Incompleteness and Intransitivity
Hector Lopez
Abstract
Eliaz and Ok[3] designed a novel procedure to identify those choice functions that can be rationalized by an incomplete and transitive preference relation. This paper shows that every choice function
rationalizable by an incomplete and transitive preference relation using Eliaz and Ok’s theory is also
rationalizable by means of a complete and intransitive preference relation.
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Introduction
Eliaz and Ok[3] showed that if a choice function satisfies a condition they call the weak axiom of revealed
non-inferiority (WARNI), then there is a transitive binary relation whose maximization is consistent with
the choice function. On the other hand, Sen[4] had shown that if a choice function satisfies the alpha and
gamma axioms, then there is a complete binary relation whose maximization is consistent with the choice
function. In the following paragraph I introduce an example to put Sen’s and Eliaz and Ok’s theorems to
work.
Assume a monkey makes his decisions based on the sweetness and brightness of fruits. If he only considers
sweetness he would rank apples over oranges and oranges over bananas. When he only looks for brightness
he ranks bananas over apples and apples over oranges. The choices of the monkey would be like this:
c{apples, oranges} = {apples} because apples are sweeter and brighter than oranges, c{apples, bananas} =
{apples, bananas} because he is indecisive (the monkey uses two orderings because he believes that sweetness
and brightness are not comparable), c{oranges, bananas} = {oranges, bananas} because of the same reason;
however c{apples, oranges, bananas} = {apples, bananas} because apples are sweeter and brighter than
oranges and he remains indecisive between apples and bananas. There is not something clearly wrong with the
choice procedure used by the monkey, however the standard theory would classify him as irrational because
his choices do not satisfy WARP. Eliaz and Ok[3] state that WARP is too demanding in environments where
incomparability arises naturally and provide WARNI to rationalize behavior with transitive binary relations.
Eliaz and Ok’s theorem provides the following incomplete and transitive binary relation: apples oranges,
bananas ./ oranges, bananas ./ apples, where ./ means unrelated. Sen’s theorem generate the following
complete and intransitive binary relation: apples oranges, bananas ∼ oranges, bananas ∼ apples.
WARNI was developed by Eliaz and Ok to provide choice theoretic foundations of incomplete preferences.
However, their whole theory is subsumed into Sen’s theory. This paper shows that this fact has nothing to
do with the Eliaz and Ok’s procedure to construct incomplete preferences, but it is a necessary consequence
of using the maximization operator. This papers shows that the maximization operator belongs to a class
of operators that I call consistent and that every consistent operator induces the alpha and gamma axioms.
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Preliminary definitions
Let X be the set of all mutually exclusive alternatives. Let Ω be any subset of 2X \ {∅}. A subset S of X is
a choice problem if S ∈ Ω. Assume that Ω contains all singletons and that it is closed under finite unions.
Definition 1. C : Ω → 2X \ {∅} is a choice function if and only if C(S) ⊆ S for every S ∈ Ω. Let C be the
set of all choice functions.
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A choice function is a full description of behavior because it describes all the elements chosen in all choice
problems. I introduce the preliminary concept of a binary relation in order to define a revealed preference.
Definition 2. B ⊂ X × X is a binary relation on X. (x, y) ∈ B is written as xBy. Let B be the set of all
binary relations on X.
I will now introduce the concept of a revealed preference.
Definition 3. R : C → B is a revealed preference. G : BA → C is an operator and BA is the set where it
is well defined.
The maximization operator is defined as follows.
Definition 4. M : Bmax → C is the maximization operator if M (B)(S) = {x ∈ S such that ¬((yBx)and¬(xBy))
∀y ∈ S} for all S ∈ Ω, and M (B) is a choice function.
Although the formal definition of a revealed preference is the one of a function, it is common to abuse of
notation. For example, if (x, y) ∈ R(C) where C ∈ C is fixed, then it is written as xRy.
Definition 5. For any binary relation B and for any x, y ∈ X, let xP y if xBy and not yBx. Let xIy if
both xBy and yBx. Let x ./ y if (x, y) 6∈ B and (y, x) 6∈ B
Sen[4] noted that there are many ways to construct revealed preferences. I will now introduce two. 1
is the textbook one[1]. 2 is from Eliaz and Ok[3].
Definition 6. For any choice function C and for any x, y ∈ X, x 1 y if and only if there exists S ∈ Ω
such that x ∈ C(S) and y ∈ S. Let x 1 y if x 1 y and not y 1 x. Let x ∼1 y if x 1 y and y 1 x
In order to define the next revealed preference it is necessary to introduce some preliminary definitions.
For a discussion about the following definition see Eliaz and Ok[3]. For any x and y, 2 has four possible
values, i.e. x can be indifferent or unrelated to y , x can be strictly preferred to y or y can be strictly
preferred to x. The previous one does not consider the possibility of being unrelated.
Definition 7. For any S ∈ Ω such that x ∈ S and y 6∈ S let Sy,−x be (S \ {x}) ∪ {y}. For any choice
function C, let (x, y) ∈ P(C) if x 6= y and {x, y} = C({x, y}). Let (x, y) ∈ I(C) if (x, y) ∈ P(C) and there
exists a finite set S ∈ Ω such that x ∈ S and y 6∈ S and at least one of the following conditions hold.
1. x ∈ C(S) and y 6∈ C(Sy,−x )
2. y ∈ C(Sy,−x ) and x 6∈ C(S)
3. C(S) \ {x} =
6 C(Sy,−x ) \ {y}
Definition 8. For any choice function C and for any x, y ∈ X, x 2 y if and only if {x} = C({x, y}). Let
x ∼2 y if (x, y) ∈ P(C) \ I(C). Let x 2 y if x 2 y or x ∼2 y or x = y. For all x, y ∈ X let x ./ y if not
x 2 y and not y 2 x
Sen’s theorem uses 1 and Eliaz and Ok’s theorem uses 2 . However, using 1 or 2 to obtain a revealed
preference for an arbitrary choice function does not guarantee that the maximization of the obtained binary
relation is consistent with the choice function. This observation points out the importance of the domain of
the operator used.
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WARNI and the alpha and gamma axioms
In general, domains are characterized by conditions on choice functions. Those conditions are called rationality axioms. I introduce the rationality axiom of Eliaz and Ok’s theorem.
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Definition 9. A choice function C satisfies the Weak Axiom of Revealed Non-Inferiority (WARNI) if and
only if for every S ∈ Ω and y ∈ S, if for every x ∈ C(S) there exist a T ∈ Ω with y ∈ C(T ) and x ∈ T , then
y ∈ C(S)
It is easy to see that WARP implies WARNI, but the converse is false. So the set of choice functions
satisfying WARNI properly contains the set of choice functions satisfying WARP (the monkey’s choice
function satisfies WARNI but not WARP).
I introduce the alpha and gamma axioms due to Sen[4].
Definition 10. (Alpha axiom) A choice function C satisfies the Alpha Axiom if and only if for all S ∈ Ω
and for all T ⊆ S such that x ∈ T and T ∈ Ω, x ∈ C(S) implies x ∈ C(T )
Definition 11. (Gamma axiom) A choice function C satisfies the Gamma Axiom if and only if for every
K ⊆ Ω and V = ∪S∈K S, such that V ∈ Ω, x ∈ C(S) for all S ∈ K implies x ∈ C(V )
The following theorem shows that WARNI implies the alpha and gamma axioms. Eliaz and Ok[3] showed
that WARNI implies alpha.
Theorem 1. Let C be a choice function. If C satisfies WARNI, the C satisfies the alpha and gamma axioms
.
Proof. Let C be a choice function satisfying WARNI. Take any S, T ∈ Ω with T ⊆ S. Let x ∈ T and
x ∈ C(S). For every y ∈ C(T ) it is true that y ∈ S and x ∈ C(S), therefore x ∈ C(T ) by WARNI. Thus
C satisfies the alpha axiom. Let K ⊆ Ω such that there is x ∈ C(S) for all S ∈ K. Let V = ∪S∈K S and
assume V ∈ Ω. For every y ∈ C(V ) there exists T ∈ K such that x ∈ C(T ) and y ∈ T , therefore x ∈ C(V )
by WARNI. Thus C satisfies the alpha and gamma axioms.
The set of choice functions satisfying WARNI is a subset of the set of choice functions satisfying the alpha
and gamma axioms. In order to show that the contention is proper I exhibit a choice function satisfying
the alpha and gamma axioms and not WARNI. Let X = {x, y, z} the set of alternatives. Consider the next
choice function, c{x, y} = {x, y}, c{x, z} = {x}, c{y, z} = {z}, c{x, y, z} = {x}. c satisfies the alpha axiom
because x ∈ c{x, y} and x ∈ c{x, z}. c satisfies the gamma axiom because x ∈ c{x, y, z}. Observe that for
{x, y, z} and y ∈ {x, y, z} there is {x, y} such that y ∈ c{x, y} y x ∈ {x, y}, but y 6∈ c{x, y, z}. Hence c does
not satisfies WARNI.
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Incompleteness or Intransitivity
I will now introduce the concept of normality due to Sen.
Definition 12. A choice function C is normal if and only if for all S ∈ Ω C(S) = {x ∈ S such that x 1 y
∀y ∈ S}
It should be noted that the concept of normality is attached to an specific revealed preference, namely
1 . I introduce the concept of consistency to allow some generality. Sen showed the equivalence between
normality and the alpha and gamma axioms.
Theorem 2. A choice function C is normal if and only if C satisfies the alpha and gamma axioms.
Proof. See Sen[4]
I state the Eliaz and Ok’s theorem.
Theorem 3. Let C be any choice function. If C satisfies WARNI, then C(S) = {x ∈ S such that ¬(y 2 x)
for all y ∈ S}. Conversely, if Ω contains all countable subsets of X and C(S) = {x ∈ S such that ¬(yP x)
for all y ∈ S} for all S ∈ Ω for some preference relation B, then C satisfies WARNI.
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Proof. See Eliaz and Ok[3]
If a choice function c belongs to Cwarni \ Cwarp , then there are two different binary relations B1 (complete
and intransitive) and B2 (incomplete and transitive) such that their maximization recovers c.
Now consider a new theory with rationality axioms A and revealed preference A . What are the necessary
conditions that this new theory must have in order to be subsumed into Sen’s theory of complete preferences?
The following theorem shows that every theory using consistent operators to recover choice functions from
revealed preferences are subsumed into Sen’s theory of complete preferences.
Definition 13. An operator G : BA → C is consistent if and only if for every B ∈ BA there is B ∗ ∈ B
such that for all S ∈ Ω, C(B)(S) = {x ∈ S such that xB ∗ y ∀y ∈ S}
Note that B ∗ is not required to be transitive, complete or to satisfy any other property. The following
theorem is the main result in the paper.
Theorem 4. Let G : BA → C be a consistent operator. Then every c ∈ CD satisfies the alpha and gamma
axioms.
Proof. Let G be a consistent operator and consider any c ∈ CD . Take S ∈ Ω and x ∈ c(S). Assume there
is T ∈ Ω such that x ∈ T and T ⊂ S. Since G is consistent there is B ∗ such that xB ∗ y for all y ∈ S, it is
clear that xB ∗ y for all y ∈ T , hence x ∈ c(T ). The alpha axiom is satisfied. Take K ⊆ Ω and V = ∪S∈K S.
Assume there is x such that x ∈ c(S) for all S ∈ K and assume that V ∈ Ω. Let y ∈ V be any alternative,
then there is a set S ∈ K such that y ∈ S and xB ∗ y. In fact we have that for all S ∈ K such that y ∈ S we
have xB ∗ y, because x ∈ c(S) for all S ∈ K. Hence x ∈ c(V ), thus c satisfies the gamma axiom.
In a previous page it was proven that Eliaz and Ok’s theory is subsumed into Sen’s theory by showing
directly that WARNI implies the alpha and gamma axioms. Theorem 4 provides another way of proving the
same. It is only necessary to observe that for every incomplete and transitive preference relation 2 , it is
possible to define B ∗ in the following manner: (x, y) ∈ B ∗ if an only if ¬(y 2 x). Thus Theorem 4 provides
a very-easy-to-check condition for a theory to be subsumed into Sen’s theory.
References
[1] Mas-Colell, Andreu and Whinston, Michael D. and Green, Jerry R., Microeconomic Theory, Oxford
University Press, 1995.
[2] Kennet J. Arrow, Rational choice functions and orderings, Economica, New Series, volume 26, number
102, 1959.
[3] Kfir Eliaz and Efe Ok, Indifference or indecisiveness? Choice theoretic foundations of incomplete preferences, Games and Economic Behavior, volume 56, 2006.
[4] Amartya K Sen, Choice functions and revealed preference, The review of economic studies, volume 38,
number 3, 1971.
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