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Mixed choice structures, with applications to binary and non-binary optimization
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Journal of Mathematical Economics 44 (2008) 242–250
Mixed choice structures, with applications to binary and non-binary optimization J.C.R. Alcantud ∗ Facultad de Econom´ıa y Empresa, Universidad de Salamanca, E 37008 Salamanca, Spain Available online 2 June 2007
Abstract We introduce the concept of mixed choice structure in order to propose an alternative model of non-binary choice behavior under certainty. Some general sufficient conditions for optimality on not-necessarily compact sets are proven. The main conclusion is that one single result incorporates as particular cases classical theorems that exemplify different approaches both to binary – [Bergstrom, T.C., 1975, Maximal elements of acyclic relations on compact sets, Journal of Economic Theory 10, 403–404; Mehta, G., 1989, Maximal elements in Banach spaces, Indian Journal of Pure and Applied Mathematics 20, 690–697; Sonnenschein, H., 1971, Demand theory without transitive preferences, with applications to the theory of competitive equilibrium. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H. (Eds.), Preferences, Utility and Demand. Harcourt Brace Jovanovich, New York; Walker, M., 1977, On the existence of maximal elements, Journal of Economic Theory 16, 470–474] – and non-binary – [Nehring, K., 1996, Maximal elements of non-binary choice functions on compact sets, Economics Letters 50, 337–340; Alcantud, J.C.R., 2002a, Characterization of the existence of maximal elements of acyclic relations, Economic Theory 19, 407–416] – optimization. © 2007 Elsevier B.V. All rights reserved. JEL Classification: D11; C60 Keywords: Mixed choice structure; Maximal element; Acyclicity; Compactness; Convexity
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 cannot be represented by assignments 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.
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The binary model includes that of choice functions that are representable by an adequate 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 different 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 theories 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 contribution 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 argument 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 Al´os-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. 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 non-empty 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 theorem. 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 Theorems 1 and 2 in Campbell and Walker (1990), where upper semicontinuity is replaced by a weaker property but a condition stronger than acyclicity is assumed.1 (a.2) The second approach utilises convexity assumptions, which favours the use of fixed point or related techniques. Fan’s (1961) Lemma 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 Prabhakar (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 appropriate 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 compactness 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 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 S´anchez, 1999), where no example is provided). The literature on this topic is virtually limited to this common approach and the aforementioned Alcantud (2002b).
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We have mentioned that the results listed above rely on the concept of compactness. However, in Alcantud (2002a) it is argued that this is a strong requirement in the search for maximality, and that it can be replaced by a certain upper order-compactness in many classical frameworks.
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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 S´anchez et al. (2003). 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 benefited from prior insights by Rodr´ıguezPalmero and Garc´ıa-Lapresta (2002). If we depart from a model by binary relations, we may construct a relevant choice structure 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-empty? 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 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 different 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 S´anchez (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 correspondence such that c(S) ⊆ S for all S ∈ B. If c(S) = ∅ 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∗ ∈ ⎝ F (S)⎠ ∩ B ⇒ x∗ ∈ c(B) for each B ∈ B
(1)
S ∈ S,S⊆B
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 assignment 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 assignments 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-empty choices are associated with the existence of optima in the set. The problem that arises is: when can we assure that choices are non-empty 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 ∈ F (x) ∩ B ⇒ x∗ ∈ c(B) for each B ∈ B (2) x∈B
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 ∈ S F (x) ⊆ F (S) for each S ∈ S, so that Axiom (2) holds for (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) = ∅. 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) = ∅. Now the rationality Axiom 1 yields c(A) = ∅. 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.
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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 S´anchez (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: (A1) (A2) (A3) (A4)
Non-emptiness. If S ∈ D is finite then C(S) = ∅. Contraction consistency or Chernoff condition. For all S, T ∈ D : T ⊆ S implies C(S) ∩ T ⊆ C(T ). Continuity. For all S ∈ D finite, {x ∈ X : x ∈ C(S ∪ {x})} is closed. 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 S´anchez (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 S : F(S) → S given by S (T ) = CTS = {x ∈ S : x ∈ C(T ∪ {x})}. This definition had been given before in Nehring (1996). Observe that C(T ) ⊆ CTS if T ⊆ S. Recall that if X and it Y are topological spaces, then a correspondence : X → Y is transfer closed valued on X if for every x ∈ X, y ∈ / (x) implies that there exists x ∈ X such that y ∈ / cl[φ(x )], where cl denotes the topological closure. C satisfies Transfer continuity if for all S ∈ D, S is a transfer closed correspondence. i.e., if for every T ∈ F(S), x∈ / CTS implies that there is T ∈ F(S) such that x ∈ / clS (CTS ). Here clS (CTS ) 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 S´anchez. 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 (A (T )) compact. Proof. Take B = {A}, c(A) = C(A) and S = F(A). For each S ∈ S, define F (S) = clA (A (S)). By A1 this set is nonempty, and it is closed in A. Eq. (1) holds: given the only element A in B, for any x ∈ S ∈ S,S⊆A F (S) = S ∈ F(A) clA (CSA ) we have x ∈ S ∈ F(A) CSA by transfer continuity and then x ∈ C(A) by A4. We have constructed a closed mixed choice structure (B, c, S, F ) on A. By assumption, there is a non-empty finite subset T of A with clA (A (T )) = F (T ) ∩ A compact. Also, LLinares and S´anchez (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 Al´os-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:
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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 , . . . , x {x} = C(x) ⊆ MT (x) by B1(a). n ). It is non-empty because Eq. (1) holds: given the only element A in B, for any x ∈ F (S) = {F ([x i , z]) : i ∈ {1, . . . , n}, z ∈ A} = S ∈ S,S⊆A {MT (xi ) : i ∈ {1, . . . , n}} we have x ∈ i=1,...,n C([xi , z]) and then x ∈ C(A) by B1(b). 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 = xn . An acyclic binary relation on X is negatively transitive if whenever x, y, z ∈ X and x y is false then either x z is false or z y is false. 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, and U(x) = {a ∈ X : a x}, the upper contour set associated with x. 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 instead, 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=1 F (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.
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Fan’s result is intended to ensure non-emptiness of ∩x ∈ X F (x), even though X ⊆ Rn is neither convex nor closed. We appeal to Lemma 1 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 assignment of closed sets satisfies Eq. (4) because co{x1 , . . . , xk } ⊆ ∪ki=1 F (xi ) for each finite subset {x1 , . . . , xk } ⊆ X, and co{x1 , . . . , xk } ⊆ X by convexity of X. Also, FX (x0 ) is compact. Thus ∩x ∈ X FX (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 ≤ 1 ≤ −x + 2y}, F (x2 ) = {(x, y) ∈ R2 : x ≤ 1, 0 ≤ y, x + 2y ≤ 1}, F (x3 ) = {(x, y) ∈ R2 : y ≤ 1, 0 ≤ x, x + y ≤ 1}. All the requirements of Fan’s Lemma are satisfied and X is closed but not convex. Despite the fact that ∩x ∈ X F (x) = {(0, 1/2)} = ∅ 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 Eq. (4) holds. Then whenever X is closed and convex and belongs to B we have c(X) = ∅. Proof. Obvious from Lemma 1 and the rationality Axiom (1). Observe that the F (x) are closed in Rn because so is X.2 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 ∈ / co U(x) (ii) is transfer lower continuous Then has a maximal element on K. Proof. Define (a) F (x) = K \ intL(x) compact and non-empty (otherwise K ⊆ int L(y) ⊆ L(y) with y ∈ K, and thus y ∈ U(y) ⊆ co U(y), which contradicts (i)) for all x ∈ K; and (b) the choice structure ({K}, c) where c(K) = ∩x ∈ K F (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 Eq. (4) holds under the aforementioned conditions. Our assumptions imply x ∈ F (x) = ∅ and thus x ∈ c({x}) for all x ∈ X such that {x} ∈ B. Of course, such apparent restriction excludes virtually no practical application. 2
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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. 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) ∈ B F (x, y) for each B ⊆ X, and c(X) = X1 ∪ {(1, 1)}. Observe that ∩(x,y) ∈ X F (x, y) = X1 . It is obvious that (P(X), c, F ) is a strong closed mixed choice structure on X. Clearly, Eq. (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. Any acyclic binary relation on a compact topological space X that is transfer lower continuous has a maximal element. 3 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 ∈ X F (x). We have produced a strong closed mixed choice structure that satisfies the requirements of Theorem 1, thus c(X) = ∅. Transfer lower continuity ensures that c(X) is contained in the set of maximal elements of X. 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 provides 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 results 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. Acknowledgements The comment by an anonymous referee helped to improve the presentation of the results. Financial support from FEDER and Ministerio de Educaci´on y Ciencia under the Research Project SEJ2005-0304/ECON, and by Junta de Castilla y Le´on under the Research Project SA098A05, is gratefully acknowledged. 3
Since the theorem of Bergstrom–Walker is a particular instance of Mehta’s statement, it follows again that we can derive it from Theorem 1. Besides, the procedure that reduces Mehta’s result to Theorem 1 permits to extend his original statement to not-necessarily compact sets too.
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References Alcantud, J.C.R., 2002a. Characterization of the existence of maximal elements of acyclic relations. Economic Theory 19, 407–416. Alcantud, J.C.R., 2002b. Non-binary choice in a non-deterministic model. Economics Letters 77, 117–123. Alcantud, J.C.R., 2006. Maximality with or without binariness: transfer-type characterizations. Mathematical Social Sciences 51, 182–191. Alcantud, J.C.R., Al´os-Ferrer, C., 2007. Nash equilibria for non-binary choice rules. International Journal of Game Theory 35, 455–464. Bergstrom, T.C., 1975. Maximal elements of acyclic relations on compact sets. Journal of Economic Theory 10, 403–404. Border, K.C., 1985. Fixed point theorems with applications to economics and game theory. Cambridge University Press, Cambridge. Browder, F.E., 1968. The fixed-point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 177, 283–301. Brown, D.J., 1973. Acyclic choice, Cowles Foundation Discussion Paper No. 360, Yale University. Campbell, D.E., Walker, M., 1990. Maximal elements of weakly continuous relations. Journal of Economic Theory 50, 459–464. Dugundji, J., 1978. Topology. Allyn and Bacon Inc., Boston. Fan, K., 1961. A generalization of Tychonoff’s Fixed Point theorem. Mathematische Annalen 142, 305–310. Fan, K., 1984. Some properties of convex sets related to fixed point properties. Mathematische Annalen 266, 519–537. Fishburn, P.C., 1984. Dominance in SSB utility utility. Journal of Economic Theory 34, 130–148. Fishburn, P.C., Rosenthal, R.W., 1986. Noncooperative games and nontransitive preferences. Mathematical Social Sciences 12, 1–7. John, R., 2000. Quasimonotone individual demand. Optimization 47, 201–209. Kreweras, G., 1961. Sur une Possibilit´e de Rationaliser les Intransitivit´es. La D´ecision, Colloques Internacionales du CNRS, Paris, pp. 27–32. LLinares, J.-V., 1998. Unified treatment of the problem of existence of maximal elements in binary relations: a characterization. Journal of Mathematical Economics 29, 285–302. LLinares, J.-V., S´anchez, M.C., 1999. Non-binary choice functions on non-compact sets. Economics Letters 63, 29–32. Mehta, G., 1984. Maximal elements for non-transitive binary relations. Economics Letters 14, 163–165. Mehta, G., 1987. Weakly lower demicontinuous preference maps. Economics Letters 23, 15–18. Mehta, G., 1989. Maximal elements in Banach spaces. Indian Journal of Pure and Applied Mathematics 20, 690–697. Mehta, G., 1990. Maximal elements of condensing preference maps. Applied Mathematics Letters 3, 69–71. Nehring, K., 1996. Maximal elements of non-binary choice functions on compact sets. Economics Letters 50, 337–340. Plott, C., 1973. Path independence, rationality and social choice. Econometrica 41, 1075–1091. Rodr´ıguez-Palmero, C., Garc´ıa-Lapresta, J.L., 2002. Maximal elements for irreflexive binary relations on compact sets. Mathematical Social Sciences 43, 55–60. S´anchez, M.C., LLinares, J.-V., Subiza, B., 2003. A KKM-result and an application for binary and non-binary choice functions. Economic Theory 21, 185–193. Shafer, W.J., Sonnenschein, H., 1975. Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics 2, 345–348. Sloss, J.L., 1971. Stable points of directional preference relations, Technical Report No. 71-7, Operations Research House, Stanford University. Sonnenschein, H., 1971. Demand theory without transitive preferences with applications to the theory of competitive equilibrium. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H. (Eds.), Preferences, Utility and Demand. Harcourt Brace Jovanovich, New York. Tarafdar, E., 1977. On nonlinear variational inequalities. Proceedings of the American Mathematical Society 67, 95–98. Tian, G., 1993. Necessary and sufficient conditions for maximization of a class of preference relations. Review of Economic Studies 60, 949–958. Walker, M., 1977. On the existence of maximal elements. Journal of Economic Theory 16, 470–474. Yannelis, N.C., Prabhakar, N.D., 1983. Existence of maximal elements and equilibria in linear topological spaces. Journal of Mathematical Economics 12, 233–245.
Journal of Mathematical Economics 44 (2008) 242–250
Mixed choice structures, with applications to binary and non-binary optimization J.C.R. Alcantud ∗ Facultad de Econom´ıa y Empresa, Universidad de Salamanca, E 37008 Salamanca, Spain Available online 2 June 2007
Abstract We introduce the concept of mixed choice structure in order to propose an alternative model of non-binary choice behavior under certainty. Some general sufficient conditions for optimality on not-necessarily compact sets are proven. The main conclusion is that one single result incorporates as particular cases classical theorems that exemplify different approaches both to binary – [Bergstrom, T.C., 1975, Maximal elements of acyclic relations on compact sets, Journal of Economic Theory 10, 403–404; Mehta, G., 1989, Maximal elements in Banach spaces, Indian Journal of Pure and Applied Mathematics 20, 690–697; Sonnenschein, H., 1971, Demand theory without transitive preferences, with applications to the theory of competitive equilibrium. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H. (Eds.), Preferences, Utility and Demand. Harcourt Brace Jovanovich, New York; Walker, M., 1977, On the existence of maximal elements, Journal of Economic Theory 16, 470–474] – and non-binary – [Nehring, K., 1996, Maximal elements of non-binary choice functions on compact sets, Economics Letters 50, 337–340; Alcantud, J.C.R., 2002a, Characterization of the existence of maximal elements of acyclic relations, Economic Theory 19, 407–416] – optimization. © 2007 Elsevier B.V. All rights reserved. JEL Classification: D11; C60 Keywords: Mixed choice structure; Maximal element; Acyclicity; Compactness; Convexity
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 cannot be represented by assignments 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.
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The binary model includes that of choice functions that are representable by an adequate 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 different 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 theories 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 contribution 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 argument 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 Al´os-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. 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 non-empty 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 theorem. 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 Theorems 1 and 2 in Campbell and Walker (1990), where upper semicontinuity is replaced by a weaker property but a condition stronger than acyclicity is assumed.1 (a.2) The second approach utilises convexity assumptions, which favours the use of fixed point or related techniques. Fan’s (1961) Lemma 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 Prabhakar (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 appropriate 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 compactness 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 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 S´anchez, 1999), where no example is provided). The literature on this topic is virtually limited to this common approach and the aforementioned Alcantud (2002b).
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We have mentioned that the results listed above rely on the concept of compactness. However, in Alcantud (2002a) it is argued that this is a strong requirement in the search for maximality, and that it can be replaced by a certain upper order-compactness in many classical frameworks.
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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 S´anchez et al. (2003). 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 benefited from prior insights by Rodr´ıguezPalmero and Garc´ıa-Lapresta (2002). If we depart from a model by binary relations, we may construct a relevant choice structure 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-empty? 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 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 different 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 S´anchez (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 correspondence such that c(S) ⊆ S for all S ∈ B. If c(S) = ∅ 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∗ ∈ ⎝ F (S)⎠ ∩ B ⇒ x∗ ∈ c(B) for each B ∈ B
(1)
S ∈ S,S⊆B
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 assignment 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 assignments 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-empty choices are associated with the existence of optima in the set. The problem that arises is: when can we assure that choices are non-empty 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 ∈ F (x) ∩ B ⇒ x∗ ∈ c(B) for each B ∈ B (2) x∈B
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 ∈ S F (x) ⊆ F (S) for each S ∈ S, so that Axiom (2) holds for (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) = ∅. 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) = ∅. Now the rationality Axiom 1 yields c(A) = ∅. 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.
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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 S´anchez (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: (A1) (A2) (A3) (A4)
Non-emptiness. If S ∈ D is finite then C(S) = ∅. Contraction consistency or Chernoff condition. For all S, T ∈ D : T ⊆ S implies C(S) ∩ T ⊆ C(T ). Continuity. For all S ∈ D finite, {x ∈ X : x ∈ C(S ∪ {x})} is closed. 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 S´anchez (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 S : F(S) → S given by S (T ) = CTS = {x ∈ S : x ∈ C(T ∪ {x})}. This definition had been given before in Nehring (1996). Observe that C(T ) ⊆ CTS if T ⊆ S. Recall that if X and it Y are topological spaces, then a correspondence : X → Y is transfer closed valued on X if for every x ∈ X, y ∈ / (x) implies that there exists x ∈ X such that y ∈ / cl[φ(x )], where cl denotes the topological closure. C satisfies Transfer continuity if for all S ∈ D, S is a transfer closed correspondence. i.e., if for every T ∈ F(S), x∈ / CTS implies that there is T ∈ F(S) such that x ∈ / clS (CTS ). Here clS (CTS ) 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 S´anchez. 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 (A (T )) compact. Proof. Take B = {A}, c(A) = C(A) and S = F(A). For each S ∈ S, define F (S) = clA (A (S)). By A1 this set is nonempty, and it is closed in A. Eq. (1) holds: given the only element A in B, for any x ∈ S ∈ S,S⊆A F (S) = S ∈ F(A) clA (CSA ) we have x ∈ S ∈ F(A) CSA by transfer continuity and then x ∈ C(A) by A4. We have constructed a closed mixed choice structure (B, c, S, F ) on A. By assumption, there is a non-empty finite subset T of A with clA (A (T )) = F (T ) ∩ A compact. Also, LLinares and S´anchez (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 Al´os-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:
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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 , . . . , x {x} = C(x) ⊆ MT (x) by B1(a). n ). It is non-empty because Eq. (1) holds: given the only element A in B, for any x ∈ F (S) = {F ([x i , z]) : i ∈ {1, . . . , n}, z ∈ A} = S ∈ S,S⊆A {MT (xi ) : i ∈ {1, . . . , n}} we have x ∈ i=1,...,n C([xi , z]) and then x ∈ C(A) by B1(b). 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 = xn . An acyclic binary relation on X is negatively transitive if whenever x, y, z ∈ X and x y is false then either x z is false or z y is false. 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, and U(x) = {a ∈ X : a x}, the upper contour set associated with x. 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 instead, 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=1 F (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.
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Fan’s result is intended to ensure non-emptiness of ∩x ∈ X F (x), even though X ⊆ Rn is neither convex nor closed. We appeal to Lemma 1 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 assignment of closed sets satisfies Eq. (4) because co{x1 , . . . , xk } ⊆ ∪ki=1 F (xi ) for each finite subset {x1 , . . . , xk } ⊆ X, and co{x1 , . . . , xk } ⊆ X by convexity of X. Also, FX (x0 ) is compact. Thus ∩x ∈ X FX (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 ≤ 1 ≤ −x + 2y}, F (x2 ) = {(x, y) ∈ R2 : x ≤ 1, 0 ≤ y, x + 2y ≤ 1}, F (x3 ) = {(x, y) ∈ R2 : y ≤ 1, 0 ≤ x, x + y ≤ 1}. All the requirements of Fan’s Lemma are satisfied and X is closed but not convex. Despite the fact that ∩x ∈ X F (x) = {(0, 1/2)} = ∅ 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 Eq. (4) holds. Then whenever X is closed and convex and belongs to B we have c(X) = ∅. Proof. Obvious from Lemma 1 and the rationality Axiom (1). Observe that the F (x) are closed in Rn because so is X.2 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 ∈ / co U(x) (ii) is transfer lower continuous Then has a maximal element on K. Proof. Define (a) F (x) = K \ intL(x) compact and non-empty (otherwise K ⊆ int L(y) ⊆ L(y) with y ∈ K, and thus y ∈ U(y) ⊆ co U(y), which contradicts (i)) for all x ∈ K; and (b) the choice structure ({K}, c) where c(K) = ∩x ∈ K F (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 Eq. (4) holds under the aforementioned conditions. Our assumptions imply x ∈ F (x) = ∅ and thus x ∈ c({x}) for all x ∈ X such that {x} ∈ B. Of course, such apparent restriction excludes virtually no practical application. 2
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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. 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) ∈ B F (x, y) for each B ⊆ X, and c(X) = X1 ∪ {(1, 1)}. Observe that ∩(x,y) ∈ X F (x, y) = X1 . It is obvious that (P(X), c, F ) is a strong closed mixed choice structure on X. Clearly, Eq. (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. Any acyclic binary relation on a compact topological space X that is transfer lower continuous has a maximal element. 3 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 ∈ X F (x). We have produced a strong closed mixed choice structure that satisfies the requirements of Theorem 1, thus c(X) = ∅. Transfer lower continuity ensures that c(X) is contained in the set of maximal elements of X. 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 provides 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 results 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. Acknowledgements The comment by an anonymous referee helped to improve the presentation of the results. Financial support from FEDER and Ministerio de Educaci´on y Ciencia under the Research Project SEJ2005-0304/ECON, and by Junta de Castilla y Le´on under the Research Project SA098A05, is gratefully acknowledged. 3
Since the theorem of Bergstrom–Walker is a particular instance of Mehta’s statement, it follows again that we can derive it from Theorem 1. Besides, the procedure that reduces Mehta’s result to Theorem 1 permits to extend his original statement to not-necessarily compact sets too.
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References Alcantud, J.C.R., 2002a. Characterization of the existence of maximal elements of acyclic relations. Economic Theory 19, 407–416. Alcantud, J.C.R., 2002b. Non-binary choice in a non-deterministic model. Economics Letters 77, 117–123. Alcantud, J.C.R., 2006. Maximality with or without binariness: transfer-type characterizations. Mathematical Social Sciences 51, 182–191. Alcantud, J.C.R., Al´os-Ferrer, C., 2007. Nash equilibria for non-binary choice rules. International Journal of Game Theory 35, 455–464. Bergstrom, T.C., 1975. Maximal elements of acyclic relations on compact sets. Journal of Economic Theory 10, 403–404. Border, K.C., 1985. Fixed point theorems with applications to economics and game theory. Cambridge University Press, Cambridge. Browder, F.E., 1968. The fixed-point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 177, 283–301. Brown, D.J., 1973. Acyclic choice, Cowles Foundation Discussion Paper No. 360, Yale University. Campbell, D.E., Walker, M., 1990. Maximal elements of weakly continuous relations. Journal of Economic Theory 50, 459–464. Dugundji, J., 1978. Topology. Allyn and Bacon Inc., Boston. Fan, K., 1961. A generalization of Tychonoff’s Fixed Point theorem. Mathematische Annalen 142, 305–310. Fan, K., 1984. Some properties of convex sets related to fixed point properties. Mathematische Annalen 266, 519–537. Fishburn, P.C., 1984. Dominance in SSB utility utility. Journal of Economic Theory 34, 130–148. Fishburn, P.C., Rosenthal, R.W., 1986. Noncooperative games and nontransitive preferences. Mathematical Social Sciences 12, 1–7. John, R., 2000. Quasimonotone individual demand. Optimization 47, 201–209. Kreweras, G., 1961. Sur une Possibilit´e de Rationaliser les Intransitivit´es. La D´ecision, Colloques Internacionales du CNRS, Paris, pp. 27–32. LLinares, J.-V., 1998. Unified treatment of the problem of existence of maximal elements in binary relations: a characterization. Journal of Mathematical Economics 29, 285–302. LLinares, J.-V., S´anchez, M.C., 1999. Non-binary choice functions on non-compact sets. Economics Letters 63, 29–32. Mehta, G., 1984. Maximal elements for non-transitive binary relations. Economics Letters 14, 163–165. Mehta, G., 1987. Weakly lower demicontinuous preference maps. Economics Letters 23, 15–18. Mehta, G., 1989. Maximal elements in Banach spaces. Indian Journal of Pure and Applied Mathematics 20, 690–697. Mehta, G., 1990. Maximal elements of condensing preference maps. Applied Mathematics Letters 3, 69–71. Nehring, K., 1996. Maximal elements of non-binary choice functions on compact sets. Economics Letters 50, 337–340. Plott, C., 1973. Path independence, rationality and social choice. Econometrica 41, 1075–1091. Rodr´ıguez-Palmero, C., Garc´ıa-Lapresta, J.L., 2002. Maximal elements for irreflexive binary relations on compact sets. Mathematical Social Sciences 43, 55–60. S´anchez, M.C., LLinares, J.-V., Subiza, B., 2003. A KKM-result and an application for binary and non-binary choice functions. Economic Theory 21, 185–193. Shafer, W.J., Sonnenschein, H., 1975. Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics 2, 345–348. Sloss, J.L., 1971. Stable points of directional preference relations, Technical Report No. 71-7, Operations Research House, Stanford University. Sonnenschein, H., 1971. Demand theory without transitive preferences with applications to the theory of competitive equilibrium. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H. (Eds.), Preferences, Utility and Demand. Harcourt Brace Jovanovich, New York. Tarafdar, E., 1977. On nonlinear variational inequalities. Proceedings of the American Mathematical Society 67, 95–98. Tian, G., 1993. Necessary and sufficient conditions for maximization of a class of preference relations. Review of Economic Studies 60, 949–958. Walker, M., 1977. On the existence of maximal elements. Journal of Economic Theory 16, 470–474. Yannelis, N.C., Prabhakar, N.D., 1983. Existence of maximal elements and equilibria in linear topological spaces. Journal of Mathematical Economics 12, 233–245.