0 ∈ C(S + )]
This is also intuitive to the extent it says that relative acceptability in a superset should
reflect relative acceptability in subsets.
To see that Sen’s β is independent of the Chernoff Condition, consider the following
violation of β with satisfaction of α:
A
C(A)
x, y, z
x
R
CS(A, R)
x, z
x, y
x
xP y
Example 4 (α 6 =⇒ β)
y, z
z
zP y
x
z
x, z
x, z
x R z, z R x
x, z
x
x
xRx
x
y
y
yRy
y
z
z
zRz
z
The relationship between β and γ is not transparent. However, it is easy to show
β
& α =⇒ γ.
HW: show that.
Thus we know that a choice function satisfying α and β is representable.
Recall
1) The base relation (“weak preference”) derived from any choice function must be
complete.
2) If a binary relation R on X satisfies our properties of completeness and acyclicity,
CS(A, R) defines a choice function. (X finite). Thus any acyclical derived strict preference
relation ensures a choice function, since a derived weak preference relation is always
complete.
11
But is it our choice function: Does CS(A, R) = C(A)?
Test #1: CS exhibits α; does C?
Comment: any CS(A, R) exhibits α [by definition of CS(A, R)].
α as a requirement of rationality. So, assume C exhibits α. But still we cannot know
unless we give more structure to C(A).The standard approach to imposing this structure
is to assume that “rational” choice is characterized by an additional rationality axiom.
The prime candidate is:
3) Expansion Consistency (Sen’s β )
Expansion Consistency (β) : Elements chosen in a set will all be chosen in a superset
whenever any one of them is.
Transitivity as Expansion Consistency
Transitivity has two components: acyclicity on triples, and a substitution property.
x P y&y P z ⇒ ¬z P x
(5)
x I y&y R z ⇒ x R z
(6)
Baigent (1990 EL) shows that when all pairs and triples are in the domain of the choice
function, these properties imply transitivity of the base relation. Clearly his proof also
implies that complete binary relation satisfying these properties is transitive. Given a
T-domain, they also imply transitivity of the revealed preference relationship.
To see the implication of transitivity, suppose we have x R y and y R z. There are four
case to consider. If either R is an I, the substitution property alone implies transitivity.
If they are both P , we must also use triple acyclicity.
Triple acyclicity is clearly implied by the Chernoff condition for any choice function
with a T-domain. It is also implied by a weaker condition, sometimes known as the Weak
12
Chernoff Condition (which is still not necessary).
x ∈ C({x, y, z}) ∩ C({x, y}) ⇒ x ∈ C({x, z})
(7)
HW: Show transitivity implies AAS.
Given the standard contraction consistency characterized by the Chernoff Condition,
transitivity of the base relation is simply a matter of the expansion consistency characterized by Sen’s β.7
Theorem 4
α & β ⇔ C(·) is representable and transitive.
Proof:
Recall α plus β implies Sen’s γ. Also, α plus γ hold iff C(·) is representable. So we just
need to prove the link to transitivity.
Necessity: Representability of C(·) and transitivity of RB is given.
We already know representability implies α, so we need to prove β. Suppose x, x0 ∈
C(S) & x0 ∈ C(S + ). Then xRB x0 and x0 RB x00
fore xRB x00
∀x00 ∈ S + , by representability. There-
∀x00 ∈ S + , by transitivity. Therefore x ∈ C(S + ), by representability.
Sufficiency: α and β are given.
We have already proved representability (since α andβ imply γ), so we need to prove
7
We can also characterize transitivity in the absence of α; see Sen (1986). Specifically, transitivity of
weak revealed preference is equivalent to a slight strengthening of Sen’s β, which Sen calls β + .
x ∈ C(S)
& y∈S
&
y ∈ C(S + ) =⇒ x ∈ C(S + )
(8)
This just weakens the antecedent in β to require only that y be present in the subset, not that it be
chosen there. Note that β + implies β and γ.
Without α we are not guaranteed that the base relation and the weak revealed preference relation
coincide, so we must separately address transitivity of the base relation. This transitivity is ensured by
β and the following condition, which Sen calls “weak α” and Baigent call the weak Chernoff Condition.
Since Sen’s formulation of “weak α” is less helpful, I will cut to the chase and characterize the condition
more simply.
x ∈ C({x, y, z} & x ∈ C({x, y}) =⇒ x ∈ C({x, z})
(9)
Note that weak α alone ensures that PB cannot cycle over triples. When coupled with β, it ensures the
base relation is transitive. See Sen (1986, p.1099) for more discussion.
13
transitivity. We will show non-transitivity implies violation of β (in the presence of α).
Suppose x00 RB x0 and x0 RB x, but ¬x00 RB x. By completeness of RB , we know xRB x00 , so
xPB x00 . Consider two cases:
Case I: x00 PB x0 . Then by acyclicity of PB , which is implied by α, we know ¬x0 PB x. So
xRB x’. Thus we have {x, x0 } = CS({x, x0 }, RB ) and {x} = C({x, x0 , x00 }), violating β.
Case II: x00 IB x0 . Then {x00 , x0 } = CS({x00 , x0 }, RB ). It follows that x0 is in C({x, x0 , x00 })
but x00 is not (since ¬x00 RB x). This violates β.
So we find that in the presence of α, the violation of transitivity implies the violation of
expansion consistency.
The Chernoff Condition is generally accepted as a requirement of rational choice. If
we think rational choice must be transitive, we are committed to β. On the other hand,
if we think rational choice must display β, we are committed to transitivity. We can
therefore for finitely complete C(·) conclude that β and α hold iff C(·) has a transitive
representation. Note that a complete and transitive binary relation is known as a weak
order. Note that on infinite X we need additional structure to assure that CS(·, RB ) is
not empty.
α
&
β ⇔ WARP
Some economists have proposed as an intuitively plausible requirement of rational choice
Houthakker’s weak axiom of revealed preference (WARP).
x, y ∈ A, B
& x ∈ C(A) & y ∈ C(B) =⇒ x ∈ C(B)
or
C(X) ∩ Y 6= ∅ & C(Y ) ∩ X 6= ∅ =⇒ C(X) ∩ Y = C(Y ) ∩ X
We can show that any choice function exhibiting α and β is also characterized by
14
WARP (and vice versa).
Theorem 5
α & β ⇔ WARP
PROOF of sufficiency: α & β =⇒ WARP
β and α are given:
β: x, y ∈ C(S) =⇒ [y ∈ C(S + ) =⇒ x ∈ C(S + )]
α: C(S + ) ∩ S ⊆ C(S)
Given β and α we need to show
[x, y ∈ A, B
& x ∈ C(A) & y ∈ C(B)] =⇒ x∈ C(B)
DEFINE: D = A ∩ B
y, x ∈ C(D) (α)
x ∈ C(B) ( β )
QED
PROOF of necessity: WARP =⇒ α and β
We are given WARP:
WARP: [x, y ∈ A, B
& x ∈ C(A) & y ∈ C(B)] =⇒ x∈ C(B)
Show β holds:
Assume x, y ∈ C(A) & A ⊂ B
WARP
Then y ∈ C(B) =⇒ x ∈ C(B)
Proving β
Show α holds:
Assume x ∈ S
& x ∈ C(S + )
Note: Since C(S) is not empty, x 6∈ C(S) =⇒ ∃y 6= x s.t. y ∈ C(S)
Contradicting WARP
Conclusion: C(·) has a transitive binary representation iff it satisfies WARP.
HW: give a direct proof that WARP implies transitivity of RB .
15
Univalence
Comment: In Houthakker’s original paper, he works with single valued choice functions.
This allows him to claim α =⇒ WARP.
HW: Show this. Soln: Since X and Y are supersets of X ∩ Y , α implies
C(X) ∩ Y ⊆ C(X ∩ Y )
C(Y ) ∩ X ⊆ C(X ∩ Y )
H concerns the case where none of the intersections are null. Single valuedness then
implies C(X) = C(X ∩ Y ) and C(Y ) = C(X ∩ Y ), so that C(X) = C(Y ) and thus
C(X) ∩ Y = C(Y ) ∩ X. QED
You might also reason as follows: persistence says that x, x0 ∈ C(S)
=⇒
[x ∈
C(S + ) =⇒ x0 ∈ C(S + )]. Single valuedness means x, x0 ∈ C(S) only if x = x0 , in which
case β must hold. So β follows directly from single valuedness.
Rational Choice: Necessary Conditions
1. Existence of a stable choice function. (Otherwise, what would we mean by rational
choice? But see Vriend (1996).)
If a choice function C(·) can be generated by a binary relation, we know the binary
relation must be complete and acyclical. Representability may not seem an obvious requirement of rational choice. Instead we may turn to more basic structural
considerations.
2. Contraction consistency (Sen’s α; Independence of Irrelevant Alternatives (Not the
same as Arrow’s.)):
x ∈ S and x ∈ C(S + ) =⇒ x ∈ C(S)
A choice in a larger set is still chosen in a smaller (sub) set.
3. Expansion Consistency (Sen’s β):
16
x, x0 ∈ C(S) =⇒ [x ∈ C(S + ) =⇒ x0 ∈ C(S + )] or C(S + ) ∩ C(S) ∈ {∅, C(S)}
If I stick with one of my choices I have to stick with all of them.
WARP =⇒ Homogeneity of Degree Zero
The Marshallian demand correspondence is homogeneous of degree zero. That is, x(αp, αw) =
x(p, w) ∀α > 0. First degree homogeneity of the demand correspondence says that consumer choice depends only on the feasible set and not on its determinants. This follows
directly from WARP (and the characterization of the choice situation).
Proof: Suppose ∃α > 0 such that x(αp, αw) 6= x(p, w). Pick x ∈ x(p, w) such that
x 6∈ x(αp, αw). Now the budget constraint implies
px ≤ w
(10)
αpx ≤ αw
(11)
which in turn implies
This means x was affordable, although it was not in x(αp, αw). WARP therefore implies
that for any x0 ∈ x(αp, αw),
px0 > w
(12)
αpx0 > αw
(13)
But this would mean
which contradicts x0 ∈ x(αp, αw).
17
Choice on <K
+
When we move from choice on finite sets to choices on infinite sets we need to attend to a
few technical issues. In particular, it will generally be important to restrict the domain.
In our development, we will take the domain to be the compact subsets of <K
+ , since
budget set are natural interpreted as falling in this domain.
One approach is simply to characterize sufficient conditions on the structure of the
choice function to assure choice on any compact set. nehring96el provides a recent example.
We will take a somewhat different tack. We will assume choice on <K
+ is generated by a
binary relation (preferences), and we will ask under what structural restrictions ensure a
binary relation implies a choice on arbitrary compact subsets of <K
+.
Necessary and sufficient conditions for maximization of preference relations have been
provided by tian93restud. However, to keep our presentation simpler and more accessible,
we will focus on sufficient conditions. There have been two primary approaches to providing sufficient conditions: convexity assumptions, and acyclicity assumptions (see Tian for
a discussion). Given the work we have already completed, it will be natural for us to focus
on acyclicity. We will do this by proving a famous existence theorem of bergstrom75jet
and walker77jet.
18
More References
1. Arrow, K., 1959, “Rational Choice Functions and Orderings,” Economica 26, 121-27.
2. Bandyopadhyay, T., and K. Sengupta, 1991, “Revealed Preference Axioms for Rational Choice,” EJ 101(404), 202–213.
3. Baigent, Nick, “Transitivity and Consistency,” Economics Letters 33, 1990, 315–17.
4. Bergstrom, Theodore C., “Maximal Elements of Acyclic Relations on Compact
Sets,” JET 10, 1975, 403–404.
5. Chernoff, H., “Rational Selection of Decsision Functions,” E 22, 1954, 422–443.
6. Hicks, J.R. and R.G.D. Allen, “A Reconsideration of the Theory of Value,” Economica 1, 1934, 52-75, 196–219.
7. Jamison, Dean, and Lawrence Lau, 1973, “Semiorders and the Theory of Choice,”
E 41(5), 901–912.
8. Kahneman, D., and A. Tversky, 1979, “Anomalies in Intertemporal Choice”, QJE.
9. Kahneman, D., and A. Tversky, 1984, “Choice, Values, and Frames,” American
Psychologist 39, 341-50.
10. Mukherji, Anjan, “The Existence of Choice Functions,” E 45(4), May 1977, 889–894.
11. Nash, John F., “The Bargaining Problem,” E 18, 1950, 155-62.
12. Nehring, K. “Maximal Elements of Non-binary choice functions on compact sets.”
Economics Letters 50(3), March 1996.
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13. Richter, M., 1971, “Rational Choice,” in J. Chipman, L. Hurwicz, and H. Sonnenschein (eds), Preferences, Utility and Demand (NY: Harcourt Brace Jovanovich,
1971).
14. Sen, Amartya, 1969, REStud 36.
15. Sen, Amartya, 1970, Collective Choice and Social Welfare (New York: Holden-Day;
as reprinted, Amsterdam: North-Holland, 1979).
16. Sen, Amartya, 1971, “Choice Functions and Revealed Preference,” REStud 38, 307317.
17. Sen, Amartya, “Behavior and the Concept of Preference.” Economica 40(158), May
1973, 241–259.
18. Sen, Amartya, 1986, “Social Choice Theory,” in K.J.Arrow and M.D. Intrilligator
(eds), Handbook of Mathematical Economics, vol. III (Elsevier Science Publishers,
1986).
19. Schwartz, Thomas, 1976, “Choice Functions, ‘Rationality’ Conditions, and Variations on the Weak Axiom of Revealed Preference,” JET 13, 414–427.
20. Tian, G., 1993, “Necessary and Sufficient Conditions for Maximization of a Class of
Preference Relations,” REStud 60, 949–58.
21. Walker, M., 1977, “On the Existence of Maximal Elements,” JET 16, 470–74.
20
Add material from Matzkin and Richter, “Testing Strictly Concave Rationality”
Since there are partial orderings as well as connected orderings, the key concept of an
ordering seems to be transitivity. (McClennen 1990).
SEMI-ORDERS Jamison and Lau (E 41, 1973; E 43, 1975) and Fishburn (E 43,
1975) offer necessary and sufficient conditions for an individual’s choices to be representable by a semi-order. They assume that observations are complete.
INCOMPLETENESS Richter (E 34, 1966) recognizes that we cannot always observe choices on each member of [X]. His Congruence Axiom is necessary and sufficient
to guarantee that an individual’s choices are explicable by an underlying weak order.
Kim (E 55,1987) extends Richter’s results to semiorders and interval preferences, but
only by assuming univalence. Gensemer (JET 54, 1991) removes the univalence restriction.
Brandt (1983, in P. Moser, Rationality in Action, p. 410) If the agent acts on his
intransivitive preferences (for example, bets accordingly), and unscrupulous adversary
can quickly drain his resources—he can become a “money-pump.” (But see McClennan
(1990, section 1.7.) This fact is a serious pragmatic reason for the requirement that
intransitive preferences be avoided. Indeed, if preferences are not transitive, the ideal
of maximizing desire-satisfaction becomes an elusive target, although intransitivities in
a preference-system as a whole are not incompatible with the rationality of particular
actions.
Strict preference: x x0 or xP x0 ⇔ [x R x0 and¬(x0 R x)]
Indifference: x ∼ x0 or xIx0 ⇔ [x R x0 andx0 R x]
We are often interested in the maximal set and the choice set on a feasible subset
S ⊂ X. For example, given prices and income, (p, y) 0 we may define the budget set
BS(p, y) = {x | x ∈ X, px ≤ y}
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E.g., 2 goods, x = <2+
BS(p, y) = {x | x ∈ X, p1 x1 + p2 x2 ≤ y}
Maximal set: M S(S, R) = {x | x ∈ S, (x0 ∈ S) =⇒ ¬(x0 P x)}
NB: Allows for incompleteness (incomparability).
E.g., ¬x R x0 and ¬x0 R x Maximal set simply contains non-dominated elements.
NB: MS may be empty. E.g., x P x0 , x0 P x00 , x00 P x.
E.g., suppose X = <2+ , S = BS(p, y), and more is better is only preference.
I.e., x R x0 ↔ x x0 , and x P x0 ↔ x x0
MS is the budget frontier, But CS is empty!
HW: Show CS(S, R) ⊂ M S(S, R) always. HW: Show CS(S, R) = M S(S, R) iff R
is connected (Prop P1, Herzberger 1973).
LOSS AVERSION Up to now, our model of decision making assumes that preferences
do not depend on any reference level such as initial assets. However, psychologists offer
evidence that reference levels are important. Kahneman and Tversky (E 1979,Am Pscyh
1984) have argued that choice under uncertainty is best modelled as including two types
of
Reference dependence: gains and losses are evaluated relative to a reference point
i. Loss aversion: losses weighted more heavily than gains
ii. Diminishing sensitivity: marginal valuation decline with distance from reference
point
This yiels their S-shaped value function around a reference point
Thus the choice function may yield different outcomes from the same set of alternatives
with a different reference state r. We have to write C(A,r). For such a dependence to be
interesting, it must be capable of generating choices that
i. cannot be rationalized without reference to the reference state
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ii. offers a better match than classical theory to the empirical evidence
Tversky and Kahneman (QJE 1991) are interested in characterizing reference dependence for choices under certainty. To do so, they introduce the concept of a “reference
structure”: a family of preference relations indexed by the reference state r ∈ X. (They
do note, however, that the reference state can “be influenced by aspirations, expectations,
norms, and social comparisons.”)
Loss Aversion (LA): Suppose x ≥ r ≥ s and y ≥ s but ∃i s.t. ri > yi . If x I (s)y,
then x P (r)y. [T&K require xi ≥ ri > yi = si and y2 > x2 : I do not see why. On p.1049
they again stress defining LA in terms of coincidence on certain dimensions. I think they
should shift to focusing on prefs: is this possible? e.g., xI(s)y with x R ()r, r R ()s, and
y R ()s but ???? implies x P (r)y.
How about: LA iff x P ()s & xI(s)y− > x P ([αs + (1 − α)x])y? But this imposes
a smoothness that rules out sign dependence. And it still requires bundles measureable
as real vectors. T&K are very similar: roughly xi P ()si
& xI(s)y =⇒ x P ([αsi +
(1 − α)xi ])y]
This allows for preference reversals: suppose xI(t)y where x >> t and y >> t but x
vs. y involves a tradeoff. Then x P (x)y and y P (y)x.
INSERT GRAPH and Empirical Work
Sign dependence: a reference structure satisfies sign dependence iff x R (r1 )y < − >
x R (r2 )y whenever sign(x − ri ) = sign(y − rj ) ∀i, j = 1, 2. That is, there is preference
independence as long as x and y are both in the same quadrant w.r.t. both r1 and r2 . So
independence is violated only when a change of reference state turns a gain into a loss or
a loss into a gain. Sign dependence rules out diminishing sensitivity.
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Reflecting this emphasis on preferential choice, let us re-interpret our conditions of
sanity as follows. For x, y ∈ D, define basic strict preference and basic indifference as
follows.8
x PB y ⇔ y 6∈ C({x, y})
(14)
x IB y ⇔ {x, y} = C({x, y})
(15)
Under this interpretation of preference, given a choice between x and y (with no other
alternatives available), x is strictly preferred to y if y is rejected, while x is indifferent
to y if neither is rejected. Then our three conditions of sanity follow directly from the
existence of a choice function.9
+++++++
rabin p.34 If people experience losses relative to a status quo as quite unpleasant, then
loss-averse behavior is rational, because people are correctly anticipating and avoiding
unpleasant sensations. And, the remembered “loss” of an owned mug may carry over
time, or in any event be substantial relative to the long-term utility consequences of owning the mug. Yet loss aversion often seems to be a judgmental bias: In decisions with
significant long-run consequences, people should put less weight than they do on their
initial experience of losses.
Self interest vs. present aims? From rabin: Utility from Fat Slice = 7 if you ate at
Blon-die’s last night Utility from Fat Slice = 5 if you ate at Fat Slice last night Utility
from Blondie’s = 4 if you ate at Fat Slice last night Utility from Blondie’s = 3 if you
ate at Blon-die’s last night On any given day, no matter your re-cent eating pattern, you
get higher util-ity from eating at Fat Slice than at Blondie’s. Yet your utility-maximizing
8
These are the asymmetric and symmetric subrelations of the base relation RB , which is defined by
x RB y ⇔ x ∈ C({x, y}).
9
schwartz76jet lists some additional implications.
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consumption program is to alternate be-tween Fat Slice and Blondie’s (thus al-ternating
between payoffs of 7 and 4, for an average of 5.5) rather than eating all the time at Fat
Slice (thus getting a payoff of 5 each period). Yet, because at each moment we tend to
ask, “Which will yield me more pleasure—Fat Slice or Blondie’s?,” we may eat too often
at Fat Slice.
rabin p.37 were framed. Money illusion provides perhaps the best example of the
importance of framing effects for economics. Kahneman, Knetsch, and Thaler (1986a)
provide survey evidence that people are very attentive to nominal rather than real changes
in wages and prices in assessing the fairness of firm behavior. A nominal wage increase of
5 percent in a period of 12 percent inflation offends people’s sense of fairness less than a
7 percent decrease in a time of no inflation. More generally, people
rabin p.38 Itamar Simonson and Tversky (1992) provide examples of context effects,
where the addition of a new option to a menu of choices may actually increase the proportion of consumers who choose one of the existing options.
elster p.66 ger 1957). An individual who is subject to several motivations that point in
different directions will feel an unpleasant feeling of tension. When on balance he favors
one action, he will try to reduce the tension by looking for cognitions that support it;
when he favors another, he will look for cognitions which stack the balance of arguments
in favor of that action (Tesser and Achee 1994, p. 104). Thus the timing of the switch in
behavior will be path-dependent. Dissonance theory is more realistic than the cost-benefit
model in that it views individuals as making hard choices on the basis of reasons rather
than on the basis of introspections about how they feel.
TESSER, ABRAHAM AND ASCHEE, JOHN. “Aggres-sion, Love, Conformity, and
Other Social Psy-chological Catastrophes,” in Dynamical systems in social psychology.
Eds.: ROBIN R. VAL-LACHER AND ANDRZEJ NOWAK. New York: Academic Press,
1994, pp. 96–109.
agi: this seems like a kind of context independence of reasons?
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Markets of this type, wrote Albert Hirschman in these pages (1982, p. 1473), are
peopled by “large numbers of price taking anonymous buyers and sellers supplied with
perfect information” . . . and “function without any prolonged human or social contact
among or between the parties.” Grocery markets approximate this ideal (a fact which may
explain why fruit stands and fish markets figure so prominently in eco-nomics textbooks).
Hirschman, A. “Rival Interpretations of Market Society: Civilizing, Destructive, or
Feeble?” J. Econ. Lit., Dec. 1982, 20(4), pp. 1463–84.
bowles Robert Lane (1991), whose The Market Experience must be the starting point
for any consideration of the psy-chology of markets, writes: In spite of the variety of
markets over time and across cultures, I believe that it is possi-ble to conceive of a market
experience that is typical, frequent, and paradigmatic for those who do market work for
pay, use money and buy—rather than make, inherit or receive from government—the
commodities with which they adorn their lives. (p. 4). . . Inevi-tably the market
shapes how humans flourish, the development of their existences, their minds, and their
dignity. (p. 17) What is psychologically distinctive about markets as opposed to other
allocation mechanisms? Max Weber ([1922]1978, p. 636) wrote “A market may be said
to exist wherever there is competition for opportunities of exchange among a plurality of
poten-tial parties.”11 Markets structure social interactions “each of which is specifi-cally
ephemeral insofar as it ceases to exist with the act of exchanging the goods.” As a result,
according to Weber, The market community as such is the most impersonal relationship
of practical life into which humans can enter with one another. This is not due to the
potentiality of struggle among the interested parties which is inher-ent in the market
relationship. . . . The rea-son for the impersonality of the market is its matter-offactness, its orientation to the com-modity and only to that. what Parsons (1967, p. 507)
calls the “two principal competitors” of the market: “requisitioning through the direct
application of political power” and “non-political solidarities and communities.” These
allocation rules contrast with markets in at least one of the characteristics— impersonality
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and ephemerality— stressed by Weber. Centralized bureaucratic allocations are in some
respects as impersonal as markets—at least ideally—but membership in the group defining
the allocation is generally given, entry and exit costs are high (often involving a change
in citizenship or at least residence), and contacts are far from ephemeral.
Markets frame choices (?, section 4).
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