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LEANDRO GORNOALESSANDRO RIVELLO Introduction
The standard model of choice in economics is the maximization of a complete andtransitive preference relation over a fixed set of alternatives. While completeness ofpreferences is usually regarded as a strong assumption, weakening it requires careto ensure that the resulting model still has enough structure to yield interestingresults. This paper takes a step in this direction by studying the class of “connectedpreferences”, that is, preferences that may fail to be complete but have connectedmaximal domains of comparability. We offer four new results. Theorem 1 identifies a basic necessary condition for acontinuous preference to be connected in the sense above, while Theorem 2 providessufficient conditions. Building on the latter, Theorem 3 characterizes the maximaldomains of comparability. Finally, Theorem 4 presents conditions that ensure thatmaximal domains are arc-connected.Methodologically, our contribution provides an incomplete preference perspectiveon a theoretical literature relating basic assumptions on preferences and the spaceof alternatives over which these preferences are defined. For example, Schmeidler(1971) shows that every nontrivial preference on a connected topological space whichsatisfies seemingly innocuous continuity conditions must be complete. In a recent ar-ticle, Khan and Uyank (2019) revisit Schmeidler’s theorem and link it to the resultsin Eilenberg (1941), Sonnenschein (1965), and Sen (1969), providing a thorough anal-ysis of the logical relations between the form of continuity assumed by Schmeidler,completeness, transitivity, and the connectedness of the space.In particular, Theorem 4 in Khan and Uyank (2019) implies a converse to Schmei-dler’s theorem: if every strongly nontrivial Schmeidler preference is complete, theunderlying space must be connected. This paper provides a different kind of con-verse: every compact space that admits at least one complete and gapless Schmeidlerpreference with connected indifference classes must be connected.2.
Preliminaries
Let X be a (nonempty) set of alternatives equipped with some topology. A prefer-ence is a reflexive and transitive binary relation on X . For the rest of this paper, weconsider a fixed preference % . Date : First draft: January 2020. This version: August 2020. Gorno (2018) examines the maximal domains of comparability of a general preorder. % is complete on a set A ⊆ X if A × A ⊆ % ∪ - . The set A is a domain if % iscomplete on A . If A is a domain such that there exists no larger domain containingit, then A is a maximal domain . % is continuous if { y ∈ X | y % x } and { y ∈ X | x % y } are closed sets for every x ∈ X . % has connected indifference classes if { y ∈ X | y ∼ x } is connected for every x ∈ X .The set A ⊆ X contains every indifferent alternative if x ∈ A , y ∈ X , and x ∼ y implies y ∈ A . A has no exterior bound if x % A % y implies x, y ∈ A .3. Connected preferences
The main concept of this paper is embedded in the following definition:
Definition 1. % is connected if every maximal domain is connected.We will restrict attention to preferences that are not only connected, but alsocontinuous. As a result, maximal domains will be necessarily closed (see Theorem 1in Gorno (2018)).3.1. A necessary condition.
A natural first step towards a characterization of con-nected preferences is to obtain a simple necessary condition.
Definition 2. % is gapless if, for every x, y ∈ X , x ≻ y implies that there exists z ∈ X such that x ≻ z ≻ y .The notion of gapless preferences is not really new; its content coincides with aspecific definition of order-denseness for sets. We now prove our first result:
Theorem 1. If % is continuous and connected, then % is gapless.Proof. Suppose, seeking a contradiction, that % is not gapless. Then, there existalternatives x, y ∈ X such that x ≻ y and no z ∈ X satisfies x ≻ z ≻ y . By Lemma1 in Gorno (2018), there exists a maximal domain D such that { x, y } ⊆ D . Define A := { z ∈ D | z % x } and B := { z ∈ D | y % z } . Clearly, A and B are nonempty, A ∩ B = ∅ , and A ∪ B = D . Moreover, since % is continuous, A and B are closed. Itfollows that D is not connected, a contradiction. (cid:3) It is easy to see that not every continuous and gapless preference is connected:
Example 1.
Let X = [ − ,
1] and % = { ( x, y ) ∈ X | x = y ∨ x = y = 1 } . Then,the preference % is continuous and gapless, but not connected (the maximal domain {− , } is not a connected set). X is said to be % -dense if for every x, y ∈ X satisfying x ≻ y there exists z ∈ X such that x ≻ z ≻ y (see Ok (2007), p. 92). Evidently, X is % -dense if and only if % is gapless. We shouldperhaps note that there are multiple distinct definitions of order-denseness in the literature and thatthe terminology has not necessarily been consistent. ONNECTED INCOMPLETE PREFERENCES 3
A sufficiency theorem.
We already know that every continuous and connectedpreference must be gapless. In this section, we provide a set of assumptions whichconstitute a sufficient condition for a preference to be connected.
Theorem 2. If X is compact and % is a continuous and gapless preference withconnected indifference classes, then % is connected.Proof. Assume first that % is antisymmetric. Suppose, seeking a contradiction, thatthere is a maximal domain D that is not connected. Then, there exist disjointnonempty closed sets A and B such that A ∪ B = D . Since % is continuous and X is compact, D is compact by Proposition 1 in Gorno (2018). It follows that A and B are also compact. Let x A and x B be the best elements in A and B , respectively.Since D is a domain and A and B are disjoint, either x A ≻ x B or x B ≻ x A . Considerthe first case (the other is symmetric). Define the set C := { x ∈ A | x % x B } . C isnonempty (as x A ∈ C ) and compact. Let x C be the worst element in C . It is easy tocheck that x C ≻ x B . Since % is gapless, there exists z ∈ X such that x C ≻ z ≻ x B .It is easy to verify that z A and z B . Hence, z D . Moreover, D ∪ { z } is adomain, contradicting the assumption that D is a maximal domain.The argument above assumes that % is antisymmetric. If this is not the case, wecan still apply it to the partial order % ∗ induced by % in the quotient space X/ ∼ . Toverify this claim, note that continuity and compactness are automatically inherited by % ∗ and X/ ∼ , respectively. Moreover, since all % -indifference classes are connectedand D must contain every indifferent alternative, D is connected if and only if D/ ∼ isconnected. It follows that there is no loss of generality in assuming the antisymmetryof % from the outset. (cid:3) The following example identifies an important class of connected preferences:
Example 2.
Let X be the set of Borel probability measures (lotteries) on a compactmetric space of prizes Z , equipped with the topology of weak convergence. FollowingDubra, Maccheroni, and Ok (2004), we say that the preference % is an expected multi-utility preference if there exists a set U of continuous functions Z → R such that x % y if and only if Z Z udx ≥ Z Z udy holds for all u ∈ U . It is easy to verify that all the assumptions of Theorem 2 hold.Thus, % is connected.4. Characterization of maximal domains
Building on Theorem 2, we can offer a useful characterization of the maximaldomains:
Theorem 3.
Assume X is compact and % is continuous, gapless, and has connectedindifference classes. Then, a set A ⊆ X is a maximal domain if and only if it isa connected domain that contains every indifferent alternative and has no exteriorbound. ONNECTED INCOMPLETE PREFERENCES 4
Proof.
We start establishing sufficiency through the following lemma:
Lemma 1.
Every connected domain that contains every indifferent alternative andhas no exterior bound is a maximal domain.Proof.
Suppose, seeking a contradiction, that D is a domain that contains everyindifferent alternative, has no exterior bound, but it is not a maximal domain, thenby Lemma 1 in Gorno (2018) exists D ′ , a maximal domain, such that D ⊂ D ′ . Take x ∈ D ′ \ D . Since D has no exterior bounds there are y, z ∈ D such that y ≻ x ≻ z .Define D := { w ∈ D | w % x } and D := { w ∈ D | x % w } . D and D are nonemptysince y ∈ D and z ∈ D . Also, D ∪ D = D because x ∈ D ′ and D ′ is a domainthat contains D . Moreover, D ∩ D = ∅ . If this intersection was not empty, therewould be w ∈ D such that x ∼ w , which would contradict that D contains everyindifferent alternative. Finally, D and D are closed. To see this, note that, since % is continuous, D is closed, but also { y ∈ X | y % x } and { y ∈ X | x % y } are bothclosed. Thus, D and D are the intersection of closed sets. It follows that { D , D } is a nontrivial partition of D by closed sets. We conclude that D is not connected,which is a contradiction. (cid:3) Now we turn to necessity. It is easy to show that every maximal domain containsevery indifferent alternative and has no exterior bound. Moreover, since % satisfiesthe assumptions of Theorem 2, every maximal domain is connected. (cid:3) We finish this section, discussing the two additional assumptions employed in The-orem 3.4.1. X is compact. Compactness of X cannot be dispensed with, as the followingexample shows. Example 3.
Let X = {− } ∪ [0 ,
1) and % = { ( x, y ) ∈ X | x = − ∨ x ≥ y ≥ } .Then, X is bounded, locally compact and σ -compact, but fails to be compact. More-over, % is complete, continuous, and gapless. However, the only maximal domain is X itself and is not connected.4.2. Connected indifferent classes.
On the one hand, the assumption that indif-ferent classes are connected is not strictly necessary for the conclusion of Theorem 3.That is, there are examples failing this condition in which the equivalence in thetheorem holds:
Example 4.
Let X = [ − ,
1] and % = { ( x, y ) ∈ X | x ≥ y } .On the other hand, it is a tight condition: there are examples that violate it, satisfythe remaining conditions, and for which the equivalence in the theorem fails to hold: Example 5.
Let X = {− } ∪ [0 ,
1] and % = { ( x, y ) ∈ X | x ≥ y } .There is a well-known axiom introduced by Dekel (1986) that ensures that indif-ference classes are connected. Assuming that X is convex, we say that % satisfies betweenness if x % y implies x % αx + (1 − α ) y % y for all x, y ∈ X and α ∈ [0 , ONNECTED INCOMPLETE PREFERENCES 5
Prominent examples of preferences satisfying betweenness include preferences satis-fying the independence axiom (such as expected utility or the expected multi-utilitypreferences studied in Dubra, Maccheroni, and Ok (2004)) and also preferences ex-hibiting disappointment aversion as in Gul (1991). The following lemma shows thatbetweenness implies connected indifference classes.
Lemma 2. If X is convex and % satisfies betweenness, then % has connected indif-ference classes.Proof. Take any x, y ∈ X such that x ∼ y and α ∈ [0 , z := αx + (1 − α ) y .Since x % y and y % x , by betweenness, we have x % z % y and y % z % x and, so z ∼ y . It follows that each indifference class is convex, thus connected. (cid:3) We should note that, if X is convex and % is a continuous preference that satisfiesbetweenness, then % does not only possess connected indifferent classes, but is alsonecessarily gapless. This fact makes the application of Theorem 2 and Theorem 3 topreferences satisfying betweenness quite direct.5. Arc-connected preferences
In some cases, it can be useful to strengthen the notion of connectedness to arc-connectedness:
Definition 3. % is arc-connected if every maximal domain is arc-connected.Every arc-connected preference is connected, but the converse does not generallyhold. To see this it suffices to take X to be any space that is connected but notarc-connected and consider % = X × X , that is, universal indifference.In the particular case of antisymmetric preferences ( i.e. , partial orders) on a metriz-able space, we can strengthen the conclusion of Theorem 2: Theorem 4. If X is a compact metrizable space and % is a continuous, gapless, andantisymmetric preference, then % is arc-connected.Proof. Let D be a maximal domain. Since % is continuous and X is compact andmetrizable, Theorem 1 in Gorno (2018) implies that D is compact and metrizable,hence second countable. Because % is complete and continuous on D , there exists acontinuous utility representation u : D → R .Since % is antisymmetric, its indifference classes are singletons, hence connected.By Theorem 2, D is connected. It follows that u ( D ) is connected and compact, thusa compact interval. Without loss of generality, we can assume that u ( D ) = [0 , % is antisymmetric, u is a continuous bijection. Since X is compact and [0 , u is actually an homeomorphism between D and [0 , D is arc-connected. as desired. (cid:3) A well-known example is the closed topologist’s sine curve, which is also compact.
ONNECTED INCOMPLETE PREFERENCES 6 Applications
First-order stochastic dominance.
Suppose X is the set of cumulative dis-tribution functions (CDFs) over a compact interval [0 , z ] (endowed with the topologyof weak convergence of the associated probability measures). Let ≥ denote the first-order stochastic dominance relation on X , that is, F ≥ G if and only if F ( z ) ≤ G ( z )for all z ∈ [0 , z ]. Proposition 1. ≥ is arc-connected. Moreover, a subset of X is a maximal domainof ≥ if and only if it is the image of a ≥ -increasing arc joining the degenerate CDFsassociated with and z .Proof. X is a compact metrizable space (it is metrized by the L´evy metric) and ≥ is continuous, gapless, and antisymmetric. Thus, by Theorem 4, ≥ is arc-connected.Since every arc-connected set is connected, Theorem 3 implies the desired equivalence. (cid:3) An analogous result holds for second-order stochastic dominance.6.2.
Schmeidler preferences.
Schmeidler (1971) shows that, in a connected space,every nontrivial preference satisfying seemingly innocuous continuity conditions mustbe complete. In this section, we explore the implications of his assumptions in spacesthat are not connected.We start by formulating the class of preferences which are the subject of Schmei-dler’s theorem:
Definition 4.
A preference % is a Schmeidler preference if it is continuous and thesets { y ∈ X | x ≻ y } and { y ∈ X | y ≻ x } are open for all x ∈ X .The following definition captures a property that generalizes the conclusion ofSchmeidler’s theorem in terms of maximal domains: Definition 5.
A preference is decomposable if every maximal domain is either aconnected component or an indifference class.Note that, when X is connected, every nontrivial decomposable preference is com-plete. More generally, any two distinct maximal domains of a decomposable prefer-ence must necessarily be disjoint. As a result, if a decomposable preference is locallynonsatiated, then no maximal domain can be trivial or, equivalently, every maximaldomain must be a connected component.We can now state the main result of this section: Proposition 2.
Let X be compact and let % be a Schmeidler preference with con-nected indifference classes. Then, % is decomposable if and only if % is gapless.Proof. To prove necessity, assume that % is gapless. Since % is a Schmeidler prefer-ence, Proposition 10 in Gorno (2018) implies that every nontrivial connected compo-nent is contained in a maximal domain. Moreover, because % is a gapless preferenceon a compact space, every maximal domain is connected by Theorem 3. It follows ONNECTED INCOMPLETE PREFERENCES 7 that every nontrivial maximal domain is a connected component. Finally, since trivialmaximal domains must be indifference classes, % is decomposable.For sufficiency, note that, since % is decomposable and has connected indifferenceclasses, every maximal domain is connected. Thus, Theorem 1 implies that % isgapless. (cid:3) Note that every continuous and complete preference is a Schmeidler preference. Inthat particular case, we have the following
Corollary 1.
Let X be compact and let % be a continuous and complete preferencewith connected indifference classes. Then, % is gapless if and only if X is connected. Schmeidler (1971) shows that if X is connected, then every nontrivial Schmeidlerpreference must be complete. Khan and Uyank (2019) prove the converse and obtainfollowing characterization: X is connected if and only if every nontrivial Schmeidlerpreference is complete. The corollary above implies a different characterization forcompact spaces: provided X is compact, X is connected if and only if there existsat least one complete, gapless, and continuous preference with connected indifferenceclasses. Acknowledgments
This study was financed in part by the Coordenao de Aperfeioamento de Pessoalde N´ıvel Superior - Brasil (CAPES) - Finance Code 001.
References
E. Dekel. An axiomatic characterization of preferences under uncertainty: Weakeningthe independence axiom.
Journal of Economic Theory , 40(2):304 – 318, 1986.J. Dubra, F. Maccheroni, and E. A. Ok. Expected utility theory without the com-pleteness axiom.
Journal of Economic Theory , 115(1):118–133, 2004.S. Eilenberg. Ordered topological spaces.
American Journal of Mathematics , 63(1):39–45, 1941.L. Gorno. The structure of incomplete preferences.
Economic Theory , 66(1):159–185,2018.F. Gul. A theory of disappointment aversion.
Econometrica , 59(3):667–686, 1991.M. Khan and M. Uyank. Topological connectedness and behavioral assumptions onpreferences: a two-way relationship.
Economic Theory , pages 1–50, 2019.E. A. Ok.
Real analysis with economic applications . 2007.D. Schmeidler. A condition for the completeness of partial preference relations.
Econo-metrica , 39(2):403–404, 1971.A. Sen. Quasi-transitivity, rational choice and collective decisions.
The Review ofEconomic Studies , 36(3):381–393, 1969.H. Sonnenschein. The relationship between transitive preference and the structure ofthe choice space.