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Mixed choice structures, with applications to binaryand non-binary optimization.

J. C. R. Alcantud 1

Facultad de Economıa y Empresa, Universidad de Salamanca,

E 37008 Salamanca, Spain

Abstract

We introduce the concept of mixed choice structure in order to propose an

alternative model of non-binary choice behavior under certainty. Some gen-

eral sufficient conditions for optimality on not-necessarily compact sets are

proven. The main conclusion is that one single result incorporates as particu-

lar cases classical theorems that exemplify different approaches both to binary

–Bergstrom (1975), Mehta (1989), Sonnenschein (1971), Walker (1977)– and

non-binary –Nehring (1996), Alcantud (2002a)– optimization.

Keywords: Mixed choice structure; maximal element; acyclicity; compact-

ness; convexity.

JEL Classification Numbers: D11, C60.

1E-mail: jcr@usal.es; tel.: +34-923-294640 ext 3180; fax: +34-923-294686. The comment by ananonymous referee helped to improve the presentation of the results. Financial support from FEDERand Ministerio de Educacion y Ciencia under the Research Project SEJ2005-0304/ECON, and by Juntade Castilla y Leon under the Research Project SA098A05, is gratefully acknowledged.

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1 Introduction

Different models of choice behavior play key roles in Economics. For example, demand

theory relies heavily on such process. The problem that a consumer in a market economy

must face is to choose some consumption vectors from the budget set, which is the set of

admissible bundles that he can afford at fixed prices for a given income. There are several

ways, with increasing generality, to formalize the criterion under which such selection is

made. One severe way to proceed is to assume that the agent has assigned a utility index,

which can be interpreted as a subjective measure of the satisfaction that consumption

vectors yield to him.But weaker rationality assumptions on his tastes permit to extend

classical results of equilibrium theory in Economics so that consumers whose tastes can

not be represented by assignements of utility levels can be incorporated.

The model by binary relations is the most common way to describe individual choice

in the social sciences. Within such model it is expected that a ‘rational’ decision maker

chooses a maximal element for the underlying binary relation in every feasible situation.

The binary model includes that of choice functions that are representable by an ad-

equate binary relation (in a wider sense, we speak of ‘binariness’). However, the generic

approach by choice functions is different from the previous ones in that choices are now

taken as the primitive concept. This permits to adopt a yet more general point of view.

The analysis in this framework depends on properties of actual choices made on differ-

ent choice sets. Consequently, the meaning of ‘rational’ changes in this context, and

it is stated typically in terms of coherence when choices among different situations are

compared.

Not surprisingly, the binariness assumption has been put aside in several choice the-

ories that were developed during the last two decades. To name but a few, procedural

considerations are the germ of Plott (1973). Nehring (1996) has made a significant con-

tribution by relaxing the binariness assumption to ‘finitariness’ in the search for maximal

elements of choice functions. He appeals here to ‘unresolvedness of preference’ as an ar-

gument to avoid the inveterate identification of rationalizable with being derived from

a binary relation. Afterwards, Alcantud (2002b) produced an altogether different model

for non-binary optimization. It precedes Alcantud (2006) and Alcantud and Alos-Ferrer

(2007). The former supposed the first characterization of existence of maximal elements

under lack of binariness. The latter is an application of part of the model in Alcantud

(2002b) to the existence of equilibria in non-cooperative games.

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We observe the following issues regarding these different approaches.

(a) In exploiting properties of binary relations, compactness has been an important issue.

An excellent survey of the sufficient conditions proposed to obtain maximal elements of

binary relations in the setting of compact sets is Border (1985), Chapter 7. Two basic

approaches can be distinguished.

a.1- The first one depends on acyclicity properties of the binary relation. In Sloss

(1971), Brown (1973), Bergstrom (1975) and Walker (1977) it is proved that every nonempty

compact subset of a space on which an upper semicontinuous acyclic binary relation is

defined contains a maximal element. This is usually known as the Bergstrom-Walker the-

orem. Continuity conditions weaker than upper semicontinuity are used e.g., by Mehta

(1989) in the same framework. Further results in the line of this latter theorem are Theo-

rems 1 and 2 in Campbell and Walker (1990), where upper semicontinuity is replaced by

a weaker property but a condition stronger than acyclicity is assumed2.

a.2- The second approach utilises convexity assumptions, which favours the use of

fixed point or related techniques. Fan’s Lemma (1961) can be interpreted in this sense,

which is further exploited by Sonnenschein (1971). A refinement of the latter result is

Theorem 7.2 in Border (1985). The usual framework involves compactness too. This is

the case of the aforementioned works and others like Mehta (1987) and Yannelis and Prab-

hakar (1983) in infinite-dimensional spaces, as well as Shafer and Sonnenschein (1975).

In Mehta’s reference fixed point theorems by Tarafdar (1977) and Schauder provide ap-

propriate maximality results. Some authors have developed ways to dispense with the

compactness assumption in the search for maximal elements. For instance, Mehta (1984)

applies Browder’s (1968) fixed point theorem to deduce the existence of maximal elements

in convex but not necessarily compact subsets of a Euclidean space. Mehta points out

that his result can be generalized to Hausdorff topological vector spaces. Some further

examples are: the use of condensing maps in Mehta (1990), which permits to replace com-

pactness with convexity plus closed boundedness in the context of Banach spaces; or the

technique in Border (1985), Chapter 7, which extends prior results to certain σ-compact

subsets of Rn.

(b) In the event that choices are the primitive concept, a choice structure is given and

its properties yield that choices on certain sets are non-empty. Ensuring non-emptiness

2We have mentioned that the results listed above rely on the concept of compactness. However, inAlcantud (2001) it is argued that this is a strong requirement in the search for maximality, and that itcan be replaced by a certain upper order-compactness in many classical frameworks.

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of the choice in large classes of choice situations is the focal problem. Some of the usual

conditions in such framework are far from appealing in cases like that of an agent faced

to budget restrictions, irrespective of its ellegance in abstract theory. This is the case of

the hypothesis that choices on all finite sets of options are non-empty (non-emptiness in

Nehring, 1996), since finite sets do not arise from budget limitations alone. However, they

seem to enable us to ensure maximization on classes wider than that of compact sets (cf.

LLinares and Sanchez, 1999, where no example is provided). The literature on this topic

is virtually limited to this common approach and the aforementioned Alcantud (2002b).

Additionally to the large literature providing sufficient conditions for maximality, some

authors have proposed different approches to unify the techniques yielding existence of

maximal elements for adequate binary or non-binary choice functions. Among them we

cite Tian (1993), who is allegedly the first contribution that unifies the convexity and

acyclicity approaches, LLinares (1998), and Sanchez, LLinares and Subiza (2000). In a

different line of inquiry, Alcantud (2002a) provides necessary and sufficient conditions in

the full class of the acyclic binary relations. The first characterization of maximality in a

non-binary model has appeared in Alcantud (2006), which benefitted from prior insights

by Rodrıguez-Palmero and Garcıa-Lapresta (2002).

If we depart from a model by binary relations, we may construct a relevant choice struc-

ture by assigning to any choice situation the set of maximal elements in it. That permits

to include the search of maximal elements (both for binary and non-binary problems)

in the following setting: given a non-decisive choice structure, under which conditions

choices out of a subset are non-emtpy?

The purpose of this paper is to unify all the aforementioned approaches. In order to

do so, we propose a general model –by Mixed choice structures– which should be fairly

intuitive due to its similarity with the model by choice structures. Examples show the

reach of our model, and interpretations are put forward. We prove some optimality results

for not-necessarily compact sets in such non-binary context. This work enables us to

derive general results on non-emptiness of non-binary choice functions (inclusive of those

by Nehring and by Alcantud), in addition to many of the aforementioned theorems in the

literature about binary relations (e.g., the Bergstrom-Walker theorem and Sonnenschein’s

theorem), from one single fundamental result. That work permits to envisage the common

structure that underlies those apparently disconnected results, which exemplify (a) the

exploitation of the two main classes of assumptions in the binary case: namely, convexity

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and acyclicity, as well as (b) the techniques that deal with the non-binary case.

Thus our contribution uncovers a common hidden pattern that underlies all the dif-

ferent approaches to optimization in abstract decision theory.

We have organized the paper as follows. Section 2 presents our model, and some

preliminary issues are discussed. A fundamental result –a sufficient condition for non-

empty choice in mixed choice structures– is stated in Section 3. It incorporates the

common background for a number of apparently disconnected results on maximality, that

are derived along the Section. In subsection 3.1 we deduce that Nehring’s (1996) result

and its generalization by LLinares and Sanchez (1999) are particular instances of that

sufficient condition for non-empty choice in mixed choice structures. The same holds for

Alcantud’s (2002b) model, as is shown in subsection 3.2. The derivation of maximality

results under the two main classes of assumptions in the binary case, namely convexity

and acyclicity, are given in subsections 3.3 and 3.4 respectively. Some Concluding remarks

put an end to this paper.

2 The model. Some comments.

We consider a decision maker whose decisions are taken on options included in a set Y .

Given a non-empty subset X ⊆ Y , a (non-decisive) choice structure on X is a pair

(B, c) where B is a collection of non-empty subsets of X and c : B −→ X is a correspon-

dence such that c(S) ⊆ S for all S ∈ B. If c(S) 6= ∅ for all S ∈ B, we say that the choice

structure is decisive.

We assume that the agent has the following primitive notions

(a) (B, c) is a non-decisive choice structure on X, and

(b) S is a collection of non-empty subsets of X, and the agent has assigned to each

S ∈ S a (possibly empty) subset F (S) ⊆ X.

Suppose further that both endowments are related through the following rationality

axiom:

x∗ ∈ (⋂

S∈S,S⊆B

F (S) ) ∩B ⇒ x∗ ∈ c(B) for each B ∈ B (1)

We then say that the agent has defined a (non-decisive) mixed choice structure (B, c, S, F )

on X ⊆ Y . If there is a topology on Y and the F (S) are closed in the topology inherited

by X for all S ∈ S, we speak of a closed mixed choice structure. A (closed) mixed choice

structure is decisive if the underlying choice structure is decisive.

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With these concepts we intend to abstract the following process. When choosing

among the subset X, the decision maker has a possibly inaccurate idea of options that

are either preferred or indifferent to a given set of options S, and that imprecise idea is

captured by the assignement of the subsets F (S). We insist that we do not intend to say

that F (S) is exactly the set of preferred or indifferent options. The F (S) can be viewed

as a selection of options that the agent perceives as definitely better than or indifferent

to S. This permits a certain degree of uncertainty, as long as these assignements need

not capture all ‘not-worse-than’ options, and thus the model does not force the agent to

make up his/her mind on all possible comparisons of options. Similarly, because c reflects

eligible options on certain choice situations, we must interpret that the rationality axiom

(1) simply states that any options that are definitely perceived as ‘not-worse-than’ any

considered set of options must be fit to be chosen. Obviously, non-emtpy choices are

associated with the existence of optima in the set. The problem that arises is: when can

we assure that choices are non-emtpy in a non-decisive mixed choice structure?

This model permits to consider a particular case, that we call strong (closed) mixed

choice structure. It consists of a (closed) mixed choice structure (B, c, S, F ) on X ⊆ Y

where S = { {x} : x ∈ X}, and thus the rationality axioms becomes

x∗ ∈ (⋂x∈B

F (x) ) ∩B ⇒ x∗ ∈ c(B), for each B ∈ B. (2)

Also, for simplicity we denote that strong (closed) mixed choice structure by (B, c, F ).

We may drop the adjective ‘strong’ when the corresponding notation indicates so.

It is plain that (decisive) choice structures can be identified with trivial (decisive)

strong mixed choice structures by setting F (x) = ∅ for all x ∈ X. If there is a topology

on X then we obtain closed, strong mixed choice structures. Here we have enriched the

model by adding information in the form of possibly non-trivial F (S) subsets. Of course

this process means that we can associate an obvious strong mixed choice structure with

any binary relation. Again, in order to define c we only need to assign the maximal

elements to any subset, and then define F (x) = ∅ for all x ∈ X. That is, we use the

choice structure that picks the maximal elements and then construct a strong mixed choice

structure as above.

Under certain conditions, a (closed) mixed choice structure (B, c, S, F ) with { {x} :

x ∈ X} ⊆ S induces a strong (closed) mixed choice structure by restricting to { {x} :

x ∈ X}. It suffices that ∩x∈SF (x) ⊆ F (S) for each S ∈ S, so that axiom (2) holds for

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(B, c, F ).

3 A general optimality result and applications.

We first present a result that consists of a sufficient condition for non-empty choice on

not-necessarily compact sets in closed mixed choice structures.

Theorem 1 Let Y be a topological space, X ⊆ Y and (B, c, S, F ) a closed mixed choice

structure on X. Let A ∈ B, and suppose that K = F (S1)∩ ...∩ F (Sn)∩A is compact for

some {S1, ..., Sn} ⊆ S with S1, ..., Sn ⊆ A. Assume further that

the family {F (S) ∩ A}S∈S, S⊆A has non-empty finite intersections. (3)

Then c(A) 6= ∅.

Proof. The finite intersection property of closed subsets characterizes compactness (cf.

Dugundji (1978), Chapter XI). It is clear that the family {F (S) ∩ K}S∈S, S⊆A of closed

sets in K has non-empty finite intersections. It follows that ∩S∈S, S⊆A(F (x) ∩ K) 6= ∅.

Now the rationality axiom (1) yields c(A) 6= ∅. �

Next we show that the two basic models for ensuring the existence of optimal choices

in non-binary situations can be endowed with suitable closed weak mixed structures, in

such way that Theorem 1 applies. This is done in subsections 3.1 and 3.2. Theorem 1

permits a particular statement for strong mixed choice structures (namely, Theorem 2

below). We prove in subsections 3.3 and 3.4 that well-known theorems due to Bergstrom

and Walker, Mehta or Sonnenschein induce strong closed mixed choice structures that

satisfy the conditions of that particular result, and therefore of Theorem 1 above, in a

natural way. As was mentioned, those classical theorems exemplify the two fundamental

approaches to binary maximization.

3.1 The non-binary axiomatization by Nehring.

Along this subsection we fix D, a domain of non-empty subsets of X that represents all the

choice situations that the agent can face. As in Nehring (1996) and LLinares and Sanchez

(1999), we assume that all finite subsets of X belong to D. Denote by C : D −→ X

a correspondence such that C(S) ⊆ S for all S ∈ D. F(S) will denote the set of all

non-empty finite subsets of the choice situation S. Following Nehring (1996) we define:

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A1. Non-emptiness. If S ∈ D is finite then C(S) 6= ∅.

A2. Contraction consistency or Chernoff condition. For all S, T ∈ D : T ⊆ S implies

C(S) ∩ T ⊆ C(T ).

A3. Continuity. For all S ∈ D finite, {x ∈ X : x ∈ C(S ∪ {x})} is closed.

A4. Finitariness. For all S ∈ D, if x ∈ S satisfies that for all T ∈ F(S), x ∈ T implies

x ∈ C(T ), then x ∈ C(S).

Nehring (1996) proves that under A1-A4, C assigns a non-empty choice to any compact

set belonging to D. LLinares and Sanchez (1999) have proven a similar result under

different hypotheses, that enlarges the class of subsets with non-empty choice. Let us

recall some issues that are needed in order to expose their contribution.

Define Axiom α∗ as follows: for all A ∈ F(X), there exists a0 ∈ C(A) such that for

all B ∈ F(A), if a0 ∈ B then a0 ∈ C(B). Observe that this axiom implies A1, which was

not the case for A2.

For any S ∈ D we define the correspondence ΓS : F(S) → S given by ΓS(T ) = CST =

{x ∈ S : x ∈ C(T ∪ {x})}. This definition had been given before in Nehring (1996).

Observe that C(T ) ⊆ CST if T ⊆ S.

Recall that if X and Y are topological spaces, then a correspondence Φ : X → Y is

transfer closed valued on X if for every x ∈ X, y 6∈ Φ(x) implies that there exists x′ ∈ X

such that y 6∈ cl[φ(x)], where cl denotes the topological closure.

C satisfies Transfer continuity if for all S ∈ D, ΓS is a transfer closed correspondence.

i.e, if for every T ∈ F(S), x 6∈ CST implies that there is T ′ ∈ F(S) such that x 6∈ clS(CS

T ′).

Here clS(CST ′) refers to the closure with respect to the relative topology of S.

Let us show that a closed mixed choice structure in the conditions of Theorem 1 permits

to derive the following extension of Nehring’s theorem due to LLinares and Sanchez.

Corollary 1 Suppose that C satisfies Axiom α∗, transfer continuity and A4. Then, C(A)

is non-empty whenever A ∈ D contains a non-empty finite subset T with clA(ΓA(T ))

compact.

Proof. Take B = {A}, c(A) = C(A) and S = F(A). For each S ∈ S, define

F (S) = clA(ΓA(S)). By A1 this set is non-empty, and it is closed in A. Equation (1)

holds: given the only element A in B, for any x ∈⋂

S∈S,S⊆A F (S) =⋂

S∈F(A) clA(CAS ) we

have x ∈⋂

S∈F(A) CAS by transfer continuity and then x ∈ C(A) by A4.

We have constructed a closed mixed choice structure (B, c, S, F ) on A.

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By assumption, there is a non-empty finite subset T of A with clA(ΓA(T )) = F (T )∩A

compact. Also, LLinares and Sanchez (1999) show that {F (S) ∩ F (T )}S∈F(A) has non-

empty finite intersections. Now Theorem 1 ensures that there is x ∈ c(A) = C(A). �

3.2 The non-binary axiomatization by Alcantud.

The model provided by Alcantud (2002b) ensures the existence of optimal choices in

non-binary contexts, under a new set of axioms. Just as Nehring’s model encompasses

the Bergstrom-Walker assumptions naturally, the case under inspection includes the SSB

model (cf. Fishburn, 1984). This fact permits to apply the model in Alcantud (2002b) to

the search for equilibria in non-cooperative games defined by choice rules: cf. Alcantud

and Alos-Ferrer (2007), where the SSB equilibrium theorem (cf. Kreweras, 1961, Fishburn

and Rosenthal, 1986) is derived as a Corollary.

Along this subsection, X will be a convex subset of a Hausdorff topological vector

space. We assume that the convex hull of every finite subset of X belongs to D, a domain

of non-empty subsets of X.

For all T ∈ D finite and x ∈ T , define MT (x) = {y ∈ co(T ) : y ∈ C([x, y])}. Then:

Corollary 2 Suppose that C satisfies

B1. Comparison with vertices. If z ∈ co(x1, ..., xn) for some x1, ..., xn ∈ X then

(a) there is i ∈ {1, ..., n} with z ∈ C([xi, z]), and also

(b) whenever z ∈ C([xi, z]) for every i ∈ {1, ..., n} then z ∈ C(co(x1, ..., xn)).

B2. Continuity. For all T ∈ D finite with cardinality 2 or higher, and for each x ∈ T ,

the set MT (x) is closed.

Then, C(co(x1, ..., xn)) is non-empty for every x1, ..., xn ∈ X.

Proof. Let us fix x1, ..., xn ∈ X. We let T = {x1, ..., xn} and A = co(x1, ..., xn).

In order to prove that an appeal to Theorem 1 yields that C(A) is non-empty, take

B = {A}, c(A) = C(A) and S = {[xi, z] : i ∈ {1, ..., n}, z ∈ A}. For each S ∈ S,

define F (S) = MT (xi), which is closed in co(x1, ..., xn). It is non-empty because {x} =

C(x) ⊆ MT (x) by B1(a). Equation (1) holds: given the only element A in B, for any

x ∈⋂

S∈S,S⊆A F (S) =⋂{F ([xi, z]) : i ∈ {1, ..., n}, z ∈ A} =

⋂{MT (xi) : i ∈ {1, ..., n}}

we have x ∈⋂

i=1,...,n C([xi, z]) and then x ∈ C(A) by B1(b).

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A is homeomorphic to a finite-dimensional Euclidean ball thus it is compact. In

order to check that Theorem 1 ensures that there is x ∈ c(A) = C(A), just note that

{F (S) ∩ A}S∈S,S⊆A = {MT (xi) : i ∈ {1, ..., n}} has non-empty finite intersections as is

argued in the proof of Alcantud (2002b), Theorem 1. �

3.3 Convexity assumptions: results and applications.

In this subsection we show that our model is adequate to derive maximality results that

appeal to fixed point or related techniques. Since we deal with binary contexts alone we

are able to state the existing results in terms of particularizations of Theorem 1 to strong

mixed choice structures. Some definitions are needed.

Let � denote an irreflexive binary relation on X. We say that � is acyclic if for each

x1, ..., xn ∈ X : x1 � x2 � ... � xn implies x1 6= xn.

An acyclic binary relation � on X is negatively transitive if whenever x, y, z ∈ X and

x � y then either x � z or z � y. The term strict preference will apply to asymmetric

and negatively transitive binary relations henceforth.

For an asymmetric binary relation � on a set X, < will denote the completion of �;

i.e., x < y means that y � x is false, for each x, y ∈ X. The indifference associated with

�, which is commonly denoted by ∼, is defined by x ∼ y iff not x � y and not y � x.

Let � be a binary relation on a set X. We denote L(x) = {a ∈ X : x � a}, the lower

contour set associated with x. We had introduced U(x) = {a ∈ X : a � x}, the upper

contour set associated with x, before.

Let τ denote a topology on X. We say that the binary relation � is upper (lower)

semicontinuous if for all x ∈ X the lower (upper) contour set L(x) (U(x)) is open; and it

is continuous if it is both upper and lower semicontinuous.

We say that a binary relation � on a set X is transfer lower continuous if: for all x ∈ X

such that ∃y′ � x ⇒ ∃y ∈ X such that x ∈ int(L(y)), as defined in Mehta (1989). This

condition is obviously implied by upper semicontinuity. Some authors (e.g. Sonnenschein

(1971)) have dealt with the following alternative expression for transfer lower continuity:

∀x � y there is x′ ∈ X and a neighborhood N(y) of y such that x′ � N(y)

where x′ � N(y) means that x′ � z for all z ∈ N(y).

By expository convenience, along the present Section we fix X ⊆ Rn for some n.

Nonetheless, Theorem 2 below can be stated in Hausdorff topological vector spaces in-

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stead, without further change in the statement. As was mentioned before, many of the

maximality results in the line under inspection rely on or depart from Fan’s Lemma (cf.

Fan, 1961, also 1984). Here we make use of the following particular form:

Lemma 1 Let X ⊆ Rn and, for each x ∈ X, let F (x) ⊆ Rn be closed. Suppose that

F (x0) is compact for some x0 ∈ X and

for each finite subset {x1, ..., xk} ⊆ X : co{x1, ..., xk} ⊆ ∪ki=1F (xi) (4)

where co{x1, ..., xk} denotes the convex hull of {x1, ..., xk}. Then whenever X is closed

and convex the set ∩x∈X(F (x) ∩X) is non-empty and compact.

Fan’s result is intended to ensure non-emptiness of ∩x∈XF (x), even though X ⊆ Rn is

neither convex nor closed. We appeal to the Lemma since we need that this intersection

meets X.

Proof of Lemma 1. Define FX(x) = F (x) ∩ X for each x ∈ X. Fan’s Lemma

applies: the new assignement of closed sets satisfies Equation (4) because co{x1, ..., xk} ⊆∪k

i=1F (xi) for each finite subset {x1, ..., xk} ⊆ X, and co{x1, ..., xk} ⊆ X by convexity of

X. Also, FX(x0) is compact. Thus ∩x∈XFX(x) = ∩x∈X(F (x) ∩X) is non-empty. �

The next example shows that the convexity assumption in Lemma 1 is not superfluous.

Example 1 Let X = {x1 = (−1, 0), x2 = (0, 1), x3 = (1, 0)} ⊆ R2, and define F (x1) =

{(x, y) ∈ R2 : x > −1, −x + y 6 1 6 −x + 2y}, F (x2) = {(x, y) ∈ R2 : x 6 1, 0 6 y, x +

2y 6 1}, F (x3) = {(x, y) ∈ R2 : y 6 1, 0 6 x, x + y 6 1}. All the requirements of Fan’s

Lemma are satisfied and X is closed but not convex. Despite the fact that ∩x∈XF (x) =

{(0, 12)} 6= ∅ as Fan’s Lemma prescribes, we have ∩x∈X(F (x) ∩X) = ∅.

We are now ready to present our next Theorem.

Theorem 2 Let X ⊆ Rn and let (B, c, F ) be a strong closed mixed choice structure on X.

Suppose that F (x0) is compact for some x0 ∈ X and Equation (4) holds. Then whenever

X is closed and convex and belongs to B we have c(X) 6= ∅.

Proof. Obvious from Lemma 1 and the rationality axiom (1). Observe that the F (x) are

closed in Rn because so is X 3. �

3Our assumptions imply x ∈ F (x) 6= ∅ and thus x ∈ c({x}) for all x ∈ X such that {x} ∈ B. Ofcourse, such apparent restriction excludes virtually no practical application.

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Remark 1 Theorem 2 addresses to Theorem 1 as well: the proof of Fan’s Lemma (as

exposed e.g., in Corollary 5.7 of Border, 1985) shows that the family of closed sets {F (x)∩X}x∈X has non-empty finite intersections.

We now comment on Theorem 2 and applications of it. Firstly, we check that it permits

to view a classical result from a new perspective (cf. Corollary 3 below, a refinement of

a theorem by Sonnenschein, 1971, as given by Theorem 7.2 in Border, 1985). Secondly,

we show that the apparently artificial requirements of Theorem 2 are satisfied in a well-

known framework in consumer theory. Thirdly, we prove that Theorem 2 applies to

non-representable models too.

Corollary 3 (Sonnenschein, Border) Let K ⊆ Rn be compact and convex. Suppose

that � is a binary relation on K such that

(i) for all x ∈ K, x 6∈ coU(x)

(ii) � is transfer lower continuous

Then � has a maximal element on K.

Proof. Define (a) F (x) = K\intL(x) compact and nonempty (otherwise K ⊆ intL(y) ⊆L(y) with y ∈ K, and thus y ∈ U(y) ⊆ coU(y), which contradicts (i)) for all x ∈ K;

and (b) the choice structure ({K}, c) where c(K) = ∩x∈KF (x). This induces a closed

mixed choice structure that satisfies requirement (4) of Theorem 2, as it is shown in

the proof of Border (1985), Theorem 7.2. This means that c(K) is not empty. But

c(K) = ∩x∈K(K \ L(x)) by transfer lower continuity, thus c(K) is the set of maximal

elements of � on K. �

Example 2 The conditions of Theorem 2 hold in the following common situation in

demand theory. Suppose that � is an upper semicontinuous, monotone and convex strict

preference on X = Rn+. Then, F (x) = {x ∈ X : x < y for all y ∈ X} are closed. If

c(B) denotes the (possibly empty) subset of maximal elements in B for each B ∈ X, then

(P(X), c, F ) is a closed mixed choice structure on X. Finally, John (2000) has proved

that Equation (4) holds under the aforementioned conditions.

Both Corollary 3 and Example 2 concern representable problems. This may settle the

wrong idea that this is the case of Theorem 2 as well. The next Example helps to clarify

the real scope of our result.

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Example 3 Let X = {(x, y) ∈ R2 : y > 0} and X1 = {(x, y) ∈ R2 : y = 0}. Now,

for each x ∈ R we set: F (x, y) = X if y > 0, and F (x, 0) = X1. Define then: c(B) =

∩(x,y)∈BF (x, y) for each B ⊆ X, and c(X) = X1∪{(1, 1)}. Observe that ∩(x,y)∈XF (x, y) =

X1. It is obvious that (P(X), c, F ) is a strong closed mixed choice structure on X.

Clearly, Equation (4) holds true. Thus the conditions of Theorem 2 are satisfied though

this is not a representable choice structure. Notice that the fact that c(X) = X1 ∪{(1, 1)}is incompatible with the existence of an underlying binary relation explaining choices, since

it is plainly false that (1, 1) ∈ c({(1, 1), (x, y)}) for each (x, y) ∈ X.

3.4 Results regarding acyclicity assumptions.

This subsection is devoted to prove that our model includes results that rely on acyclicity

or related assumptions, such as the Theorem of Bergstrom-Walker and generalizations of

it.

In fact, we have proven that a decision agent in the conditions of the Bergstrom-Walker

theorem induces a closed mixed structure in the conditions of Theorem 1: the agent

generates a choice structure in the conditions of Nehring’s theorem (see the Corollary

in Nehring, 1996), which in turn induces a convenient closed mixed structure. That

possibility is now exploited further: we show that other classical theorems that appeal

to the use of acyclicity-related assumptions induce an underlying closed mixed choice

structure in the conditions of Theorem 1. The fact that we may associate strong mixed

structures appeals once again to the binariness of the problem. Consider the following

result in Mehta (1989).

Corollary 4 (Mehta) 4 Any acyclic binary relation � on a compact topological space X

that is transfer lower continuous has a maximal element.

Proof. Define F (x) = X \intL(x), a closed and non-empty set for each x ∈ X (otherwise

X ⊆ L(x), contradicting acyclicity). A well-known argument involving acyclicity yields

that {F (x)}x∈X has non-empty finite intersections (see e.g., the proof of Theorem 4 in

Mehta, 1989). Define now c(X) = ∩x∈XF (x). We have produced a strong closed mixed

choice structure that satisfies the requirements of Theorem 1, thus c(X) 6= ∅. Transfer

lower continuity ensures that c(X) is contained in the set of maximal elements of X. �

4Since the theorem of Bergstrom-Walker is a particular instance of Mehta’s statement, it follows againthat we can derive it from Theorem 1. Besides, the procedure that reduces Mehta’s result to Theorem 1permits to extend his original statement to not-necessarily compact sets too.

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4 Concluding remarks.

We have introduced a model by (closed) mixed choice structures which enriches that

by choice structures. We have shown that it is a powerful tool of analysis since it pro-

vides maximality results which apply to different contexts. Some sufficient conditions

for non-emptiness in closed mixed choice structures have been proven (cf. Theorem 1 in

infinite-dimensional topological spaces; Theorem 2 is stated in Euclidean spaces though

it may be easily generalized to infinite-dimensional Hausdorff topological vector spaces).

These conditions are not limited to the case of compact spaces. The aforementioned re-

sults do not imply binariness, which widens the applicability of our sufficient conditions.

Theorems due to Bergstrom and Walker, Mehta, Nehring, Sonnenschein, and Alcantud

among others, have been derived by uncovering underlying closed mixed choice structures

that satisfy the requirements of a single result: namely, Theorem 1. We consider that this

is a significant achievement as long as it establishes a common ground for very different

approaches to maximality both in the binary and the non-binary contexts.

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