A Maximum Theorem for Incomplete Preferences
aa r X i v : . [ ec on . T H ] S e p A MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES
LEANDRO GORNO ALESSANDRO RIVELLO
Abstract.
We extend Berge’s Maximum Theorem to allow for incomplete prefer-ences ( i.e. , reflexive and transitive binary relations which fail to be complete). Wefirst provide a simple version of the Maximum Theorem for convex feasible sets anda fixed preference. Then, we show that if, in addition to the traditional continuityassumptions, a new continuity property for the domains of comparability holds, thelimits of maximal elements along a sequence of decision problems are maximal ele-ments in the limit problem. While this new continuity property for the domains ofcomparability is sufficient, it is not generally necessary. However, we provide condi-tions under which it is necessary and sufficient for maximality and minimality to bepreserved by limits.
Keywords: incomplete preferences, maximum theorem, maximal elements, continuity.
JEL classifications:
C61, C62. Introduction
An important issue arising in the study of models involving optimization is whetheroptimal choices depend continuously on parameters affecting the objective functionand the constraints. The main tool to address this question is the Maximum Theoremby Berge (1963), which can be stated as follows:
Maximum Theorem.
Let X and Θ be topological spaces; let u : X × Θ → R be a con-tinuous function; let K : Θ ⇒ X be a continuous and compact-valued correspondence.Then, the correspondence M : Θ ⇒ X defined by setting M ( θ ) := arg max x ∈ K ( θ ) u ( x, θ ) for each θ ∈ Θ is upper hemicontinuous and compact-valued. This result can be easily modified to dispense with utility functions and deal directlywith complete (continuous) preferences and also with incomplete preferences with openasymmetric parts (see Walker (1979)). However, to the best of our knowledge, noneof the existing generalizations of the Maximum Theorem applies to some of the moststandard types of incomplete preferences such as Pareto orderings based on continuousutility functions, preferences over lotteries admitting an expected multi-utility repre-sentation as in Dubra, Maccheroni, and Ok (2004), or ordinal preferences possessing acontinuous multi-utility representation studied by Evren and Ok (2011).The following example shows that obtaining a maximum theorem that covers thesetypes of preferences requires additional conditions:
Date : First draft: January 2020. Current version: September 2020.
Example 1.
Consider a consumer choosing bundles of two goods: apples (A) andbananas (B). Her preferences are fixed, but incomplete. They can be represented withtwo utilities: u ( q A , q B ) = q A + q B and u ( q A , q B ) = q A + 2 q B , in the sense that abundle ( q A , q B ) is considered at least as good as another bundle ( q ′ A , q ′ B ) if and onlyif u ( q A , q B ) ≥ u ( q ′ A , q ′ B ) and u ( q A , q B ) ≥ u ( q ′ A , q ′ B ). The price of apples, p A , isnormalized to 1 and there is sequence of prices for bananas p B,n = 1 + 1 /n . Note that,if the consumer’s wealth is w = 1, bundle (1 ,
0) is optimal for every n ∈ N : there isno feasible bundle that the consumer strictly prefers to (1 , n → + ∞ , the bundle (1 ,
0) is no longer optimal because the consumer strictly prefersthe bundle (0 , p A , p B, + ∞ ) = (1 , Preliminaries
Let (
X, d ) be a metric space. In this paper, a preference , generically denoted by % , is a reflexive and transitive binary relation on X . As usual, ∼ and ≻ denote thesymmetric and asymmetric parts of % , respectively. For every x ∈ A ⊆ X , the set { y ∈ A | y ∼ x } is the indifference class of % in A .We say that % is complete on a set A ⊆ X if either x % y or y % x holds for all x, y ∈ A . The set A is a % - domain if % is complete on A . If A ⊆ B ⊆ X and A is a % -domain such that there exists no % -domain contained in B and strictly containing A , then A is a maximal % -domain relative to B . Denote by D ( % , B ) the collection ofall maximal % -domains relative to B .A point x ∈ A is % -maximal in A if, for every y ∈ A , y % x implies x % y . The setof all % -maximal elements in A is denoted by M ax ( % , A ). Analogously, a point x ∈ A is % -minimal in A if, for every y ∈ A , x % y implies y % x . The set of all % -minimalelements in A is denoted by M in ( % , A ).A preference % is continuous if it is a closed subset of X × X . Let P be the collectionof continuous preferences on X . A set U ⊆ R X is a multi-utility representation for % whenever, for every x, y ∈ X , x % y holds if and only if u ( x ) ≥ u ( y ) for all u ∈ U . MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 3
Let K X be the collection of nonempty compact subsets of X . Consider both K X and P equipped with the Hausdorff metric topology derived from X and X × X ,respectively. Finally, for any sequence {A n } n ∈ N of nonempty subsets of K X , denoteby LS n → + ∞ A n the collection of accumulation points of all sequences { A n } n ∈ N , where A n ∈ A n for each n ∈ N .3. A Simple Maximum Theorem
In this section, we introduce a new continuity condition that is compatible withinteresting classes of incomplete preferences and allows us to prove a simple MaximumTheorem. Throughout this section, we will assume that X is convex. Definition 1.
A preference % is midpoint continuous if, for every x, y ∈ X satisfying y ≻ x , there exists α ∈ [0 ,
1) and open sets
V, W ⊆ X such that ( αx + (1 − α ) y, x ) ∈ V × W and z ′ ≻ x ′ for all ( z ′ , x ′ ) ∈ V × W .Every complete and continuous preference satisfies midpoint continuity. The fol-lowing result establishes that a significant class of incomplete preferences also does:
Proposition 1. If % admits a finite multi-utility representation U ⊆ R X with each u ∈ U continuous and strictly quasiconcave, then % satisfies midpoint continuity.Proof. Take x, y ∈ X such that x ≻ y . By definition of multi-utility representation,we have u ( x ) ≥ u ( y ) for all u ∈ U . Define z := (1 / x + (1 / y . Since each u ∈ U isstrictly quasi-concave, u ( z ) > u ( y ) holds for all u ∈ U . For each u ∈ U , there are opensets V u , W u such that ( z, y ) ∈ V u × W u and u ( z ′ ) > u ( y ′ ) for all ( z ′ , y ′ ) ∈ V u × W u .Define V := T u ∈U V u and W := T u ∈U W u . Clearly ( z, y ) ∈ V × W , so V and W are nonempty. Moreover, since U is finite, V and W are open. By construction, wehave u ( z ′ ) > u ( y ′ ) for all z ′ ∈ V , y ′ ∈ W , and u ∈ U . Since U is a multi-utilityrepresentation, ( z ′ , y ′ ) ∈ V × W implies z ′ ≻ y ′ , showing that % satisfies midpointcontinuity. (cid:3) Using the concept of midpoint continuity, we can establish the first major result ofthis paper, a simple Maximum Theorem:
Theorem 1.
Let % ∈ P and let { ( K n , x n ) } n ∈ N be a sequence in K X × X such that(1) { ( K n , x n ) } n ∈ N converges to ( K, x ) ∈ K X × X as n → + ∞ .(2) x n ∈ M ax ( % , K n ) for every n ∈ N .(3) K n is convex for every n ∈ N .(4) % satisfies midpoint continuity.Then, x ∈ M ax ( % , K ) .Proof. Suppose, seeking a contradiction, that x / ∈ M ax ( % , K ). Then, there exists y ∈ K such that y ≻ x . By midpoint continuity, there must exists α ∈ [0 ,
1) and open sets
V, W ⊆ X satisfying ( αx + (1 − α ) y, x ) ∈ V × W and z ′ ≻ x ′ for all ( z ′ , x ′ ) ∈ V × W .Define z := αx + (1 − α ) y . Note that, being the limit of a sequence of convex sets, K If a preference is complete and continuous, its asymmetric part is open in X × X . As a result, wecan always take α = 0 to satisfy the definition of midpoint continuity. MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 4 must be convex. It follows that z ∈ K . Since x = lim n → + ∞ x n , there must be N ∈ N such that x n ∈ W for all n ≥ N . Since K = lim n → + ∞ K n and z ∈ K , there must be N ∈ N such that K n ∩ V = ∅ for all n ≥ N . Take N := max { N , N } . Then, for all n ≥ N , there exists z n ∈ K n such that z n ≻ x n . This implies x n
6∈ M ax ( % , K n ), acontradiction. We conclude that x ∈ M ax ( % , K ). (cid:3) Note that the preferences in Example 1 violate midpoint continuity. In the followingexample, we apply Proposition 1 and Theorem 1 to establish the upper hemicontinuityof Pareto efficient allocations with respect to resource endowments.
Example 2.
Consider a simple exchange economy with L ∈ N goods and N ∈ N agents. Here X = R L × N + denotes the set of all possible allocations or social outcomes.Each agent has a continuous and strictly convex preference on R L + . By definition, thePareto relation for this economy admits a finite multi-utility representation composedby continuous and strictly quasiconcave utilities. It thus follows from Proposition 1that the Pareto relation on X satisfies midpoint continuity. Now suppose that we havea sequence of endowments { ω n } n ∈ N that converges to ω and a sequence of allocations { x n } n ∈ N that converges to an allocation x . By Theorem 1, if each x n is Pareto efficientwhen the endowment is ω n , then x is Pareto efficient when the endowment is ω .Theorem 1 deals with a fixed preference % . It would be desirable to relax thisrequirement, allowing for a sequence of preferences converging to % . However, thefollowing example shows that Theorem 1 becomes false with this modification. Example 3.
Let X = [0 ,
1] and consider the sequence { % n } n ∈ N of preferences de-fined by setting % n := { ( x, y ) ∈ X × X | x = y } ∪ { ( x, y ) ∈ Y n × Y n | u ( x ) ≥ u ( y ) } foreach n ∈ N , where Y n := [1 / ( n + 1) , − / ( n + 1)] and u : X → R is given, foreach x ∈ X , by u ( x ) := ( x − / . It is easy to check that % := lim n → + ∞ % n isthe complete preference represented by u . Moreover, both % and all the % n satisfymidpoint continuity. However, if we consider K n = { , } , x n = 0 for each n ∈ N , K := lim n → + ∞ K n = { , } and x := lim n → + ∞ x n = 0, we have x n ∈ M ax ( % n , K n ) forall n ∈ N even though x
6∈ M ax ( % , K ).4. A General Maximum Theorem
A significant limitation of Theorem 1 is that it requires a fixed preference and convexfeasible sets. The second major result of this paper replaces these restrictions with acontinuity condition on maximal domains of comparability:
Theorem 2.
Let { ( % n , K n , x n ) } n ∈ N be a sequence in P × K X × X such that(1) { ( % n , K n , x n ) } n ∈ N converges to ( % , K, x ) ∈ P × K X × X as n → + ∞ .(2) x n ∈ M ax ( % n , K n ) for every n ∈ N .(3) LS n → + ∞ D ( % n , K n ) ⊆ D ( % , K ) .Then, x ∈ M ax ( % , K ) . A preference is said to be strictly convex if, for every x, y, z ∈ X , y % x , z % x , y = z , and α ∈ (0 ,
1) imply αy + (1 − α ) z ≻ x . See Mas-Colell, Whinston, and Green (1995), p. 44. MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 5
Proof.
For each n ∈ N , there exists D n ∈ D ( % n , K n ) such that x n ∈ D n . Since K n ∈ K X converges to K ∈ K X and D n is closed in K n , we have D n ∈ K X . ByLemma 1, there exists a convergent subsequence ( D n h ) h ∈ N . As a result, there is no lossof generality in assuming that { D n } n ∈ N itself converges. Define D := lim n → + ∞ D n . ByLemma 3, x n ∈ D n for all n ∈ N and lim n → + ∞ ( D n , x n ) = ( D, x ) together imply that x ∈ D . Moreover, condition (3) implies that D ∈ D ( % , K ).We now claim that x is a % -best in D . To prove this, suppose, seeking a contradic-tion, that there exists y ∈ D such that y ≻ x . Since lim n → + ∞ D n = D , there must exista sequence { y n } n ∈ N such that lim n → + ∞ y n = y and y n ∈ D n for every n ∈ N . Moreover,since lim n → + ∞ % n = % , by the second part of Lemma 3, there must exist N ∈ N suchthat y N ≻ N x N . This contradicts that x N is % N -maximal in K N , as assumed.Since x is % -best in D ∈ D ( % , K ), Theorem 1 of Gorno (2018) implies that x is % -maximal in K . (cid:3) Theorem 2 generalizes the upper-hemicontinuity of the arg max correspondence inBerge’s Maximum Theorem by weakening the completeness implied by the existenceof a utility representation to condition (3). Roughly, this condition says that limits ofmaximal % n -domains should be maximal % -domains, relative to the relevant feasiblesets. In the particular case in which all preferences in the sequence { % n } n ∈ N arecomplete, the limit preference % must also be complete and condition (3) holds trivially.However, condition (3) is also compatible with incomplete preferences. Example 4.
Suppose there is a finite partition D ∗ of X such that D ( % n , X ) = D ∗ for all n ∈ N . Note that this assumption nests the case of complete preferences asthe particular case in which D ∗ = { X } . Convergence of preferences implies that D ( % , X ) = D ∗ as well. Moreover, since all maximal domains relative to X are disjoint,we also have D ( % , K ) = { D ∩ K | D ∈ D ∗ } and D ( % n , K n ) = { D ∩ K n | D ∈ D ∗ } forevery n ∈ N . We conclude that LS n → + ∞ D ( % n , K n ) ⊆ D ( % , K ) and condition (3)holds. 5. A Characterization
Even though condition (3) in Theorem 2 constitutes a general sufficient conditionfor optimality to be preserved by limits, it is not necessary:
Example 5.
Let X = [0 , % = (cid:8) ( x, y ) ∈ [0 , . (cid:12)(cid:12) x = y (cid:9) ∪ [0 . , . Consider the sequence { K n } n ∈ N , where K n := [0 . − . /n,
1] for each n ∈ N . Onthe one hand, D n = { . − . /n } is a maximal % -domain relative to K n , while thesequence { D n } n ∈ N converges to D = { . } , which is not a maximal % -domain relativeto K := lim n → + ∞ K n = [0 . , M ax ( % n , K n ) = M in ( % n , K n ) = K n for all n ∈ N and M ax ( % , K ) = M in ( % , K ) = K , so all convergent sequencescomposed by % -maximal and % -minimal elements in each K n converge to % -maximaland % -minimal elements in K .However, condition (3) is indeed necessary and sufficient for maximal and minimalelements to be preserved by limits in more specific settings. In this section, we obtain MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 6 a characterization by restricting attention to limit preferences that are antisymmetric( i.e. , partial orders) and sets that are “order dense”.Formally, a set A ⊆ X is % - dense if, for every x, y ∈ A , x ≻ y implies that thereexists z ∈ A such that x ≻ z ≻ y . % -dense sets are quite common in applications.For instance, if % is a preference over lotteries that admits an expected multi-utilityrepresentation , then every convex set of lotteries is % -dense. We can now state themain result of this section. Theorem 3.
Denote by
G ⊆ P the collection of continuous partial orders on X . Let { ( % n , K n ) } n ∈ N be a converging sequence in P × K X with limit ( % , K ) ∈ G × K X andsuch that, for every n ∈ N , K n is a % n -dense set and all indifference classes of % n in K n are connected. Then, K is % -dense. Moreover, the following are equivalent:(1) LS n → + ∞ D ( % n , K n ) ⊆ D ( % , K ) (2) LS n → + ∞ M ax ( % n , K n ) ⊆ M ax ( % , K ) and LS n → + ∞ M in ( % n , K n ) ⊆ M in ( % , K ) .Proof. (1) ⇒ (2). The convergence of the maximal elements is a direct implication ofTheorem 2. To see the convergence of the minimal elements let { y n } n ∈ N be a convergentsequence such that lim n → + ∞ y n = y and, for all n ∈ N , y n is a % n -minimal element.If we define % ∗ n := { ( x, y ) ∈ X × X | y % n x } , then D ( % ∗ n , K n ) = D ( % n , K n ) for every n ∈ N . Moreover, every y n is a % ∗ n -maximal. Thus we can apply Theorem 2 to concludethat y is a % ∗ -maximal which is equivalent to it be a % -minimal.(1) ⇐ (2). Take a sequence { D n } n ∈ N such that D n ⊆ K n , D n ∈ D ( % n , K n ), andlim n → + ∞ D n = D . Since lim n → + ∞ % n = % , D is a % -domain. For each n ∈ N , % n isa continuous preference such that % n ∩ ( K n × K n ) has connected indifference classesand K n is % n -dense. Thus, by Lemma 4, D n is connected for every n ∈ N . It followsthat D is also connected by Lemma 5 and % -dense by Lemma 6.We claim that D has no exterior bounds. Suppose, seeking a contradiction, thereis x ∈ K such that x % ˜ y for every ˜ y ∈ D and x / ∈ D . In fact, we must have x ≻ ˜ y because % is a partial order. For each D n take y n ∈ D n such that y n % n z for all z ∈ D n . By Theorem 1 in Gorno (2018) each y n is a maximal element in K n . Notethat { y n } ⊂ K n and { y n } is compact for every n ∈ N , thus by Lemma 1 there is no lossof generality in assuming that { y n } n ∈ N converges to some y ∈ K . Since y n ∈ D n forevery n and { ( D n , y n ) } n ∈ N converges to ( D, y ), we must have y ∈ D . By hypothesis wehave x ≻ y and y is a maximal element in K , a contradiction. An analogous argumentguarantees that there is no x ∈ K such that y % x for every y ∈ D and x / ∈ D . Thismeans that D has no exterior bounds.Furthermore, since % is a partial order, D contains all its indifferent alternatives.Thus, by Lemma 4, we conclude that D ∈ D ( % , K ). (cid:3) Theorem 3 provides assumptions under which condition (3) in Theorem 2 is necessaryand sufficient for all limits of maximal or minimal elements to be maximal or minimal,respectively. An immediate application is to show that, in the case of a fixed preference, Dubra, Maccheroni, and Ok (2004) show that a preference over lotteries has an expected multi-utility representation if and only if it is continuous and satisfies the independence axiom.
MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 7 antisymmetry and midpoint continuity combined with mild additional assumptionsimply condition (3).
Corollary 1.
If, in addition to the conditions of Theorem 1, % is a partial order andthere exists x ∈ T n ∈ N K n such that x % x for every x ∈ S n ∈ N K n , then condition (3)holds. This means that Theorem 2 does generalize Theorem 1 for partial orders in contextssuch as consumer theory (in which consuming zero of every good is always feasible andthere is no worse bundle).A clear limitation of Theorem 3 is that the limit preference % is required to be apartial order. Example 5 above shows that the characterization does not hold withoutthis assumption even if X = [0 , % n -densenessof the K n and connectedness of the relative indifference classes, refer to the spe-cific sequence { ( % n , K n ) } n ∈ N under consideration. However, if we restrict attentionto convex feasible sets and preferences over lotteries that admit an expected multi-utility representation , these assumptions are automatically satisfied for all sequences { ( % n , K n ) } n ∈ N . This is the content of the following corollary: Corollary 2.
Let X be the space of (Borel) probability measures on a separable metricspace equipped with the topology of weak convergence. Suppose further that:(1) For each n ∈ N , K n ∈ K X is convex and % n is a preference that admits anexpected multi-utility representation,(2) % is a partial order that admits an expected multi-utility representation.Then, the equivalence in the conclusion of Theorem 3 holds. Finally, a more subtle aspect of Theorem 3 is that condition (2) requires that every convergent sequence of % -maximal (resp. % -minimal) elements converges to a % -maximal (resp. % -minimal) element. The next example illustrates this point. Example 6.
Let % be the natural vector order on X = [0 , . % is a continuouspartial order and X is % -dense. Now, for each n ∈ N , consider K n := { ( x , x ) ∈ X | x ≤ n (1 − x ) } . Note that K n is nonempty and compact for each n ∈ N . Moreover, lim n → + ∞ K n = K := [0 , and (1 , ∈ M ax ( % , K n ). However, (1 , / ∈ M ax ( % , K ). It follows fromTheorem 3 that there must be at least one convergent sequence { D n } n ∈ N , where, foreach n ∈ N , D n is a maximal % -domain relative to K n , but such that lim n → + ∞ D n is not a maximal % -domain relative to K . In fact, taking D n = { } × [0 ,
1] yields aspecific example of such { D n } n ∈ N . Let C be a separable metric space of consequences and let X be the set of (Borel) probabilitymeasures (lotteries) over C equipped with the topology of weak convergence of probability measures.Similarly to Dubra, Maccheroni, and Ok (2004), we say that a set U of bounded continuous functions C → R constitutes an expected multi-utility representation for % whenever, for every two lotteries x, y ∈ X , x % y is equivalent to R C u ( c ) dx ( c ) ≥ R C u ( c ) dy ( c ) for all u ∈ U . MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 8 Related literature
Considerable work has been devoted to the study of incomplete preferences. . De-spite the fact that continuous weak preferences are one of the two central classes ofpreferences in this literature, to the best of our knowledge, this is the first paper toprovide positive results on the continuity properties of maximal elements for this class.Continuity of optimal choices is an important problem and has been studied exten-sively. The central result is the Maximum Theorem in Berge (1963), which is concernedwith the behavior of value functions and the maximizers that attain them as param-eters change continuously. Our results, in particular Theorem 2, extend this work byallowing for incomplete preferences. Even though its key condition is trivially satisfiedwhen preferences are complete, Theorem 2 is not truly a generalization of the originalMaximum Theorem because we assume that X is a metric space, whereas Berge’s re-sult is proven in a general topological space. Moreover, since assuming that preferencesadmit a utility representation would imply completeness, our results focus exclusivelyon maximal elements and make no statements about value functions.Walker (1979) proves a generalized maximum theorem for a strict relation ≻ θ thatdepends on a parameter θ ∈ Θ and has open graph (as a correspondence Θ ⇒ X × X ).Lemma 9 in Evren (2014) shows that if X is a space of lotteries and ≻ is open, then M ax ( % , K ) is relatively closed in K and the correspondence K ⇒ M ax ( % , K ) is up-per hemicontinuous, even if K is neither convex nor compact. However, neither of thesetwo results bears significance for the class of incomplete continuous weak preferencesconsidered in the present paper. The reason is that, as long as X is connected, everyincomplete continuous weak preference that has an open strict part must be trivial (seeSchmeidler (1971)) and, as a result, satisfy M ax ( % , K ) = K for every K ⊆ X .7. Discussion
The present paper provides three major results that expand the scope of Berge’sMaximum Theorem to allow for incomplete preferences. Theorem 1 is based on asimple continuity condition, but its applicability is somewhat limited since it requiresconvex feasible sets and a fixed preference.Theorem 2 does not have the aforementioned limitations. The result depends cru-cially on its condition (3), a form of upper hemicontinuity of the mapping betweenpreferences-feasible sets pairs and the corresponding collection of maximal domainsof comparability. Since Gorno (2018) shows that every maximal element is the bestelement in some maximal domain and vice-versa, convergence of maximal domains per-mits the application of a Berge-type of argument to ensure the convergence of maximalelements through the convergence of local best elements, where the term “local” heremeans “relative to a maximal domain”. The list is long. A few examples in chonological order are Aumann (1962),Peleg (1970),Ok(2002), Dubra, Maccheroni, and Ok (2004), Eliaz and Ok (2006), Dubra (2011), Evren and Ok(2011), Ok, Ortoleva, and Riella (2012), Evren (2014), Riella (2015), Gorno (2017), and Gorno (2018).
MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 9
Finally, Theorem 3 describes a more specific setting in which condition (3) in Theo-rem 2 is necessary and sufficient for minimality and maximality to be preserved whentaking limits.We believe that these results constitute a step forward towards understanding con-vergence of maximal elements without completeness and open at least three avenuesfor future research. First, the abstract nature Theorem 2 suggests to look for addi-tional sets of assumptions which are sufficient for its condition (3) to hold. Second,the equivalence in Theorem 3 might be true under weaker assumptions. Third, wecurrently do not know whether our results remain true in a general topological space(which is the environment in which Berge’s Maximum Theorem is formulated).
Appendix A. Technical lemmas
The proof of Theorem 2 requires some results about Hausdorff convergence. In thefollowing two lemmas (
M, d ) is any metric space and K M (resp. F M ) is the collectionof all nonempty compact (resp. closed) subsets of M . Lemma 1.
Let { K n } n ∈ N be a convergent sequence in K M with limit K ∈ K M . Then,every sequence { A n } n ∈ N in K M such that A n ⊆ K n for all n ∈ N has a subsequencewhich converges to a nonempty compact subset of K .Proof. Let d H : K M × K M → R + denote the Hausdorff distance and let K K be thecollection of all nonempty compact subsets of K . Note that (cid:0) K M , d H (cid:1) is a metricspace, K K is compact in the (relative) Hausdorff metric topology, and d H ( A n , · ) iscontinuous on K K . For each n ∈ N , let B n ∈ arg min ˜ B ∈K K d H ( A n , ˜ B ). I now claimthat, for each n ∈ N , we have d H ( A n , B n ) ≤ d H ( K n , K ) . To prove this claim, note that, since { y } ∈ K K for all y ∈ K , we have d H ( A n , B n ) ≤ d H ( A n , { y } ) = max x ∈ A n d ( x, y )for all y ∈ K . Defining y ∗ ( x ) ∈ arg min y ∈ K d ( x, y ) for each x ∈ A n , we have d H ( A n , B n ) ≤ max x ∈ A n d ( x, y ∗ ( x )) = max x ∈ A n min y ∈ K d ( x, y ) , ≤ max x ∈ K n min y ∈ K d ( x, y ) ≤ d H ( K n , K )as desired. It follows that lim n → + ∞ d H ( A n , B n ) ≤ lim n → + ∞ d H ( K n , K n ) = 0.Since K K is compact, the sequence { B n } n ∈ N has a convergent subsequence, say { B n h } h ∈ N . Let A := lim h → + ∞ B n h ∈ K K . We claim that { A n h } h ∈ N converges to A .To prove this, note that, since lim n → + ∞ d H ( A n , B n ) = 0 and lim h → + ∞ B n h = A , thetriangle inequality d H ( A n h , B ) ≤ d H ( A n h , B n h ) + d H ( B n h , B ) . implies lim h → + ∞ d H ( A n h , B ) = 0. (cid:3) MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 10
Lemma 2.
Denote by F M the collection of all nonempty closed subsets of M and let { ( F n , x n ) } n ∈ N be a convergent sequence on F M × M with limit ( F, x ) ∈ F M × M andsuch that x n ∈ F n for every n ∈ N . Then, x ∈ F .Proof. Suppose, seeking a contradiction, that x / ∈ F . Since F is closed, there exists ǫ > { y ∈ M | d ( x, y ) ≤ ǫ } ∩ F = ∅ . Hence, inf y ∈ F d ( x, y ) > ǫ/
2. By thetriangule inequality, we have d ( x, y ) ≤ d ( x, x n ) + d ( x n , y )for all y ∈ F and all n ∈ N . Thereforeinf y ∈ F d ( x, y ) ≤ inf y ∈ F { d ( x, x n ) + d ( x n , y ) } = d ( x, x n ) + inf y ∈ F d ( x n , y ) ≤ d ( x, x n ) + d H ( F n , F )for all n ∈ N , where we usedinf y ∈ F d ( x n , y ) ≤ sup x ∈ F n inf y ∈ F d ( x, y ) ≤ d H ( F n , F )Taking limits we conclude that inf y ∈ F d ( x, y ) = 0, a contradiction. (cid:3) Lemma 3.
Consider a convergent sequence { ( % n , K n , x n , y n ) } n ∈ N in P × K X × X × X with limit ( % , K, x, y ) ∈ P × K X × X × X . Then:(1) x n ∈ K n for all n ∈ N implies x ∈ K .(2) x n % n y n for all n ∈ N implies x % y .Proof. The first part follows from Lemma 1 by taking M = X and A n = { x n } foreach n ∈ N , since every subsequence of { x n } n ∈ N converges to x ∈ K . The secondpart follows from Lemma 2 by taking M = X × X and noting that x n % n y n means( x n , y n ) ∈ % n . (cid:3) The next three lemmas are used in the proof of Theorem 3. In what follows, K is anelement of K X and % is a continuous preference on X . A set A ⊆ K has no exteriorbound in K if, for every x, y ∈ K , x % A % y implies x, y ∈ A . Lemma 4.
Assume that K is % -dense and the indifference classes of % in K areconnected. Then, a subset of K is a maximal % -domain relative to K if and only if itis a connected % -domain relative to K which has no exterior bound in K .Proof. Since K is compact and % ∩ ( K × K ) is a continuous preference on K withconnected indifference classes, the result follows from Theorem 4 in Gorno and Rivello(2020). (cid:3) Lemma 5.
Let { K n } n ∈ N be a sequence in K X such that K n is connected for every n and lim n → + ∞ K n = K , then K is connected.Proof. Suppose, seeking a contradiction, that K is not connected. Then, there existdisjoint nonempty sets A and B which are closed in K and satisfy A ∪ B = K . Forany ǫ >
0, define A ǫ := { x ∈ X | d ( x, A ) < ǫ } and ¯ A ǫ as its closure. Define B ǫ and ¯ B ǫ analogously. Define K ǫ := A ǫ ∪ B ǫ and ¯ K ǫ as its clousure.Since K is closed, A and B are also closed in X , which is a normal space. Then,there exists ¯ ǫ > ǫ ∈ (0 , ¯ ǫ ], we have A ǫ ∩ B ǫ = ∅ . Fix ǫ = ¯ ǫ/ MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 11 then ¯ A ǫ ∩ ¯ B ǫ = ∅ . Because lim n → + ∞ K n = K there is N ǫ ∈ N such that n ≥ N ǫ implies K n ⊆ ¯ K ǫ . Now define A n := K n ∩ ¯ A ǫ and B n := K n ∩ ¯ B ǫ . It is easy to see that A n ∩ B n = ∅ , A n ∪ B n = K n , and A n , B n ∈ K X . It follows that K n is not connected, acontradiction. (cid:3) Lemma 6.
Let { ( % n , K n ) } n ∈ N be a converging sequence in P × K X with limit ( % , K ) ∈G × K X . If, for every n ∈ N , K n is % n -dense and all indifference classes of % n in K n are connected, then K is % -dense.Proof. Suppose, seeking a contradiction, that K is not % -dense. Then, there exist x, y ∈ K such that x ≻ y and there is no z ∈ K that satisfies x ≻ z ≻ y . Take { x n } n ∈ N and { y n } n ∈ N such that x n , y n ∈ K n for every n ∈ N , lim n → + ∞ x n = x , andlim n → + ∞ y n = y . Define M n := { z ∈ K n | x n % n z % n y n } . Note that M n ∈ K X and M n ⊆ K n for every n ∈ N , so Lemma 1 implies that { M n } n ∈ N has a convergentsubsequence. Thus, we can assume without loss of generality that { M n } n ∈ N itselfconverges and define M := lim n → + ∞ M n .We claim that M n is connected for each n ∈ N . Suppose, seeking a contradiction,that M n is not connected for some n ∈ N . Then there should exist disjoint nonemptysets A and B which are closed in M n and satisfy A ∪ B = M n . Without loss, assumethat x n ∈ A . Since B is compact and % n is continuous, there exists at least one % n -maximal element in B , call it x B . Define C := { z ∈ A | z % n x B } . Note that C is also compact and nonempty ( x n ∈ C ), so we can take x C , one of its % n -minimalelements. We will now show that x C ≻ n x B . Define I := { z ∈ K n | z ∼ n ¯ x B } . Since I ⊆ M n , both I ∩ B and I ∩ A are closed sets which satisfy ( I ∩ A ) ∪ ( I ∩ B ) = I and( I ∩ A ) ∩ ( I ∩ B ) = ∅ . Since I is assumed to be connected and ¯ x B ∈ I ∩ B it mustbe that I ∩ A = ∅ , proving that x C ≻ n x B . Define D := { z ∈ M n | x C ≻ n z ≻ n x B } . Ifthere is z ∈ D , then z ≻ n x B implies z / ∈ B . Moreover x C ≻ n z ≻ n x B implies that z / ∈ C , so z / ∈ A either. It follows that D must be empty, which is a contradiction with K n being % n -dense. We conclude that M n is connected.On the one hand, since { M n } n ∈ N is a sequence of connected sets in K X , M is con-nected by Lemma 5. On the other hand, we claim that M = { x, y } . It is easy to seethat { x, y } ⊆ M . To prove the other inclusion take any sequence { z n } n ∈ N convergingto z ∈ M and such that z n ∈ M n for every n ∈ N . Then x n % n z n % n y n implies x % z % y . But, since we initially assumed that there is no z ∈ K that satisfies x ≻ z ≻ y , we must necessarily have that either x ∼ z or z ∼ y . Furthermore, % antisymmetric implies that x = z or z = y . Hence, M must be equal to { x, y } .We conclude that M must simultaneously be connected and equal to { x, y } , whichyields the desired contradiction. (cid:3) Acknowledgments
We thank participants at various seminars for their helpful comments. This studywas financed in part by the Coordenao de Aperfeioamento de Pessoal de N´ıvel Superior- Brasil (CAPES) - Finance Code 001.
MAXIMUM THEOREM FOR INCOMPLETE PREFERENCES 12
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