Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot
aa r X i v : . [ ec on . T H ] J u l Binary Relations in Mathematical Economics: On the Continuity,Additivity and Monotonicity Postulates inEilenberg, Villegas and DeGroot ∗ M. Ali Khan † and Metin Uyanık ‡ July 3, 2020
Abstract:
This chapter examines how positivity and order play out in two important questionsin mathematical economics, and in so doing, subjects the postulates of continuity, additivity and monotonicity to closer scrutiny. Two sets of results are offered: the first departs from Eilenberg’s(1941) necessary and sufficient conditions on the topology under which an anti-symmetric, com-plete, transitive and continuous binary relation exists on a topologically connected space; andthe second, from DeGroot’s (1970) result concerning an additivity postulate that ensures a com-plete binary relation on a σ -algebra to be transitive. These results are framed in the registers oforder, topology, algebra and measure-theory; and also beyond mathematics in economics: theexploitation of Villegas’ notion of monotonic continuity by Arrow-Chichilnisky in the context ofSavage’s theorem in decision theory, and the extension of Diamond’s impossibility result in socialchoice theory by Basu-Mitra. As such, this chapter has a synthetic and expository motivation,and can be read as a plea for inter-disciplinary conversations, connections and collaboration.(164 words) Classification Numbers: 91B55, 37E05.
Journal of Economic Literature
Classification Numbers: C00, D00, D01
Key Words: continuity, additivity, monotonicity, ordered space, weakly ordered space
Running Title:
Continuity, Additivity and Monotonicity Postulates ∗ This work was initiated during Khan’s visit to the Department of Economics, University of Queensland, July27 to August 13, 2018. In addition to the hospitality of the Department, Khan also thanks Rabee Tourky foremphasizing the importance of Eilenberg (1941) during a most pleasant visit at the trimester program “StochasticDynamics in Economics and Finance” held by Hausdorff Research Institute for Mathematics (HIM) in August2013. The authors are still trying to track down Lerner (1907). The authors also thank Youcef Askoura,Ying Chen, Aniruddha Ghosh and Eddie Schlee for conversation and collaboration. This paper draws its basicconception and composition to an invited plenary talk titled “The Role of Positivity in Mathematical Economics:Monotonicity and Free-Disposal in Walrasian Equilibrium Theory”, and delivered by Khan at
Positivity X heldin Pretoria, July 8-12, 2019. He thanks Jan Harm and his team of Organizers for the invitation, and for theirhelp in the logistics. He also thanks Jacek Banaciak, Bernard Cornet, Jacobus Grobler, Malcolm, King, SonjaMouton, Asghar Ranjbari, Eric Schliesser and Nicoll`o Urbinati for stimulating conversation and encouragementafter his talk. † Department of Economics, Johns Hopkins University, Baltimore, MD 21218. ‡ School of Economics, University of Queensland, Brisbane, QLD 4072. ontents
Contents 11 Introduction 22 Mathematical and Conceptual Preliminaries 73 On the Existence of a Continuous Binary Relation 8 t has often happened that a theory designed originally as a tool for the study of a physicalproblem came subsequently to have purely mathematical interest. When that happens thetheory is generalized way beyond the point needed for applications, the generalizations makecontact with other theories (frequently in completely unexpected directions), and the subjectbecomes established as a new part of pure mathematics. Physics is not the only externalsource of mathematical theories; other disciplines (such as economics and biology) can playa similar role. Halmos (1956)
It is also possible that algebra, as a separate discipline within mathematics may not survive.The 20th century was a period of unification, with algebra invading other areas of math,and they counter-invading it. If I am engaged in studying a family of functions on multi-dimensional manifolds, those families having a group structure, am I working in analysis(the functions), topology (the manifolds) or algebra (the groups)? Derbyshire (2006)
In this chapter revolving around the ideas of positivity and order in mathematical economics,one can do worse than begin with Garett Birkhoff’s review of Eilenberg (1941): it is well-worthquoting in full.
An “ordered topological space” is, in effect, a simply ordered set whose topology is ob-tainable by a weakening of its intrinsic topology. The author proves that a topologicalconnected space X can be ordered if and only if the subset of its square X obtained bydeleting the diagonal of points (x, x) is not connected; the same condition also characterizesthose connected locally connected separable topological spaces which are homeomorphicwith subsets of the linear continuum. In this, his paper on “ordered topological spaces,” Eilenberg (1941) is justly celebrated forposing two questions of seminal importance for economic theory. First, can a continuous binaryrelation on a set be represented by a continuous function on the same set? Second, what are theconditions on the set under which a complete and continuous relation is necessarily transitive?Both questions, the second perhaps more than the other, investigate how technical topologicalconditions, assumed for tractability, necessarily translate into behavioral consequences. However,Eilenberg limited himself to the study of anti-symmetric relations, and thereby to studyingagency in a context wherein distinct elements in the choice set are necessarily preferred one toanother, a kind of extreme decisiveness. It remained for Debreu (1954, 1960) to place the first (Halmos, 1956, p.419) The part of pure mathematics so created does not (and need not) pretend to solve thephysical problem from which it arises; it must stand and fall on its own merits. (Derbyshire, 2006, p.319) and for Sonnenschein (1965, 1967) the second, in a setting where the symmetric partof the given binary relation is not an equality, which is to say, the set of indifferent elements ofthe relation are not singletons. They and their followers have by now given rise to a rich andmature body of work.Eilenberg also asked, and answered, two other questions that seem to have had less tractionin economic theory, at least in the way that they were initially posed. He asked for conditionson the topology under which there exist “nice” relations (in the sense of being anti-symmetric,transitive, complete and continuous) on a given set, and furthermore, turning the matter onits head, how such relations disallow sets that are “rich” in the meaning endowed to the termthrough the topological and/or algebraic sructures on the set over which they are defined. Weshall think of these as Eilenberg’s third and fourth questions. Both questions are again naturalones. The third is in some sense analogous to the question concerning conditions on a topologyunder which non-constant continuous functions exist. If the topology is too ”sparse” then everycontinuous function is necessarily constant, and every reflexive, transitive and continuous relationis necessarily trivial in sense that no element is preferred to another. In the context of his fourthquestion, Eilenberg showed that the existence of a “nice” relation defined on a connected, locallyconnected and separable space necessarily renders the space to be a linear continuum. Theseresults then are a testimony to the mutual imbrication of assumptions on a relation and thespace on which the relation is defined, a two-way relationship that in recent work, Khan-Uyanık(2019) see and study as the Eilenberg-Sonnenschein (ES) research program.In terms of the third and fourth questions concerning “nice” relations, to be sure, topol-ogists have understood this mutual imbrication very well. Thus, for there to be a rich supplyof continuous linear functions, the topology on the common domain of the functions must, ofnecessity, satisfy some properties, and cannot be too sparse. Alternatively, the only continuousfunctions on a set endowed with an indiscrete topology are the constant functions; and digging alittle deeper, there is a plethora of (say) non-locally convex spaces with no continuous functionat all other than the zero function. The question of the existence of a supporting hyperplane isexplicitly studied by Klee (1963) in the context of an algebraic structure, and in the context oftopological vector spaces, Kalton, Peck, and Roberts (1984, p. vii) write: The role of the Hahn-Banach theorem may be said to be that of a universal simplifierwhereby infinite-dimensional arguments can be reduced to the scalar case by the use of the In his reproduction of Debreu’s theorem [Proposition 1] on the sufficiency of connectedness of a choice setin a finite-dimensional Euclidean space, Koopmans (1972) for example, observes that “Debreu credits a paper ofEilenberg (1941) as containing the mathematical essence of [his] Proposition 1.” Eilenberg’s third question is entirely analogous to the existence of a one-to-one continuous function sincehe requires the anti-symmetry property. In Section 3 we introduce a result for binary relations that is exactlyanalogous to the existence of a non-constant continuous function. biquitous linear functional. Thus the problem with non-locally convex spaces is that of“getting off the ground.” The point is that there is some hiddenness in the mutual interaction of a function and set thatneeds to be flushed out. In terms of the origins, Urysohn (1925) studies the problem of deter-mining the most general class of topological spaces in which non-constant real-valued continuousfunctions exist. Hewitt (1946) provides an example of a countable, connected Urysohn space in which every continuous function is constant. Following Hewitt’s work, there are results onthe class of topological spaces on which every continuous function is constant; see Chittenden(1929) for the original paper, and the following, for example, for more modern work. Wallace(1962), Herrlich (1965), Lord (1995) and Iliadis and Tzannes (1986).The question is of substantive consequence for functional analysis but also beyond it foreconomic theory and mathematical economics. In terms of this register, the problem gets trans-lated into the question of the sustaining of technologically efficient program as in value max-imization programs. Majumdar (1974) furnishes a complete characterization and refers to hisresult as follows: One should recall that a major motivation behind research in this area comes from the needto determine whether efficient allocations can be attained by the use of a price mechanism ina decentralized system achieving economy of information and utilizing individual incentives.The implications of any result on complete characterization should be seriously consideredin this context, and as far as [the result] goes, they seem to be somewhat negative incharacter. The equivalence established indicates that, in general, one would need a familyof price systems to specify an efficient program. Indeed, the applicability of the criterionis rather restricted since one has to know too many prices. It is then to this literature that we connect Eilenberg’s third and fourth questions. Wesee him asking this question: rather than the existence of a a function from a “nice” classof functions, does there exist a binary relation from a “nice” class of binary relations? Andthe first contribution of this chapter is that it generalizes Eilenberg’s answer to this question byrelaxing connectedness and anti-symmetry assumptions: in a nutshell, we do to Eilenberg in thiscontext what Sonnenschein did in another and Debreu did in yet another. This is to say that wegeneralize Eilenberg’s result by dropping the anti-symmetry assumption, and then extend the A topological space in which any two distinct points can be separated by closed neighborhoods. Majumdar continues, “But being a complete characterization,[the result] provides a new angle from whichthe difficulties faced by the earlier approaches can be viewed and tends to suggest that simpler criteria involvingfewer price systems, in particular, the use of just one price system as is typically the case, may be incapable ofisolating the set of efficient programs unless restrictive assumptions on technology are introduced.” For furtherwork on the problem. see Stephan (1986), and the references to his chapters in Faber (1986). k -connected spaces, and then to a setting that substitutes k -connectedness withlocal-connectedness. Finally, we note that our first two results can be analogously generalized togeneral preferences, and connect our results to the literature on the non-existence of non-constantcontinuous functions.But we also make another connection that has been missed in the economic literature. Thisis the application of our results on the existence of a “nice” preference relation to Diamond’s(1965) impossibility theorem: what this economic literature sees an an impossibility result, wesee simply as a question of the existence of a nice binary relation where the adjective nice hasbeen given a meaning and an elaboration in terms of intergenerational equity. In introducing hisown paper, Zame (2007, p. 188) documents the trajectory of this substantial economic literature. Diamond (1965) shows that a complete transitive preference relation that displays inter-generational equity and respects the Pareto ordering cannot be continuous in the topologyinduced by the supremum norm. Basu and Mitra (2003) show that such a preferencerelation – whether continuous or not – cannot be represented by a (real-valued) util-ity function. On the other hand, Svensson (1980) proves that such preference relationsdo exist. Fleurbaey and Michel (2003), Hara, Suzumura, and Xu (2006), Basu and Mitra(2007), and Bossert, Sprumont, and Suzumura (2007) provide further results, both positiveand negative.
Moreover, as already illustrated in Toranzo-Herv´es-Beleso(1995), there are continuous, completeand transitive relations on non-separable spaces which are not representable. This connectionthat we make is important in that it sights Eilenberg (1941) as one of the originating papersof this substantial economic literature. This concludes our discussion of the first substantivesection, Section 3, of the paper.Section 4 of the paper returns to Eilenberg’s second question: to find a suitable topologicalcondition which ensures the transitivity of a complete, reflexive and continuous binary relations.Khan and Uyanık (2019, 2020) frame this question in settings that remain squarely remainwithin the purely topological registe, but go considerably beyond Eilenberg. In his considerationof the relationship however, Sonnenschein (1965) move to a a setting that also embrace linearstructures. In a complementary result, Galaabaatar-Khan-Uyanık (2018), henceforth GKU, showthe existence of a mixture-continuous, anti-symmetric, transitive and complete relation definedon a mixture space renders the setting to be isomorphic to either a greater-than-or-equal-torelation, or its inverse, defined on the interval! These results are of substantive consequence for It is worth noting that both Eilenberg (1941) and Sonnenschein (1965) limit their attention to one way of thetwo-way relationship, in that they examine the implication of assumptions on the choice set on the properties ofthe relation defined on that set; the backward direction exploring the implication of the properties of a class ofpreferences on the choice set over which they are defined is the signature of the Khan-Uyanık work. desiderata must of necessity be sparse and impoverished: alinear continuum in the case of Eilenberg and an interval in the case of GKU. Note that theseresults, while bearing obvious implication for results on the representation of binary relations,belong to an entirely different register. They concern the dove-tailing and mutual imbricationof a set of assumptions on one object for those on a different but not unrelated object.The second contribution of this chapter is to make a further move from the register ofmixture-spaces to a more abstract algebraic one. Our point of departure now is Villegas (1964,1967): this work studied countably additive qualitative probability representations and showedthat given a finite additive qualitative probability, monotone continuity is necessary and suffi-cient for a countably additive representation It remained for DeGroot (1970) to flush out theabstract algebraic register grounding this result. The contribution of Section 4 below is (i) tointroduce an sharper additivity postulate, one supplemented by monotone continuity postulates,on abstract algebraic structures that are analogous to Villegas’ additivity postulate, (ii) to ob-tain an equivalence result between additivity and transitivity without referring to completenessor continuity of the binary relation, (iii) to show that under additivity, different variants of themonotonicity concepts are equivalent, (iv) to relate our result to Villegas, DeGroot, de Finetti,Arrow and Chichinisky. In particular we highlight the hiddenness and redundancy of the tran-sitivity assumption as these desiderata are emphasized in Khan and Uyanık (2019). As such,it contributes to the depth and maturity of the ES program. This concludes our discussion ofSection 4 of the paper.We began this this introduction by reading Kakutani (1941) as a text revolving aroundfour questions concerning binary relations: leaving Section for notational and conceptual prelim-inaries, we shall focus on the third and the fourth in Section 3, and on the second in Section 4.The reader may well wonder our silence about the (first) question that mathematical economistsand economic theorists know him by. As mentioned, this pertains to the representation of abinary relation by a function, of a continuous relation by a continuous function, of a monotonicrelation by a monotonic function, and of a concave relation by a quasi-concave function. Westate the question in all these elaborated way simply to allude the river of work that has accu-mulated in mathematical economics and mathematical psychology on this question. But to keepto Eilinberg’s parameters except that of his singleton indifference sets, we quote from (Beardon,2020, p. 3). See Krantz, Luce, Suppes, and Tversky (1971, Section 5.4.2) for further discussion. n this expository essay we consider how much of the theory can be developed from a purelytopological perspective. We focus on those ideas which provide a link between utility theoryand topology, and we leave the economic interpretations to others. Briefly, we give priorityto results that seem to be topologically important, so we pay more attention to the quotientspace of indifference classes than is usual, and more attention to the order topology thanother topologies. We refer the reader to the above article and to Bosi, Campi´on, Candeal, and Indurain (2020),the book of which it is a chapter, and move on. Let X be a set. A subset < of X × X denote a binary relation on X. We denote an element( x, y ) ∈ < as x < y. The asymmetric part ≻ of < is defined as x ≻ y if x < y and y < x , and its symmetric part ∼ is defined as x ∼ y if x < y and y < x. The inverse of < is defined as x y if y < x . Its asymmetric part ≺ is defined analogously and its symmetric part is ∼ . We providethe descriptive adjectives pertaining to a relation in a tabular form for the reader’s conveniencein the table below. reflexive x < x ∀ x ∈ X complete x < y or y < x ∀ x, y ∈ X non-trivial ∃ x, y ∈ X such that x ≻ y transitive x < y < z ⇒ x < z ∀ x, y, z ∈ X semi-transitive x ≻ y ∼ z ⇒ x ≻ z and x ∼ y ≻ z ⇒ x ≻ z ∀ x, y, z ∈ X anti-symmetric x < y and y < x ⇒ x = y ∀ x, y ∈ X Table 1: Properties of Binary RelationsLet < be a binary relation on a set X . For any x ∈ X , let A < ( x ) = { y ∈ X | y < x } denotethe upper section of < at x and A ( x ) = { y ∈ X | y x } its lower section at x . Now assume X is endowed with a topology. We say < is continuous if its upper and lower sections are closedat all x ∈ X and the upper and lower sections of its asymmetric part ≻ are open at all x ∈ X .A topological space X is said to be connected if it is not the union of two non-empty, disjointopen sets. The space X is disconnected if it is not connected. A subset of X is connected if it isconnected as a subspace. We say X is locally connected if for all x ∈ X , every open neighborhoodof x contains a connected and open set containing x . A component of a topological space is amaximal connected set in the space; that is, a connected subset which is not properly contained The reader interested in this (first) Eilenberg question can also see Wold (1943–44), Nachbin (1965),Bridges and Mehta (1995) Herden (1989b); we shall return to Wold (1943–44) in Section 5.
7n any connected subset. For any natural number k, a topological space is k -connected if it hasat most k components. The concept of k -connectedness provides a quantitative measure ofthe degree of disconnectedness of a topological space. It is easy to see that 1-connectedness isequivalent to connectedness and that any k -connected space is l -connected for all l ≥ k. Eilenberg (1941, Theorem I) provides a necessary and sufficient condition for the existence of ananti-symmetric, complete, transitive and continuous binary relation on a connected topologicalspace X. In this section we start with introducing a generalization of Eilenberg’s result to k -connected spaces, and then show that when the space is locally connected, then cardinalityof the components of the space does not matter. We continue by presenting a result whicheliminates the anti-symmetry requirement in Eilenberg’s theorem. We end the section with abrief discussion of our results.Before presenting our result, we need the following notation: for any set X, define P ( X ) = { ( x, y ) ∈ X × X : x = y } . Eilenberg (1941) calls a topological space ordered if there exists an anti-symmetric, complete,transitive and continuous binary relation on it. He then presents
Theorem (Eilenberg).
A connected topological space X which contains at least two elementscan be ordered if and only if P ( X ) is disconnected. The following theorem generalizes Eilenberg’s theorem to k -connected spaces. Theorem 1.
For any natural number k, a k -connected topological space X can be ordered if andonly if P ( C ) is disconnected for each non-singleton component C of X .Proof of Theorem 1. Let { C i } ℓi =1 be the collection of the components of X where ℓ ≤ k . Firstnote that Theorem (Eilenberg) implies that for each component C i of X which contains at leasttwo elements, there exists an anti-symmetric, complete, transitive and continuous binary relation < i on C i if and only if P ( C i ) is disconnected.In order to prove the forward direction, assume < is an anti-symmetric, complete, transitiveand continuous binary relation on X . Then, for each C i , the restriction of < on C i , defined as See Khan and Uyanık (2019) for a detailed discussion on k -connectedness. i = < ∩ ( C i × C i ) , is an anti-symmetric, complete, transitive and continuous binary relation on C i . Then, Theorem (Eilenberg) implies that P ( C i ) is disconnected for all non-singleton C i .In order to prove the backward direction, assume P ( C i ) is disconnected for each non-singleton C i . It follows from Theorem (Eilenberg) that there exists an anti-symmetric, complete,transitive and continuous binary relation < i on every non-singleton C i . If C i is a singleton, thendefine < i = C i × C i . Then define a binary relation < on X as follows: S ℓi =1 < i ⊆ < , and forall i > j , C i × C j ⊆ < . Then, < is anti-symmetric, complete, and transitive. Since each C i isclosed in X, therefore < i has closed sections in both C i and X , and hence < has closed sections.Therefore, < is continuous. Theorem 2.
A locally connected topological space X can be ordered if and only if P ( C ) isdisconnected for each non-singleton component C of X .Proof of Theorem 2. Let { C i } i ∈ I be the collection of the components of X . The proof of theforward direction is identical to the proof of the forward direction of Theorem 1. In order to provethe backward direction, assume P ( C i ) is disconnected for each non-singleton C i . It follows fromTheorem (Eilenberg) that there exists an anti-symmetric, complete, transitive and continuousbinary relation < i on each non-singleton C i . If C i is a singleton, then define < i = C i × C i .The well-ordering theorem (Munkres, 2000, Theorem, p.65) implies that there exists an anti-symmetric, complete and transitive binary relation ˆ < on I. Then define a binary relation < on X as follows: S i ∈ I < i ⊆ < , and for all i ˆ ≻ j , C i × C j ⊆ < . Then, < is anti-symmetric, complete,and transitive. Since X is locally connected, each C i is both open and closed. Then, each < i has closed sections in both C i and X . Note that for all C i and all x ∈ C i ,A < ( x ) = A < i ( x ) ∪ [ j ˆ ≻ i C j = A < i ( x ) ∪ \ i ˆ < j C cj . Then, it follows from C i is open for all i ∈ I that < has closed upper sections. An analogousargument implies that < has closed lower sections. Therefore, < is continuous. This subsection provides a necessary and sufficient condition for the existence of a non-trivial,complete, transitive and continuous binary relation on a connected topological space. Thisresult is analogue to Theorem (Eilenberg), except that the binary relation is not necessarilyanti-symmetric.
Definition 1.
A topological space is weakly ordered if there exists a non-trivial, complete, tran-sitive and continuous binary relation on it. heorem 3. A connected topological space X which contains at least two elements can be weaklyordered if and only if P ( X | ∼ ) is disconnected for some equivalence relation ∼ on X .Proof of Theorem 3. Let X be a topological space with at least two elements. Assume thereexists a non-trivial, complete, transitive and continuous binary relation < on X . Let ∼ denotethe symmetric part of < . Since X is connected, the quotient space X | ∼ is connected. It iseasy to show that the induced binary relation ˆ < on X | ∼ , defined as ([ x ] , [ y ]) ∈ ˆ < if and only if( x ′ , y ′ ) ∈ < for all x ′ ∈ [ x ] and all y ′ ∈ [ y ], is non-trivial, anti-symmetric, complete, transitiveand continuous. Then, it follows from Theorem (Eilenberg) that P ( X | ∼ ) is disconnected.In order to prove the backward direction, assume there exists an equivalence relation ˜ ∼ on X such that P ( X | ˜ ∼ ) is disconnected. Then, X | ˜ ∼ contains at least two elements. Since X | ˜ ∼ is connected, it follows from Theorem (Eilenberg) that there exists an anti-symmetric,complete, transitive and continuous binary relation ˆ < on X | ˜ ∼ . Define a binary relation < on X as ( x, y ) ∈ < if and only if ([ x ] , [ y ]) ∈ ˆ < . Then the symmetric part ∼ of < is identical to ˆ ∼ . Itfollows from A ˆ < ([ x ]) and A ˆ ([ x ]) are closed in X | ˜ ∼ and the definition of the quotient topologythat the sections A < ( x ) = [ [ y ] ˆ < ([ x ] [ y ] and A ( x ) = [ [ y ] ˆ [ x ] [ y ]of < are closed in X, hence < is continuous. The non-triviality, completeness and transitivityof < directly follow from its construction .Note that in an ordered space, the indifference relation ∼ in Theorem 3 is assumed tobe the equality relation. Hence, as expected, the requirement for the existence of an order isstronger than the requirement for that of a weak order. The following example illustrates aweakly ordered topological space which cannot be ordered. Example.
Let X = [0 ,
2] and the following define a basis for the topology on X : [0 , x ) for all x ∈ (1 , , ( x,
2] for all x ∈ [1 , , and ( x, y ) for all x, y ∈ [1 , . Note that the smallest closed setcontaining any point in [0 ,
1] is [0 , X is connected. Since the topology is notHausdorff, Eilenberg (1941, 1.4) implies that there does not exist an anti-symmetric, completeand continuous binary relation on X . However, the following is a non-trivial, complete, transitiveand continuous binary relation on X : ( x, y ) ∈ for all x, y ∈ [0 , x, y ) ∈ for all x, y ∈ X with x < y. Finally, the methods of proofs presented in Theorems 1 and 2 can be used to providegeneralizations of this result to disconnected spaces.10 .3 Discussion of the Results
We can apply our results to the literature on the non-existence of a non-constant function ontopological spaces as follows. First, note that every non-constant continuous function induces anon-trivial, complete, transitive and continuous binary relation. Therefore, by Hewitt’s (1946)result we know that there does not exist a non-trivial, complete, transitive and continuousrelation. Moreover, note that the space in Hewitt’s paper is countable, hence separable, andconnected. Therefore, every non-trivial, complete, transitive and continuous relation has a non-constant, continuous real-valued representation. Therefore, Theorem 3 provides an equivalencecondition for the existence of a non-constant function in Hewitt’s setting. Hence, Theorem 3may provide a new perspective on Hewitt’s theorem and on the subsequent work in this line ofwork. Moreover, Miller (1970), Golomb (1959), Kirch (1969) and Jameson (1974) provide count-able spaces that are connected and satisfy the Hausdorff separation axiom. Since continuousfunctions take connected sets to connected sets, therefore there cannot exist a non-constantcontinuous function on these spaces. We next show that there does not exist a continuous,non-trivial, semi-transitive relation with a transitive symmetric part on these spaces. First, byappealing to the current authors’ earlier work, any such relation is complete and transitive.Since the space is countable, it is separable. Therefore, it follows from Debreu (1954, TheoremI) that there exists a continuous real-valued function representing the binary relation. Since therelation is non-trivial, therefore the function is non-constant. This furnishes us a contradiction.The literature has focused on the existence, or non-existence, of a non-constant continuousfunction. For binary relations, different continuity postulates has been introduced and used inmathematical economics. The existence of a non-trivial binary relation satisfying different con-tinuity assumptions may be of interest; see Uyanık and Khan (2019a) for an extended discussionon the continuity postulate. In this section we provide two results on the implications of the additivity postulate. Wefirst show that a strong form of additivity postulate is equivalent to the transitivity postulate.Then we define three monotone continuity postulates on partially ordered sets, inspired by thepioneering work of Villegas on qualitative probability, and then show that under the additivity See Khan and Uyanık (2019, Theorem 2). We refer the reader to Khan and Uyanık (2020) andUyanık and Khan (2019b) for generalizations to bi-preference structures and general parametrized topologicalspaces. There is a literature on different continuity postulates for functions; see Ciesielski and Miller’s (2016) recentsurvey on this.
A binary relation < on an Abelian group ( X, +) is called additive if for all x, y, z ∈ X , x < y implies x + z < y + z . Moreover, we say < is strongly additive if for all x , x , y , y ∈ X , x i < y i for i = 1 , implies x + x < y + y . We first present a result on the relationship between additivity and strong additivity.
Proposition 1.
Every reflexive and strongly additive relation on an Abelian group is additive.Proof of Proposition 1.
Assume < is strongly additive relation on an Abelian group ( X, +).Pick x, y, z ∈ X such that x < y . Then z < z , by reflexivity, and strong additivity of < imply x + z < y + z . Hence < is additive.Along with this observation, the next result shows that when a reflexive binary relation istransitive, the two additivity postulates are equivalent. Moreover, it shows that the transitivityof the relation is implied by strong additivity. Theorem 4.
An additive binary relation < on an Abelian group ( X, +) is transitive if and onlyif it is strongly additive.Proof of Theorem 4. Let < be an additive binary relation on an Abelian group ( X, +). Assume < is transitive. Pick x , x , y , y ∈ X such that x i < y i for i = 1 ,
2. Then it follows fromadditivity that x + x < y + x and x + y < y + y . Then commutativity of + andtransitivity of < implies that x + x < y + y .Now assume < is strongly additive. Pick x, y, z ∈ X such that x < y < z . Then strongadditivity implies x + y < y + z . Then additivity of < imply x + y + ( − y ) < y + z + ( − y ).Therefore x < z .The following is a direct corollary of Proposition 1 and Theorem 4. Corollary 1.
Every reflexive and strongly additive relation on an Abelian group is transitive.
Let ( X, ≥ ) be a partially ordered set. We say X is order-complete if every non-empty subset of X with an upper bound has a least upper bound. Note that a poset X is order-complete if andonly if every non-empty subset of X with a lower bound has a greatest lower bound; see Fremlin(3.14B, vol3I). 12 efinition 3. Let ( X, ≥ ) be an order-complete poset and < a binary relation on X . We definethe following monotone continuity axioms for < . [C1 ′ ] For all y ∈ X and all bounded below sequence { x i } i ∈ N in X , x i ≥ x i +1 and x i < y for all i imply inf { x i } i ∈ N < y . [C2 ′ ] For all y ∈ X and all bounded above sequence { x i } i ∈ N in X , x i +1 ≥ x i and y < x i for all i imply y < sup { x i } i ∈ N . [C3 ′ ] For all y ∈ X and all bounded above sequence { x i } i ∈ N , x i +1 ≥ x i and y ≺ sup { x i } i ∈ N implythere exists an integer N > such that, for i ≥ N , we have y ≺ x i . Theorem 5.
For any complete and strongly additive binary relation on an Abelian group whichis also an order-complete poset, the continuity axioms C1 ′ , C2 ′ and C3 ′ are equivalent.Proof of Theorem 5. Let ( X, + , ≥ ) be an order-complete poset on an Abelian group and < acomplete and strongly additive binary relation on X. It follows from Corollary 1 that < istransitive.First, we show that C1 ′ is equivalent to C2 ′ . Note that additivity implies x < y if andonly if − y < − x . Assume C1 ′ . Pick a bounded above sequence { x i } i ∈ N and y in X such that x i +1 ≥ x i and y < x i for all i . Then, − x i ≥ − x i +1 and − x i < − y for all i . It follows formC1 ′ that inf {− x i } i ∈ N < − y . We now show that additivity implies inf {− x i } i ∈ N = − sup { x i } i ∈ N .Define x =inf {− x i } i ∈ N and ¯ x = − sup { x i } i ∈ N Assume towards a contradiction that x > ¯ x . Bydefinition, x ≤ − x i for all i . Then additivity implies − x ≥ x i for all i . Then − x is an upperbound of { x i } i ∈ N , hence − x ≥ ¯ x . This contradicts the assumption that x > ¯ x . An analogousargument yields a contradiction for x < ¯ x . Therefore, x = ¯ x . Then − sup { x i } i ∈ N < − y , henceby additivity, y < sup { x i } i ∈ N . Therefore, C2 ′ holds. The proof of the converse relationship isanalogous.We next show that C2 ′ is equivalent to C3 ′ . Assume C2 ′ . Assume towards a contradictionthat there exists a bounded above { x i } i ∈ N and y in X such that x i +1 ≥ x i and y ≺ sup { x i } i ∈ N ,but for all N >
0, there exists j ≥ N such that y < x i . Then there exists a subsequence { x ik } k ∈ N such that for all k , x i k +1 ≥ x i k and y < x i k . Then C2 ′ implies y < sup { x i k } k ∈ N . It is easy to seethat sup { x i k } k ∈ N = sup { x i } i ∈ N . This contradicts the assumption that y ≺ sup { x i } i ∈ N . HenceC3 ′ holds. The converse relationship immediately follows from the definitions. Villegas (1964) introduced the following additivity concept for binary relations on a σ -algebra. Definition 4.
A preference relation < on a σ -algebra X on a set X is Villegas-additive if for all A , A , B , B ∈ X with A ∩ A = B ∩ B = ∅ , A i < B i for i = 1 , implies A ∪ A < B ∪ B . f, in addition, A ≻ B or A ≻ B , then A ∪ A ≻ B ∪ B . First note that the union operation is similar to the additivity operation but it does notsatisfy all properties the addition in an Abelian group satisfies. Moreover, the usual additivityassumption is neither stronger nor weaker than Villegas-additivity: the latter imposes restrictionon a smaller class of elements whereas additivity does not impose a restriction on the strictrelation. DeGroot (1970, Theorem 1, p. 71) followed Villegas and proved a result analogous toTheorem 4 where the space is a σ -algebra with the usual inclusion relation. Theorem (DeGroot).
Every complete and Villegas-additive binary relation on a σ -algebra istransitive. We next apply our results to de Finetti’s expected utility representation theorem. Let X = R n which is endowed with the usual topological, algebraic and order structures. A realvalued function u is called monotone if for all x, y ∈ R n such that x > y (i.e. x i ≥ y i for all i and x = y ), u ( x ) > u ( y. ) A preference relation < on R n is monotone if for all x, y ∈ R n , if x > y, then x ≻ y. The following theorem is due to de Finetti (1937, 1974). Theorem (de Finetti).
Let < be a binary relation on R n . The following are equivalent. (a)
The binary relation < is complete, transitive, additive and continuous. (b) There exist positive ( p i ) ni =1 , summing to one, such that u ( x ) = P i p i x i represents < . The equivalence theorem of de Finetti can be restated as
Corollary 2.
Let < be a binary relation on R n . The following are equivalent. (a)
The binary relation < is complete, strongly additive and continuous. (b) There exist positive ( p i ) ni =1 , summing to one, such that u ( x ) = P i p i x i represents < . Therefore, we can drop the transitivity assumption in de Finetti’s theorem by replacing additivitywith strong additivity, which are equivalent in the presence of the transitivity postulate. We canalso drop the completeness assumption; see Uyanık and Khan (2019a) for a detailed expositionon the hiddenness and redundancy in mathematical economics. We next move to monotone continuity. The second subsection above is an attempt tounderstand this postulate introduced in Villegas, DeGroot, Arrow and Chichilnisky in order tostudy qualitative/subjective probability. As we illustrate above, monotone continuity neither Note that Fishburn (1986, p. 336) calls Villegas-additivity the additivity axiom . See Wakker (1989, Theorem A.2.1, p.161) for the statement and further details. See also Krantz, Luce, Suppes, and Tversky (1971, Section 5.4.2) for an interesting discussion on hiddennessand redundancy. Moreover, it may be of interest to generalize this result to groupoids or semigroups; see Fishburn(1972, Chapter 11). Villegas (1964) and DeGroot (1970) provide the following monotonecontinuity postulates for binary relations defined on σ -algebras. Definition 5.
Let X be a σ -algebra on a set and < a binary relation on X . We define thefollowing monotone continuity axioms for < . [C1] For all sets { A i } i ∈ N , B in X , A ⊇ A ⊇ · · · and A i < B for all i imply T i A i < B . [C2] For all sets { A i } i ∈ N , B in X , ‘ A ⊆ A ⊆ · · · and B < A i for all i imply B < S i A i . [C3] For all sets { A i } i ∈ N , B in X , A ⊆ A ⊆ · · · and B ≺ S i A i imply there exists an integer N > such that, for i ≥ N , we have B ≺ A i . The following is a result analogous to Theorem 5 above for the special case of σ -algebras. Theorem 6.
For any complete and Villegas-additive binary relation on a σ -algebra, the mono-tone continuity postulates C1 , C2 and C3 are equivalent. The equivalence between C2 and C3 is due to Villegas (1964, Theorem) and between C1 and C2is due to DeGroot (1970, Theorem 5).Villegas (1964, 1967) studied countably additive qualitative probability representation andshowed that given a finitely additive qualitative probability, monotone continuity is necessary andsufficient for countably additive representation; see Krantz, Luce, Suppes, and Tversky (1971,Section 5.4.2) for further discussion. In particular, the following result is quoted. Theorem (Villegas).
A finitely additive probability representation of a structure of qualitativeprobability, on a σ -algebra, is countably additive if and only if the structure is monotonicallycontinuous. Finally, the following monotone continuity postulate is due to Arrow (1971).
Definition 6.
Given a and b , where a ≻ b , a consquence c and a vanishing sequence { E i } ,suppose sequence of actions satisfy the conditions that ( a i , s ) yield the same consequences as ( a, s ) for all s ∈ E ci , and the consequence c for all s ∈ E i , while ( b i , s ) yield the same consequences See Khan and Uyanık (2019) and Ghosh, Khan, and Uyanık (2020) for a discussion on the relationship amongdifferent continuity postulates. Moreover, DeGroot’s assumption SP “There exists a random variable which hasa uniform distribution on the interval [0 , For definitions, we refer the reader to Krantz, Luce, Suppes, and Tversky (1971). s ( b, s ) for all s ∈ E ci , and the consequence c for all s ∈ E i . Then, for all i sufficiently large, a i ≻ b and a ≻ b i . Chichilnisky (2010) interpreted Arrow’s definition as follows and showed that it is equivalent tothe continuity postulate C1.
Definition 7.
Let X be a σ -algebra on a set and < a binary relation on X . We call < satisfiesMonotone Continuity Axiom 4 (C4) if for all { A i } i ∈ N , F, G in X , A ⊇ A ⊇ · · · , T ∞ i =1 A i = ∅ and F ≻ G imply there exists N > such that altering arbitrarily the events F and G on theset A i , where i > N , does not alter the ranking of the events, namely F ′ ≻ G ′ , where F ′ and G ′ are the altered events. We began this essay with Halmos’ take on how applied mathematics transits to pure mathe-matics; and Derbyshire’s take on how an important sub-field, with increasing importance, getsincorporated into the larger field of which it is a part, and thereby changes the identity of thelarger field and loses its own. In this concluding section to this chapter on binary relationsin mathematical economics, we read, against the grain, these two texts and their claims on theincorporation of positivity and order-theoretic methods in Walrasian general equilibrium theory.In classical Walrasian general equilibrium theory, as brought to fruition in Koopmans(1957), Debreu (1959), Nikaido (1968), Arrow and Hahn (1971) and McKenzie (2002), the agentsin the economy are categorized as consumers and producers, with the former parametrized bypreferences (a binary relation) defined on a (consumption) set and endowments being elementsof such a set; and the latter drawing their signature simply by having an access to a productionset. The vernacular of order and positivity is relevant in so far as it is relevant to its constituentconceptions of a consumer and a producer. The idea of monotonicity enters the theory of In his participation in the composition of this section, Khan should like to acknowledge his indebtedness toconversations with Malcolm King, and Niccol`o Urbinati, and to JJ Grobler’s inspiring talk titled
101 years ofvector lattice theory: A general form of integral: PJ Daniell (1918) at the Conference. He should also like toacknowledge the stimulus received from Schliesser’s readings of Foucault (2008 (French edition (2004). Our choice of these four texts should perhaps be justified. For the texts of Debreu and McKenzie, wecan appeal to D¨uppe and Weintraub (2014) who argue that the 1954 papers of Arrow-Debreu and McKenziewrought fundamental changes in economic theory, a claim contested in Khan (2020) who urged the inclusionof Uzawa, Nikaido and Gale also as fellow-pioneers of what we are calling here “Walrasian general equilibriumtheory.” The naming of the Arrow-Debreu model or the Arrow-Debreu-Mckenzie model facilitated a homogeneousmonolithic view and added to the cofusion and to an unfortnate haste in canonization; see Footnote 22 below.A confounding factor in this is that many of the pioneers of Walrasian general equilibrium theory were alsopioneers of linear and non-linear programming; Uzawa breing one of the leaders. For this line of work, seeDorfman, Samuelson, and Solow (1958), and the recent application of Uzawa’s consequential extension of theKuhn-Tucker-Karush theorem in Khan and Schlee (2019). free disposal, an assumption delineated by Debreu (1954)in the context of a production set, say in a ordered normed space whose positive cone has anon-empty interior.
The assumption of free disposal for the technology means that if an input-output com-bination is possible, so is one where one where some outputs are smaller or some inputslarger; it is implied that a surplus can be freely disposed of. With this assumption, if theproduction set is non-empty, it has an interior point.
It is the existence of an interior that proves crucial for the sustainablity of technologicallyefficent production plans and Pareto optimal allocations through individual value and profitmaximization. As far as the theory of the consumer is concerned, the ideas of order and positivity enterthrough the assumption of monotonicity of preferences which gets translated into “more is alwayspreferred to less.” To be sure, it factors into the Eilenberg questions regarding binary relationswith which we began the introduction. Thus (Arrow and Hahn, 1971, p. 106) write:
Wold seems to have been the first to see the need of specifying assumptions under whichthe representation of the continuous utility functions exists. Wold assumed that the [con-sumption set] is the entire non-negative orthant [of finite-dimensonal Euclidean space] andthat preference is strictly monotone in each commodity. A very considerable generalization,based on a mathematical paper by Eilenberg (1941), was achieved with the deeper methodsof Debreu (1954); he assumed only the continuity of preferences and the connectedness ofthe [consumption set] (a property weaker than convexity).
This is an important passage: its irony lies in the fact that it comes from two of the more distin-guished and senior Walrasian theorists at the time who could not refrain from drawing arbitraryand needless distinctions between mathematicians and economists, and between mathematicaland economic papers, and thereby in sighting Eilenberg (1941), and devaluing it at the sametime. The point is that Eilenberg and Wold were independent pioneers of what later assumedthe identity of an important subfield of “choice and decision theory.” But returning to trajectories being implicitly charted by Halmos and to Derbyshire, thepoint is that the monotonicity assumption for consumers in the Walrasian conception comesrather late in its development: it is not there, for example, in Debreu (1959), or in McKenzie We invite the reader to compare Debreu’s definition with corresponding definitions of the concept in the fivetexts to which Footnote 19 refers. The idea of “free disposal” is intimately tied to the non-negativity of prices;see Hara (2005), and compare Debreu (1959) and McKenzie (2002) on this issue. We can recommend Fishburn (1972) Luce (2000) Gilboa (2004, 2009). Moscati (2016) and their referencesfor this subject, which branches off also into mathematical psychology. The more important question,however is where the subject is in terms of these, their trajectories. This is a question that meritsan investigation of its own, and is outside the scope of this technical essay: it suffices to maketwo observations. With respect to Halmos, classical Walrasian general equilibrium theory hasneglected, by its very definitional conception, interdependencies between the parametrizations ofwhat it sees as the relevant agents in the economy; and classical game theory, again by virtue ofits definitional conception, has neglected the market in its formalizations. The applied problemsof our time cry out for a formalization of these interdependencies in what perhaps ought to bea synthetic view of both subjects. Thus even after 70 years, mathematical economics (includinggame theory) has very much retained its dependence on both economics and mathematics. Thisis to say that it has remained pure and applied. As to Derbyshire on algebra, in terms of thealgebraic approach to these subjects, it has yet to be incorporated into both Walrasian generalequilibrium theory and in non-cooperative game theory. In the authors’ judgement, this cannotbut be a fruitful task.There is another, perhaps narrower, way to view the substance of these results. Thequestion of the “right” commodity space for general Walrasian general equilibrium, or the “rightsetting” of the individual action sets in game theory, has not been explicitly posed. There hasbeen little need to do so. Given the substantive questions at issue, the economic or game-theoretic formulations assume a strong-enough structure on the payoff functions and the choicesets by setting them either in a finite-dimensional Euclidean space, or in the context of gametheory, a finite number of actions, to allow the question to be investigated and determinativelyanswered. When this rather arbitrary limitaion is removed, the question becomes of consequence,and notions of order and positivity began to take on colours that one may not have previouslyimagined. This introduction has framed the results to follow as stemming from Eilenberg’s (1941) This also suggests how much a reader of Walrasian general equilibrium theory loses by ascribing to it amonolithic conception. Each of these pioneers had their own ways of looking at their subject. In this connectionone may also refer the interested reader to McKenzie’s conception of production in his (McKenzie, 2002, Section2.8, pages 77-82) on an “Economy of Activities.” It is also perhaps worth noting that Debreu’s resistance to themonotonicty assumption on consumers may be due to his having relaxed the montonicity assumption in Wold(1943–44). To the authors knowledge, his first recourse to the assumption is in connection with the Debreu-Scarftheorem in 1963, and to be sure the auumption irrevocably enters into the field with Aumann and his Israeli schoolof Walrasian theory; see Debreu (1983) for the relevant papers and references. As emphasized in Khan (2020), theerasure of the production sector can also be ascribed to this school, and it becomes folded into the iideologicaldivide between the “two Cambridges,” those of the UK and the US. Foucault’s (2008 (French edition (2004)emphasis on “governmentality” in the formulation of perfect competition and its normative properties is clearlyrelevant here. This investigation remains an ongoing project of Khan and Urbinati, and in his talk in Pretoria, Khan madesome room to expand at some length to report on Nikaido’s contributions to this question in keeping with thisproject. oeuvre, written almost as a fragmen-tary passing thought, proves to be of such decisive and sustainable consequence in what may havebeen perceived at the time of its writing to be an unrelated discipline. Eilenberg’s paper, alongwith Kakutani’s (1941) fixed point theorem, coincidentally published in the same year, may wellbe two canonical examples. 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