Expected utility theory on mixture spaces without the completeness axiom
aa r X i v : . [ ec on . T H ] F e b Expected utility theory on mixture spaces without thecompleteness axiom ∗ David McCarthy † Kalle Mikkola ‡ Teruji Thomas § February 16, 2021
Abstract
A mixture preorder is a preorder on a mixture space (such as a convex set) that is compat-ible with the mixing operation. In decision theoretic terms, it satisfies the central expectedutility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it mustsatisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity issufficient for a mixture-preserving multi-representation when the dimension of the mixturespace is countable, but not when it is uncountable. Our strongest positive result is that mix-ture continuity is sufficient in conjunction with a novel axiom we call countable domination,which constrains the order complexity of the mixture preorder in terms of its Archimedeanstructure. We also consider what happens when the mixture space is given its naturalweak topology. Continuity (having closed upper and lower sets) and closedness (having aclosed graph) are stronger than mixture continuity. We show that continuity is necessarybut not sufficient for a mixture preorder to have a mixture-preserving multi-representation.Closedness is also necessary; we leave it as an open question whether it is sufficient. Weend with results concerning the existence of mixture-preserving multi-representations thatconsist entirely of strictly increasing functions, and a uniqueness result.
Keywords.
Expected utility; incompleteness; mixture spaces; multi-representation; conti-nuity; Archimedean structures.
JEL Classification.
D81.
The importance of allowing for incomplete preferences is by now beyond dispute. In the contextof expected utility, von Neumann and Morgenstern (1953, p. 630) themselves remarked, of thecompleteness axiom, that it is “very dubious, whether the idealization of reality which treats thispostulate as a valid one, is appropriate or even convenient”. In the first systematic treatment ofexpected utility without the completeness axiom, Aumann (1962, p. 446) wrote that while all ∗ We are grateful for extremely helpful advice from two anonymous reviewers. We also thank Efe Ok forinvaluable discussion of the history of the subject. David McCarthy thanks the Research Grants Council ofthe Hong Kong Special Administrative Region, China (HKU 750012H) for support. Teruji Thomas thanks theLeverhulme trust for funding through the project ‘Population Ethics: Theory and Practice’ (RPG-2014-064). Thisis a heavily revised version of a preprint ‘Representation of strongly independent preorders by sets of scalar-valuedfunctions’ MPRA. Paper No. 79284 (2017). Declarations of interest: none. † Corresponding author, Dept. of Philosophy, University of Hong Kong, Hong Kong, [email protected] ‡ Dept. of Mathematics and Systems Analysis, Aalto University, Finland, kalle.mikkola@iki.fi § Global Priorities Institute, University of Oxford, United Kingdom, [email protected] M together with a mixing operation, so that for any elements x and y in M and α ∈ [0 , xαy of M is understood to be a mixture of x and y in which x is given weight α and y weight 1 − α . We give the standard axiomatization of mixture spaces in section 2. For now, thebest known example involving uncertainty is when M is the set of probability measures on someoutcome space, and xαy is taken to be the probability measure αx + (1 − α ) y . More generally,any convex set, and thus any vector space, is a mixture space, with the mixing operation definedby the same formula.Given a possibly incomplete preorder % on mixture space M , a multi-representation is anonempty set U of functions M → R such that x % y if and only if, for all u ∈ U , u ( x ) ≥ u ( y ). It is natural to require that the functions u respect the mixing operation. A function u : M → M ′ between mixture spaces is mixture preserving when u ( xαy ) = u ( x ) αu ( y ). In amulti-representation, as we have defined it, M ′ is the vector space of real numbers. So thequestion we consider is under what conditions a preorder % on M has a mixture-preservingmulti-representation ; that is, under what conditions does it satisfy MR There is a nonempty set U of mixture-preserving functions M → R , such that for all x, y ∈ M , x % y ⇐⇒ u ( x ) ≥ u ( y ) for all u ∈ U . It is well known that any mixture space is isomorphic to a convex set. Using this fact, ourquestion is mathematically equivalent to the question of when a preorder on a convex set has amulti-representation consisting of affine (or even linear) functionals on the ambient vector space,restricted to the convex set. We will exploit this equivalence in proofs (see section 4.1), but wefollow Mongin (2001) in thinking that mixture spaces are conceptually more fundamental fordecision theory. For example, it is often easier to verify that an algebraic structure of interest todecision theorists is a mixture space than to show directly that it is isomorphic to a convex set.Much of the literature on mixture-preserving multi-representations has focussed on specifictypes of mixture spaces. Besides sets of probability measures (with different possible assumptionsabout the underlying measurable space), examples include sets of Savage-acts, at least given mildstructural assumptions (Ghirardato, Maccheroni, Marinacci and Siniscalchi, 2003); Anscombe-Aumman acts; charges (i.e. finitely additive measures); and vector-valued measures representingimprecise probabilities. Mixture-preserving multi-representations themselves come in a variety offorms. In the popular Anscombe-Aumman setting, for example, incomplete preferences may bea matter of incomplete beliefs, incomplete tastes, or both, and multi-representations can reflectthese distinctions. The concept of a multi-representation of a preorder was introduced in Ok (2002), but the general idea goesback much further. In decision theory, Bewley (1986) is perhaps the earliest explicit example, but in the guiseof a single vector-valued function, rather than a family of scalar-valued functions, multi-representations wereenvisioned but not developed in von Neumann and Morgenstern (1953, pp. 19–20). There is no reason, however,why the general concept of a multi-representation has to stipulate that the codomain is the real numbers. For anexample in which it is taken to be a linearly ordered abelian group, see Pivato (2013). For examples involving incomplete beliefs, see Bewley (1986, 2002); Ghirardato et al (2003). For tastes, seeDubra, Maccheroni and Ok (2004); Eliaz and Ok (2006); Evren (2008, 2014); Gorno (2017); Hara, Ok and Riella(2019); Borie (2020). For beliefs and tastes, see Seidenfeld, Schervish and Kadane (1995); Nau (2006);Ok, Ortoleva and Riella (2012). For closely related examples, see Manzini and Mariotti (2008) (interval-valuedrepresentations), Galaabaatar and Karni (2012, 2013) (nonstandard preorders), and Heller (2012) (justifiablechoice). First, thepreorder must be what we call a ‘mixture preorder’: it must satisfy what is arguably the centralaxiom of expected utility theory, strong independence. Strong independence is not in general anatural assumption for preferences on mixture spaces; few people’s preferences satisfy it on thesimplex whose points denote different proportions of coffee, milk, and sugar. But it is a plausiblenormative requirement in the examples of mixture spaces introduced above, which all involveuncertainty. Second, it is not hard to show that if a mixture preorder has a mixture-preservingmulti-representation, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953).The result which sets the stage for our discussion, Theorem 2.1, shows, we think rather surpris-ingly, that mixture continuity is not sufficient for a mixture preorder to have a mixture-preservingmulti-representation. However, mixture preorders that satisfy mixture continuity without havinga mixture-preserving multi-representation must be rather complicated; for example, Theorem 2.3shows that they must have uncountably infinite dimension. This raises the question of whetherthere are normatively natural ways of strengthening or supplementing mixture continuity thatdo guarantee MR.Our strongest positive result, Theorem 2.4, shows that, in combination with mixture conti-nuity, an axiom we call ‘countable domination’ is sufficient for a mixture preorder to satisfy MR.We provide two interpretations of this axiom. First, it is a member of a natural but apparentlynovel family of decision-theoretic axioms that constrain what we call the ‘Archimedean struc-ture’ of the preorder. Another axiom in this family is the standard Archimedean axiom, whichis much stronger than countable domination. Second, countable domination may be seen as adimensional restriction on mixture preorders that is much less demanding than the requirementof countable dimension.Our strongest negative result, Theorem 2.5, considers what happens if we impose a topologyon mixture spaces and upgrade mixture continuity to a stronger continuity condition. It notesthat any mixture preorder that satisfies MR must be both continuous and closed in the weaktopology, understood as the coarsest topology on the mixture space in which the real-valuedmixture-preserving functions are continuous. However, more surprisingly, it also shows thatbeing continuous is not sufficient for MR. We leave it as an open question whether being closedis sufficient.Section 2 states our axioms more formally and presents our main results. Section 2.1 re-lates them to the most immediately relevant literature, showing how they extend results ofShapley and Baucells (1998) and answer a question posed by Dubra et al (2004). Section 3 dis-cusses the interpretation of countable domination. Section 4 provides proofs of our main results;it emphasizes the central ideas, appealing to a series of auxiliary results whose proofs we deferto appendix C. Section 5 refines our results by considering two topics. Section 5.1 presents re-sults concerning the existence of mixture-preserving multi-representations that consist entirelyof strictly increasing functions, and relates them (in section 5.1.1) to results by Aumann (1962);Dubra et al (2004); Evren (2014) and Gorno (2017). Section 5.2 presents a uniqueness resultfor mixture-preserving multi-representations that is an abstract version of the uniqueness re-sult of Dubra et al (2004). Appendix A explains the connection between our independence and We define these axioms in section 2. Slightly different versions of the two axioms are common in the literature;we clarify some of the relationships in appendix A. A mixture space is a nonempty set M together with a mixing operation m : M × M × [0 , → M that satisfies axioms shortly to be described. As is customary, when the mixing operation isunderstood, we write xαy for m ( x, y, α ). The axioms are then: (i) xαy = y (1 − α ) x ; (ii) xαx = x ;(iii) if xαz = yαz for some α = 0, then x = y ; and (iv) xα ( yβz ) = ( x αα + β − αβ y )( α + β − αβ ) z if α and β are not both zero. These axioms abstract features of convex subsets of vector spaces,where the mixing operation is given by xαy = αx + (1 − α ) y . The first three are self-explanatory,and the last is an associativity axiom.We will need the notion of the dimension of a mixture space. The standard definition(Hausner, 1954) reduces to the case of convex sets (see section 4.1). However, it is more inthe spirit of our focus on mixture spaces to provide a characterisation directly in terms of themixture-space structure. Given a mixture space M , say that M ′ ⊂ M is a mixture subspace of M if it is a mixture space under the mixing operation inherited from M . For any nonempty A ⊂ M , let M ( A ) be the smallest mixture subspace of M containing A . Say that A is mixtureindependent if, for any nonempty A , A ⊂ A , A ∩ A = ∅ = ⇒ M ( A ) ∩ M ( A ) = ∅ . Wedefine the dimension of M , written dim M , to be | A | − A . In section 4.2.1 we show this is well defined and equivalent to the customary definition.A mixture preorder is a preorder % on a mixture space M that is compatible with the mixingoperation in that it satisfies the following axiom: SI For x , y , z ∈ M , and α ∈ (0 , x % y ⇐⇒ xαz % yαz. A preordered mixture space is a pair ( M, % ) where M is a mixture space and % is a mixturepreorder on M . When M is a convex set of probability measures, SI is strong independence ,arguably the central axiom of expected utility theory.We are interested in the question: when does a mixture preorder have a mixture-preservingmulti-representation?Consider the following axiom, introduced by Aumann (1962). MC For x , y , z ∈ M , if xαy ≻ z for all α ∈ (0 , y % z . These are a reordering of the axioms given by Hausner (1954). Mixture sets , as used for expected utility theoryin e.g. Herstein and Milnor (1953) and Fishburn (1970, 1982) are more general. Terminology varies; Mongin(2001) uses ‘non-degenerate mixture sets’ for what we are calling mixture spaces. In our terminology, despitethe greater generality of mixture sets, Mongin recommends focussing on mixture spaces for the development ofdecision theory. This is analogous to the following characterisation of linear independence of a subset B of a vector space: forany B , B ⊂ B , B ∩ B = ∅ = ⇒ span( B ) ∩ span( B ) = { } . However, Aumann (1962) regarded MC as too strong for his purposes, and instead focussed on, in our labelling: Au For x , y , z ∈ M , if xαy ≻ z for all α ∈ (0 , z y .This axiom is strong enough to rule out, for example, the lexicographic ordering of the unit square. But as well asbeing weaker than MC, for mixture preorders, Au is also weaker than the axiom Ar discussed below. We discussAu further in section 5.1.1.
4s Aumann noted, for mixture preorders, MC is equivalent to the well-known mixture conti-nuity axiom of Herstein and Milnor (1953), that { α ∈ [0 ,
1] : xαy % z } and { α ∈ [0 ,
1] : z % xαy } are closed in [0 ,
1] for all x, y, z ∈ M . Our interest in the axiom MC is prompted by the trivial observation, recorded in the following,that MC is necessary for MR. However, to our surprise, MC is not sufficient:
Theorem 2.1.
For any preordered mixture space ( M, % ) ,MR = ⇒ MC , but the implication is not reversible. The failure of reversibility is in fact quite general.
Theorem 2.2.
Every mixture space of uncountable dimension has a mixture preorder that sat-isfies MC but violates MR.
This raises the question: how might MC be strengthened to guarantee a mixture-preservingmulti-representation? We will consider a range of conditions that are stronger than MC. Some wewill show are sufficient for a mixture-preserving multi-representation, but not necessary. Othersare necessary, but not sufficient. We do not know of a nontrivial condition that is necessary andsufficient, but one of our results will suggest a natural candidate. A first sufficient condition for MR is suggested by Theorem 2.2: we simply strengthen MCby assuming in addition that dim M is countable. (Recall that countable means either finite orcountably infinite.) Theorem 2.3.
For any preordered mixture space ( M, % ) ,MC & dim M is countable = ⇒ MR , but the implication is not reversible. However, the assumption of countable dimension is clearly much stronger than necessary. Wewill give some examples in section 3: in particular, Example 3.6 provides two simple ways inwhich a preordered mixture space of countable dimension that satisfies MC, and consequentlyMR, can be blown up to one of arbitrarily large dimension that still satisfies both MC and MR.Instead, our weakest sufficient condition involves an apparently novel axiom that we call countable domination (CD). We state it now but will discuss its significance at length in section3; in short, it strictly weakens the assumption that dim M is countable, and can also be seen asa much weaker form of the standard Archimedean axiom.Let Γ % ⊂ M × M be the graph of the mixture preorder % : it consists of pairs ( x, y ) with x % y . For any ( x, y ) and ( s, t ) in Γ % , say that ( x, y ) weakly dominates ( s, t ) if xαt % yαs for some α ∈ (0 , % (see appendix B). Anatural interpretation is that when ( x, y ) weakly dominates ( s, t ), the (weakly positive) differencein value between s and t is at most finitely many times greater than that between x and y . Ouraxiom is See section 2.1 and appendix A for further clarification of the connection between MC, the Herstein-Milnoraxiom HM, and the related axiom WCon used by Shapley and Baucells (1998) and Dubra et al (2004). Inparticular, we explain in Remark A.2 why they are all equivalent for mixture preorders. We note in passing that, if ( M, % ) is a preordered mixture space, then the quotient M/ ∼ is also naturallya mixture space with a mixture preorder % ′ , and % ′ is actually a partial order ( x ∼ ′ y = ⇒ x = y ). For manypurposes it suffices to consider M/ ∼ rather than M . In particular, it is not hard to see that % satisfies MR if andonly if % ′ does. But we will focus on M itself. D There is a countable set D ⊂ Γ % such that each ( s, t ) ∈ Γ % is weakly dominated by some( x, y ) ∈ D .Our strongest positive result is Theorem 2.4.
For any preordered mixture space ( M, % ) ,MC & CD = ⇒ MR , but the implication is not reversible. Instead of adding to MC a condition such as CD, we might impose a topology on the mixturespace, and upgrade MC to a stronger continuity condition.Given an arbitrary topological space M , we say that a preorder % on M is continuous if,for all x ∈ M , the sets { y ∈ M : y % x } and { y ∈ M : x % y } are closed in M . A strongercontinuity-like condition that is sometimes used is that the graph Γ % is closed in the producttopology on M × M ; in this case we simply say that % is closed . Thus we study the followingaxioms.
Con % is continuous. Cl % is closed.Specific examples of mixture spaces (like sets of probability measures) may suggest specifictopologies (see section 2.1). However, we will focus on what we call the weak topology , whichmakes sense for any mixture space. By definition, it is the coarsest topology (i.e. containingthe fewest open sets) such that all the mixture-preserving functions M → R are continuous.See Remark 2.6 below for more on our terminology. The interest of the weak topology comesfrom the fact that it makes both Cl and Con into necessary conditions for MR, as the followingelaboration of Theorem 2.1 explains. Theorem 2.5.
For any preordered mixture space ( M, % ) in which M has the weak topology,MR = ⇒ Cl = ⇒ Con = ⇒ MC , but the second and third implications are not reversible. As before, the displayed implications are easily proved and essentially well known; the noveltylies in the failures of reversibility. In particular, Theorem 2.5 shows that Con is still not sufficientfor MR. This is our strongest negative result; it is somewhat delicate because Con, unlike MC, does entail MR when, for example, M is a vector space (see Remark 4.13). For us, it is an openquestion whether Cl and MR are equivalent. Of course, by Theorem 2.4, all four conditions areequivalent when CD holds. Remark 2.6.
A vector space V is a mixture space, so, as we have defined it, the weak topologyon V is the coarsest one that makes every mixture-preserving function V → R continuous.This is equivalent to the more standard definition of the weak topology on a vector space asthe coarsest one that makes every linear functional on V continuous, since a function on V ismixture preserving if and only if it is affine (i.e. linear plus a constant). In the study of arbitrary preorders on topological spaces, the distinction between these two forms of continuityis standard, but terminology varies. For example, Evren and Ok (2011) use ‘semicontinuous’ and ‘continuous’ forour ‘continuous’ and ‘closed’ respectively. Bosi and Herden (2016) use ‘semi-closed’ and ‘closed’.
6n the vector space case, there are, of course, a variety of weak topologies, each inducedby a given subspace of linear functionals. Similarly, there are a variety of weak topologies onmixture spaces, corresponding to subspaces of mixture-preserving functions. But unless otherwisestated, we will not be discussing other weak topologies, hence our use of the term the weaktopology. Other basic features of the weak topology on a mixture space are noted in Lemma C.2in appendix C.Following discussion of our axiom CD in section 3, section 4 presents proofs of the aboveresults, while relegating technical work to appendix C. Section 5 refines the picture in twoways. First, if ( M, % ) is a preordered mixture space, we say that a function f : M → R is increasing if x % y implies f ( x ) ≥ f ( y ), and strictly increasing if, in addition, x ≻ y implies f ( x ) > f ( y ). A mixture-preserving multi-representation clearly consists of functions that areincreasing, but they need not be strictly increasing. Section 5.1 gives results concerning the exis-tence of mixture-preserving multi-representations that contain only strictly increasing functions.Second, section 5.2 provides a uniqueness result for mixture-preserving multi-representationsthat is essentially an abstract version of the uniqueness result given by Dubra et al (2004). In section 1 we noted the wide variety of types of mixture spaces, and forms of mixture-preserving multi-representations, that have been discussed. While it would be desirable toconsider whether our abstract results have applications in all of those areas, that project lieswell beyond the scope of this article. Instead, we will first discuss how our results improve onthose of Shapley and Baucells (1998), and then present one application: we explain how one ofour results solves a problem left open by the influential work of Dubra et al (2004).Our basic objects of study are preorders on mixture spaces that satisfy SI and MC. It iscommon—and is done so specifically by Shapley and Baucells, and Dubra et al —to focus on aslightly different set of basic axioms; we refer to these as ‘independence’ (Ind), which is strictlyweaker than SI, and ‘weak continuity’ (WCon), which is strictly stronger than MC. However,our axioms SI and MC are together equivalent to their axioms Ind and WCon. This equivalenceseems to have been known already by Shapley and Baucells (see their note 1), but since formalstatements and proofs are hard to find, we provide details in appendix A. For ease of comparison,we take the liberty of presenting their results in terms of our axioms and terminology.Shapley and Baucells used a standard embedding theorem to show that any mixture pre-order is naturally associated with an essentially unique convex cone. We explain this technique,which we will also use, in section 4.1. They called a mixture preorder ‘proper’ if its cone hasa nonempty relative algebraic interior; see section 4.2.4 for the definition. Their main result onmixture-preserving multi-representations showed that every proper mixture preorder that satis-fies MC also satisfies MR. As Shapley and Baucells observed, properness holds automaticallywhen the mixture space is finite-dimensional. Thus they effectively proved a weaker versionof our Theorem 2.3, in which ‘countable’ is replaced by ‘finite’. More importantly, our Theo-rem 2.4 strengthens their main result, as our axiom CD is much weaker than their assumptionof properness. Indeed, properness is equivalent to a strengthening of CD that we call ‘singletondomination’ (SD), to be introduced in section 3.The assumption of properness was criticized by Dubra et al (2004, p. 127): “Unfortunately,it is not at all easy to see what sort of a primitive axiom on a preference relation would supportsuch a technical requirement.” Our axioms CD and SD are not subject to this kind of criticism.They are formulated directly in terms of the preorder, and, as we explain in section 3, they aremembers of a natural family of axioms that place limits on the complexity of the preorder in7erms of its Archimedean structure. The standard Archimedean axiom is a much stronger axiomof this type.Dubra et al (2004) consider the mixture space M = P ( X ) of Borel probability measures ona compact metric space X . Let C ( X ) be the set of continuous functions X → R . They endow P ( X ) with the narrow topology (or what Dubra et al call the topology of weak convergence):the coarsest topology such that all the functions P ( X ) → R , defined by integrating againstfunctions in C ( X ), are continuous. Their expected multi-utility theorem shows that Cl isenough to ensure that any mixture preorder on M has a mixture-preserving multi-representationthat consists of expectational functions: functions of the form p R X u d p for some u ∈ C ( X ). They raise the question of whether this result would hold if Cl was weakened to Con or MC,noting only that MC is enough when X is a finite set. Our Theorem 2.2 shows that Cl cannot be weakened to MC in their expected multi-utility theorem, since when X is infinite, P ( X ) hasuncountable dimension. We do not know whether Cl can be weakened to Con in their result,but Theorem 2.5 shows that there can be no general inference from Con to Cl.There is large body of literature on the general question of when a preorder on an arbitrarytopological space has a continuous multi-representation (a condition we call CMR). In requiringa mixing-structure, along with mixture-preserving multi-representations, the focus of this articlehas been different. In the general setting, it is well-known that being closed is not sufficient forCMR. One source of counterexamples is a topological vector space (and hence mixture space): L p [0 , < p <
1, which has no non-zero continuous linear functionals(Rudin, 1991, § We now discuss our axiom CD, and provide some examples. First, we show that it is a naturalweakening of the well-known Archimedean axiom, and connect it with the idea of Archimedeanclasses. Second, we explain how it weakens the assumption that M has countable dimension. To better understand CD, we now introduce two more axioms that are in the same naturalclass. As we will explain, the axioms in this class can be interpreted as constraining the ordercomplexity of mixture preorders. When X is finite, the narrow topology is equal to what we have called the weak topology; when X is infinite,it is more coarse, i.e. contains fewer open sets, strengthening Con and Cl. As well as by Dubra et al , thisstrengthened form of Cl is used in the context of multi-representations by e.g. Ghirardato et al (2003); Ok et al (2012) and Gorno (2017). Their result contains more detail than this. For discussion and further elaboration, see Evren (2008) andHara et al (2019). This follows from the result about finite dimensionality due to Shapley and Baucells (1998) noted above, sinceevery P ( X ) with X finite is a finite-dimensional mixture space (of dimension | X | − P ( X ). Forexample, (0 ,
1) is a one-dimensional mixture space but it is not isomorphic to P ( { , } ) ∼ = [0 , M, % ), let Γ ≻ ⊂ Γ % consists of pairs ( x, y ) with x ≻ y .Our first axiom is the following. Ar Every ( x, y ) ∈ Γ ≻ weakly dominates every ( s, t ) ∈ Γ % .Recall the Archimedean axiom , stated by von Neumann and Morgenstern (1953): if x ≻ y and y ≻ z , then xαz ≻ y and y ≻ xβz for some α and β in (0 , singleton domination . SD There is some ( x, y ) ∈ Γ % that weakly dominates every ( s, t ) ∈ Γ % .Both of these axioms are stronger than CD:Ar = ⇒ SD = ⇒ CD . The first implication is trivial when Γ ≻ is nonempty. When it is empty, both Ar and SD holdautomatically, in the latter case because every ( x, x ) in Γ % weakly dominates every ( s, t ) in Γ % .For the second implication, notice that SD is the special case of CD when D is a singleton.The implications, however, are irreversible, as shown by the next example. Further examplescontrasting Ar, SD and CD will be given below. Example 3.1.
Let S be a non-empty set, and M = R S , the vector space of finitely-supportedfunctions S → R . As a vector space, it is also a mixture space. Define a mixture preorder on M by f % g ⇐⇒ f ( s ) ≥ g ( s ) for all s ∈ S. Then % satisfies MC. It satisfies Ar if and only if | S | = 1. It satisfies SD if and only if | S | isfinite; it satisfies CD if and only if | S | is countable. To illustrate when | S | is countable, define D = { (1 A ,
0) : A ⊂ S, A finite } , where 1 A ∈ M is the characteristic function of A . Then D is acountable subset of Γ % , and each ( f, g ) ∈ Γ % is weakly dominated by the element (1 supp( f − g ) , D .The axioms Ar, SD, and CD can also be reformulated in terms of ‘Archimedean classes’, anidea usually developed in the context of ordered groups or vector spaces (see e.g. Hausner and Wendel,1952). In the present context of preordered mixture spaces, let us say two pairs ( x, y ) and ( s, t )in Γ % are in the same Archimedean class if each weakly dominates the other (this is an equiv-alence relation, since weak domination is a preorder). Write [( x, y )] for the Archimedean classof ( x, y ), and let Π % be the set of Archimedean classes in Γ % . What we call the Archimedeanstructure of a mixture preorder % is the partially ordered set (Π % , ≥ ) where [( x, y )] ≥ [( s, t )] ifand only if ( s, t ) weakly dominates ( x, y ). Note that Π % always contains a maximal element,the single Archimedean class consisting of all pairs ( x, y ) with x ∼ y . As the following easilyproved equivalences show, Ar, SD, and CD can all be seen as placing limits on the complexityof the Archimedean structure.(a) % satisfies Ar if and only if (Π % , ≥ ) has at most two elements.(b) % satisfies SD if and only if (Π % , ≥ ) contains a minimum element.(c) % satisfies CD if and only if (Π % , ≥ ) contains a countable coinitial subset. The direction of the inequality may be surprising, but it is standard in the related literature on valuationtheory, and may be thought of as saying that ( s, t ) comes earlier in order of importance than ( x, y ). Recall that a subset S ′ of a preordered set ( S, % S ) is coinitial if and only if, for every s ∈ S , there exists s ′ ∈ S ′ with s % S s ′ . D ⊂ Γ % , then { [( x, y )] : ( x, y ) ∈ D } is acountable coinitial subset of Π % .There are of course many other ways of limiting the complexity of Archimedean structures,but these are the ones of immediate interest. Appendix B provides more formal discussion ofArchimedean structures; here we illustrate with some examples. Example 3.2.
In Example 3.1, for ( f, g ), ( h, k ) ∈ Γ % , ( f, g ) weakly dominates ( h, k ) if andonly if supp( f − g ) ⊃ supp( h − k ). Therefore [( f, g )] supp( f − g ) is an isomorphism betweenthe Archimedean structure (Π % , ≥ ) and the set of finite subsets of S , partially ordered by ⊂ .The results of appendix B yield a different description. Consider the convex cone of positivefunctions, C = { f ∈ R S : f ≥ } . For each finite A ⊂ S , F A = { f ∈ C : supp( f ) ⊂ A } is a faceof C . If f ∈ C , then F supp( f ) is the smallest face containing f ; this shows that the faces of theform F A , with A finite, are what we call the regular faces of C . Clearly A ⊂ B ⇐⇒ F A ⊂ F B .So we conclude that (Π % , ≥ ) is isomorphic to the set of regular faces of C , partially orderedby ⊂ . Proposition B.1 generalizes this description. It also notes that (Π % , ≥ ) has at most oneminimal element, corresponding to the largest (thus ⊂ -minimal) face C , if it is regular. In thepresent example, it has a minimal element only if S is finite.The following example of a lexicographically ordered vector space makes the structure of(Π % , ≥ ) particularly clear (but MC is not generally satisfied): Example 3.3.
Let ( S, ≥ ) be an ordered set, and as in Example 3.1, let M = R S be the set offinitely supported functions S → R . For distinct f and g in M , let s ( f, g ) = min { s ∈ S : f ( s ) = g ( s ) } . Define a mixture preorder on M by f % g ⇐⇒ either f = g, or f ( s ( f, g )) ≥ g ( s ( f, g )) . Let Π ≻ ⊂ Π % be the set of Archimedean classes of strictly positive pairs, i.e. the [( f, g )] with f ≻ g . It merely omits the maximal element of Π % . One can then see that [( f, g )] s ( f, g ) isan isomorphism of ordered sets between Π ≻ and S . Thus Ar holds if and only if | S | ≤
1; SDholds if and only if S contains a minimal element, e.g. if S = N ; and CD holds if and only if S contains a countable coinitial subset, e.g. if S = R . Remark 3.4.
Most of our examples in this section concern vector spaces. However, this is onlyfor simplicity. Indeed, if ( M, % ) is a preordered mixture space (a vector space or otherwise), and M ′ is any mixture space of the same dimension, then there is a mixture preorder on M ′ with thesame Archimedean structure as % , and which satisfies MC or MR if and only if % does. (Thisfollows from Propositions B.1(iii) and 4.1 below.) As already mentioned, CD strictly weakens the requirement that the dimension of M be count-able; we prove the following in appendix C: Proposition 3.5.
If a preordered mixture space has countable dimension, then it satisfies CD.The converse does not hold, even for mixture preorders that satisfy MC.
We first illustrate why the converse of Proposition 3.5 fails, in particular for mixture preordersthat satisfy MC. One reason is that the dimension of a mixture space can always be increased byintroducing extra dimensions of indifference or incomparability, as the following example shows. Convex cones are defined and discussed in section 4.1. xample 3.6. Let ( M , % ), ( M , % ), ( M , % ) be preordered mixture spaces. Assume that % is complete indifference ( x ∼ y for all x, y ∈ M ), and % is complete incomparability ( x % y only if x = y for x, y ∈ M ). Note that % and % both satisfy MC. Define a preordered mixturespace ( M, % ) by letting M be the product M = M × M × M , with the mixture operationdefined component-wise, and % be the product preorder. Thus in this case( x , x , x ) % ( y , y , y ) ⇐⇒ x % y and x = y . It is easy to check that % satisfies MR, MC, or CD if and only if % does, and that dim M =dim M +dim M +dim M . Suppose that % satisfies MC and that M has countable dimension.By Theorem 2.3 and Proposition 3.5, % will satisfy MR and CD. Thus % will also satisfy MC,MR and CD, but M may have arbitrarily high dimension.However, the following example shows that we can have MC and CD (and hence MR), andarbitrarily high dimension, even if there is no decomposition of the type just illustrated. Example 3.7.
Let W be a nontrivial normed vector space, and M = W × R : as a vector space, M is also a mixture space. Define a mixture preorder on M by( v, a ) % ( w, b ) ⇐⇒ | v − w | ≤ a − b. Then % satisfies MC, SD, and hence CD, but not Ar. We can take D = { (0 ,
1; 0 , } . In this case,however, dim M = dim W + 1, which can be arbitrarily large. There is no nontrivial indifference( x ∼ y = ⇒ x = y ). Although there is incomparability, note that, for any x, y ∈ M , there issome z ∈ M with z % x and z % y . This would not be true if M were a product of preorderedmixture spaces with a nontrivial, completely incomparable factor.For a similar example in which MC and CD hold, but SD (and hence Ar) does not, take the M just described and let M ′ = M N , the set of finitely supported functions N → M . Define a mixturepreorder % ′ on M ′ by f % ′ g if and only if f ( n ) % g ( n ) for all n . In analogy to Example 3.1 (in thecase of S countably infinite), CD holds for % ′ with respect to D = { (0 , A ; 0 ,
0) : A ⊂ N , A finite } .Turning to ways in which Proposition 3.5 may be strengthened, Example 3.6 may suggestthe conjecture that CD holds if, for every x ∈ M , the mixture sets { y ∈ M : y % x } and { y ∈ M : x % y } have countable dimension. However, Example 4.12 will provide a counterexampleto this conjecture; in it, those sets even have finite dimension. Nevertheless, as we explain inRemark 4.8, there is a precise sense in which CD is a dimensional restriction. Our motivation for studying mixture spaces was given in the introduction. However, at a techni-cal level, we will use a standard method to reduce questions about mixture spaces to equivalent,but mathematically more convenient, questions about vector spaces. In the context of multi-representations, this reduction was first used in Shapley and Baucells (1998). It follows from a standard embedding theorem that any mixture space M can be embeddedin a (real) vector space V , in such a way that V is the affine hull of M (so V = span( M − M )), Besides Shapley and Baucells (1998), we refer the reader to Mongin (2001) for a careful study of the embeddingit relies on, and to a text such as Ok (2007) for the vectorial concepts. See Hausner (1954, § M coincides with that on V : xαy = αx + (1 − α ) y . M is, therefore,a convex subset of V , and from this it is easy to show V = { λ ( x − y ) : λ > , x, y ∈ M } . (4.1)We follow Shapley and Baucells in calling such an embedding efficient . Efficient embeddings areessentially unique: if M ⊂ V and M ⊂ V ′ are efficient embeddings, then there is a unique affineisomorphism V → V ′ that is the identity map on M .Recall that a linear preorder % V on a vector space V is a preorder on V that is compatiblewith vector addition and positive scalar multiplication; that is, v % V v ′ ⇐⇒ λv + w % V λv ′ + w for all v, v ′ , w ∈ V and λ >
0. Let M ⊂ V be an efficient embedding. (Considering V as a mixturespace, a linear preorder is the same as a mixture preorder.) As Shapley and Baucells explain,there are natural one-to-one correspondences between mixture preorders % on M , convex cones C ⊂ V , and linear preorders % V on V , such that, for all x, y ∈ M , x % y ⇐⇒ x − y ∈ C ⇐⇒ x % V y. (4.2)This formula explicitly defines % in terms of % V or C , while the next formulae explicitly define C in terms of % , and % V in terms of C : C = { λ ( x − y ) : λ > , x % y } v % V ⇐⇒ v ∈ C. (4.3)We then call C the positive cone of % , and % V the linear extension of % .Finally, mixture-preserving functions u : M → R correspond one-to-one with affine functions˜ u : V → R , in such a way that ˜ u extends u , that is, ˜ u | M = u . Moreover, a set U of mixture-preserving functions M → R is a multi-representation of % if and only if { ˜ u : u ∈ U} is amixture-preserving multi-representation of % V . It follows from (4.3) that an equivalent conditionin terms of C is C = \ u ∈U { v ∈ V : ˜ u ( v ) ≥ ˜ u (0) } . (4.4) We now prove our main results in terms of a series of auxiliary results. We outline the ideason which the auxiliary results are based, but unless otherwise stated, we defer their full proofsto appendix C. Given the existence of efficient embeddings, our positive results mainly rely onstandard extension and separation techniques in vector spaces. The proofs of the negative resultsare more striking, and we describe the counterexamples on which they are based.
Recall that a subset S of a vector space V is algebraically closed if v ∈ S whenever ( v, w ] ⊂ S .(In standard notation, ( v, w ] = { (1 − α ) v + αw : α ∈ (0 , } .) We say that S ⊂ V is weaklyclosed in V if it is closed in the weak topology on V (see Remark 2.6). We prove the followingproposition in appendix C. Since terminology varies slightly: C ⊂ V is a convex cone if and only if C is nonempty, convex and [0 , ∞ ) C = C . We note that although Shapley and Baucells start with axioms that are different from ours (see appendix A),they first derive SI from their axioms, then use SI to construct the correspondences we describe here. Thecorrespondence between % and C is stated in their equations (11) and (12); the well-known correspondencebetween C and % V follows if we consider V as a mixture space. roposition 4.1. Let ( M, % ) be a preordered mixture space, M ⊂ V an efficient embedding,and C ⊂ V the positive cone.(i) dim M equals the vector-space dimension of V .(ii) % satisfies MC if and only if C is algebraically closed.(iii) % satisfies MR if and only if C is weakly closed in V . Part (i) shows that our definition of the dimension of M in section 2 is equivalent to a more stan-dard characterisation (see e.g. Mongin, 2001). Part (ii) is almost the same as Shapley and Baucells(1998, Thm. 1.6), but since our axioms are slightly different, we provide a proof. In fact, we willuse (ii) to show that our axioms are equivalent to theirs, in Appendix A. Part (iii) is a routineapplication of the strong separating hyperplane theorem. The proof that MR implies MC is straightforward. Indeed, supposethat % has a mixture-preserving multi-representation U . Suppose given x, y, z ∈ M such that xαy ≻ z for all α ∈ (0 , u ∈ U , u ( xαy ) ≥ u ( z ). But u ( xαy ) = αu ( x )+(1 − α ) u ( y ).In the limit α →
0, we find u ( y ) ≥ u ( z ). Since this is true for all u ∈ U , we must have y % z , asrequired for MC.The fact that the converse fails is immediate from Theorem 2.2, to which we now turn.The proof of Theorem 2.2 appeals to the following proposition, further discussed below. Proposition 4.2.
Let V be a vector space of uncountable dimension. There exists a convex conein V that is algebraically closed but not weakly closed in V . Proof of Theorem 2.2.
Let M ⊂ V be an efficient embedding of a mixture space M of un-countable dimension, so that, by Proposition 4.1(i), V also has uncountable dimension. ByProposition 4.2, V contains a convex cone that is algebraically closed but not weakly closed.Using (4.2), this cone defines a mixture preorder on M . By Proposition 4.1 parts (ii) and (iii),this mixture preorder satisfies MC but not MR.We prove Proposition 4.2 in appendix C. The proof relies on following example, which isbased on Klee (1953). Klee showed that if a vector space has uncountable dimension, then itcontains a convex subset that is algebraically closed but not weakly closed (see K¨othe (1969,pp. 194–195) for a discussion in more modern terminology). We modify Klee’s construction toobtain a convex cone with similar properties. Example 4.3.
Let V be a vector space with an uncountable basis B . Endow V with the weaktopology. Given a subset S of V , we write cone( S ) for the convex cone in V generated by S ,that is, the smallest convex cone that contains S . It consists of all linear combinations of S withnon-negative coefficients. Choose b ∈ B , and let B = B \ { b } . For each finite, non-emptysubset A ⊂ B , let y A = | A | − P b ∈ A b . Define a convex cone K = cone { y A + b : A ⊂ B is nonempty and finite } . The proof of Proposition 4.2 shows that K is algebraically closed but not closed. In fact, thisgeneralizes slightly: the same argument, using separating hyperplanes, shows that K is not closedwith respect to any locally convex topology on V .13 .2.3 Theorem 2.3 The proof rests on the following, which provides a converse to the result of Klee just mentioned.
Proposition 4.4.
Let V be a vector space of countable dimension. Every convex set in V thatis algebraically closed is weakly closed in V . This was proved using the algebraic version of the separating hyperplane theorem in K¨othe(1969, (3) on p. 194). In appendix C we provide a slightly different proof: to apply the separatinghyperplane theorem, we use a result of Klee (1953), that in a vector space of countable dimension,the finite topology is locally convex.
Proof of Theorem 2.3.
Suppose that MC holds and that M has countable dimension. Givenan efficient embedding M ⊂ V , V also has countable dimension, by Proposition 4.1(i). ByProposition 4.1(ii), the positive cone C is algebraically closed, so, by Proposition 4.4, it is weaklyclosed. Therefore, by Proposition 4.1(iii), % satisfies MR.For a counterexample to the converse implication, let M be an uncountable-dimensionalvector space with the preorder of complete indifference: x ∼ y for all x, y ∈ M . This satisfies MRdespite having uncountable dimension. (Examples 3.6, 3.7 and 4.7 provide less trivial examples.) We first interpret CD and, for future reference, SD, in terms of the positive cone. For furtherdiscussion of Archimedean structure along similar lines, see appendix B. Let V be a vector spacewith linear preorder % V ; let C be any subset of V . Recall that the relative algebraic interior of C consists of those v ∈ C with the following property: for every w ∈ aff( C ), the affine hull of C ,there is some ǫ > v, v + ǫw ] ⊂ S .Recall also that a set S is cofinal in C (with respect to % V ) if S ⊂ C and, for all v ∈ C , thereis some s ∈ S with s % V v . Proposition 4.5.
Let ( M % ) be a preordered mixture space, M ⊂ V an efficient embedding, C the positive cone, and % V the linear extension.(i) % satisfies SD if and only if C has a nonempty relative algebraic interior.(ii) % satisfies CD if and only if there is a countable set that is cofinal in C . We will also use the following standard extension theorem, due to Kantorovich (1937). For aproof, see Aliprantis and Tourky (2007, Thm. 1.36).
Theorem 4.6 (Kantorovich) . Let V be a vector space with a linear preorder % V . Let W be alinear subspace that is cofinal in V . Then any increasing linear functional on W extends to anincreasing linear functional on V . Proof of Theorem 2.4.
We first give a counter-example to the reverse implication; that is, wegive an example of a mixture preorder that satisfies MR (and therefore MC) but not CD.
Example 4.7.
Let M = R N , the vector space of functions N → R . Define a mixture preorderon M by f % g ⇔ f ( n ) ≥ g ( n ) for all n ∈ N . This clearly satisfies MR (the canonical projections R N → R provide a multi-representation), but it violates CD. Proof:
In this case, the positive cone C consists of the f ∈ M with f ( n ) ≥ n . Suppose that CD holds; by Proposition 4.5(ii),there is a countable subset { f , f , . . . } cofinal in C . Let f ( k ) = f k ( k ) + 1 ∈ C . Then for no k isit true that f k % f , a contradiction. 14ow let ( M, % ) be a preordered mixture space, satisfying MC and CD; we have to show itsatisfies MR. Let M ⊂ V be an efficient embedding, C the positive cone, and % V the linearextension. For any subspace W ⊂ V we let % W be the restriction of % V to W , a linear preorderwith positive cone C W = C ∩ W .By Proposition 4.5(ii), there is a countable set Z cofinal in C . Given w ∈ V \ C , set Z w = span( Z ∪ { w } ). By Proposition 4.1(ii), C is algebraically closed. It follows that C Z w isalso algebraically closed. Since Z w has countable dimension, C Z w is weakly closed in Z w , byProposition 4.4. By the strong separating hyperplane theorem (Aliprantis and Border, 2006,Cor. 5.84), there is a linear functional L ′ w on Z w such that L ′ w ( C Z w ) ⊂ [0 , ∞ ) and L ′ w ( w ) < L ′ w ( C Z w ) ⊂ [0 , ∞ ), L ′ w is an increasing linear functional on Z w .Let Y w = span( C ∪ { w } ). We claim that Z w is cofinal in Y w . Indeed, let y ∈ Y w . We canwrite it in the form y = λw + P c ∈ C λ c c , with λ, λ c ∈ R and finitely many λ c being non-zero.Since Z is cofinal in C , we can find, for each c ∈ C , some z c ∈ Z with z c % V c . Since c % V
0, itfollows that | λ c | z c % V λ c c . Therefore λw + P c ∈ C | λ c | z c % V y . Since the left-hand side of thisformula is an element of Z w , Z w is cofinal in Y w .By Theorem 4.6, L ′ w extends from Z w to an increasing linear functional L ′′ w on Y w . Arbitrarilyextend L ′′ w to a linear functional L w on V . By construction, L w ( C ) ⊂ [0 , ∞ ) and L w ( w ) < C = T w ∈ V \ C { v ∈ V : L w ( v ) ≥ } . It follows from (4.4) that U = { L w | M : w ∈ V \ C } is a mixture-preserving multi-representation of % . Remark 4.8.
The following variation on Proposition 4.5(ii), also proved in appendix C, explainsthe sense in which CD is a dimensional restriction, generalizing the countable dimensionalitycondition used in Theorem 2.3.
Corollary 4.9.
Let ( M % ) be a preordered mixture space, M ⊂ V an efficient embedding, C the positive cone, and % V the linear extension. Then % satisfies CD if and only if there is asubspace that is cofinal in span C and that has countable dimension. To illustrate: in Example 3.7, span( C ) = M , which may have arbitrarily high dimension, butspan { (0 , } is a one-dimensional cofinal subspace. We begin with a mostly well-known observation that generalizes some of the claims in Theo-rem 2.5. Say that a preorder % on an arbitrary topological space M has a continuous multi-representation if it satisfies CMR
There is a nonempty set U of continuous functions M → R , such that for all x, y ∈ M , x % y ⇐⇒ u ( x ) ≥ u ( y ) for all u ∈ U . Lemma 4.10.
Let % be a preorder on a topological space M . Then CMR = ⇒ Cl = ⇒ Con . Moreover, suppose M is a mixture space such that, for each x, y ∈ M , the map f x,y : [0 , → M given by α xαy is continuous. Then Con = ⇒ MC . The proof of Lemma 4.10 is in appendix C. Here we use it to deduce Theorem 2.5.
Proof of Theorem 2.5. If M is a mixture space with the weak topology, then every mixture-preserving function M → R is continuous; therefore MR implies CMR. Moreover, for each x, y ∈ M , the map f x,y : [0 , → M given by α xαy is continuous. The implications stated inTheorem 2.5 are therefore immediate from Lemma 4.10.15o show that the third implication in Theorem 2.5 cannot be reversed, we need an examplethat satisfies MC but not Con. We again appeal to Example 4.3. We take M = V and let % be the mixture preorder with positive cone C = K . Recall that K is algebraically closed butnot weakly closed (as shown in proving Proposition 4.2). By Proposition 4.1(ii), % satisfies MC.Since K = { x ∈ M : x % } , % violates Con.Finally, we need to show that Con does not imply Cl. We isolate this claim as the followingproposition and prove it separately. Proposition 4.11.
There is preordered mixture space ( M, % ) such that % is continuous but notclosed in the weak topology on M . The proof of Proposition 4.11, given in appendix C, involves the following modification of Ex-ample 4.3.
Example 4.12.
Let V , B , and K be as in Example 4.3. Let V + = cone( B ). For any v ∈ V + ,let A v ⊂ B be the set of elements of B with respect to which v has strictly positive coefficients.Let V v = span( A v ∪ { b } ), and M = { ( v, w ) : v ∈ V + , w ∈ V v } ⊂ V × V. This M , it is easy to check, is a convex set. Let % be the mixture preorder on M with thepositive cone K ′ = { (0 , w ) : w ∈ K } ⊂ V × V . That is, for all ( x, y ) , ( v, w ) ∈ M × M ,( x, y ) % ( v, w ) ⇐⇒ x − v = 0 , y − w ∈ K ∩ V v . (4.5)Equip M with the weak topology. The proof of Proposition 4.11 consists in the verification that % is continuous but not closed. Remark 4.13.
Let ( M, % ) be a preordered mixture space with the weak topology. As alreadynoted, by Theorems 2.4 and 2.5, the conditions MR, Cl, Con and MC are equivalent when CDholds. In addition, when M is a vector space, the conditions MR, Cl, and Con (but not MC)are equivalent. To show the equivalence, it is sufficient, by Theorem 2.5, to show that Conentails MR. Since M is a vector space, % is a linear preorder, with corresponding positive cone C = { x ∈ M : x % } . But Con implies that C is closed, implying MR by Proposition 4.1(iii). We now briefly discuss two standard topics concerning mixture-preserving multi-representations.
The pioneering study of expected utility without the completeness axiom of Aumann (1962)focussed on the existence of a single real-valued, strictly increasing, mixture-preserving function(as defined in section 2); see also Fishburn (1982). But such a function does not fully charac-terize an incomplete preorder, and interest turned to the existence of mixture-preserving multi-representations, which do. One can try to combine these approaches by considering mixture-preserving multi-representations that consist entirely of strictly increasing functions:
SMR
There is a nonempty set U of strictly increasing mixture-preserving functions M → R ,such that for all x, y ∈ M , x % y ⇐⇒ u ( x ) ≥ u ( y ) for all u ∈ U . Proposition 5.1.
Let ( M, % ) be a preordered mixture space. Then % satisfies SMR if and onlyif it satisfies MR and there exists a strictly increasing mixture-preserving function M → R . The second result extends the picture given by Theorems 2.2 and 2.3 to representations bystrictly increasing functions.
Proposition 5.2.
Let M be a mixture space.(i) If dim M is countable, any mixture preorder on M that satisfies MR also satisfies SMR.(ii) If dim M is uncountable, there is a mixture preorder on M that satisfies MR but not SMR. In common with our earlier results, these results show a sharp difference between the casesof countable and uncountable dimension. Theorem 2.3 and Proposition 5.2 together show that,when dim M is countable, MC is equivalent to SMR. But when dim M is uncountable, MC isnot sufficient even for MR; and even if MR is satisfied, SMR may not be.The proof of Proposition 5.1 is very simple. For Proposition 5.2, the main idea of the proofof (i) is that countable dimension enables us to focus on multi-representations with countablymany elements, as the following lemma shows. Such a countable multi-representation can beused to construct a strictly increasing function, and Proposition 5.1 applies. Lemma 5.3.
Let ( M, % ) be a preordered mixture space. If % has a mixture-preserving multi-representation U , then it has a mixture-preserving multi-representation U ′ ⊂ U such that |U ′ | ≤ max( ℵ , dim M ) . The proof of Proposition 5.2(ii) rests on the following example.
Example 5.4.
Assume that dim M is uncountable. Let M ⊂ V be an efficient embedding, sodim V is uncountable. For some uncountable ordinal κ , we can choose a basis { v α : α < κ } ⊂ M for V indexed by ordinals α smaller than κ . For each β < κ , let π β be the unique linear functionalon V such that π β ( v α ) = 1 if α = β and π β ( v α ) = 0 otherwise. For each α < κ , define a mixture-preserving function u α on M by u α ( x ) = P β ≤ α π β ( x ). This is well-defined, since for each x in V , and hence M , π β ( x ) is nonzero for only finitely many β . Let U = { u α : α < κ } , and let % be the mixture preorder on M that it represents. The proof of Proposition 5.2(ii) shows that % does not have a strictly increasing function, mixture-preserving or otherwise. Suppose that M is a preordered mixture space of uncountable dimension. Aumann (1962) showedthat the continuity condition Au (see note 6), which is weaker than MC, is not sufficient for theexistence of a strictly increasing, mixture-preserving function. Propositions 5.1 and 5.2 togetherstrengthen this result: the existence of a mixture-preserving multi-representation (a conditionstronger than MC, and also stronger than Con for the weak topology) is not sufficient either.As we discussed in section 2.1, Dubra et al (2004) consider mixture preorders on the setof probability measures on a compact metric space, and assume Cl with respect to the nar-row topology. Besides proving the existence of a mixture-preserving, and indeed expectational,multi-representation, they also prove in their Proposition 3 the existence of a strictly increasing17xpectational function. Gorno (2017) uses this to prove the existence of a multi-representationby strictly increasing expectational functions. Our proof of Proposition 5.1 is based on a similartechnique.Evren (2014) also considers probability measures on a compact metric space. He does notfocus on multi-representations in our sense, but nonetheless gives conditions under which apreorder can be represented by a set of strictly increasing functions in a different sense, whichmay have some advantages. We note that Evren’s approach is essentially incompatible with ours(and with the one of Dubra et al ), insofar as his main continuity axiom, ‘open-continuity,’ rarelyholds when MC does: a mixture preorder that satisfies both is either complete or symmetric. Finally, we give a uniqueness result for mixture-preserving multi-representations. It is verysimilar to the uniqueness theorem of Dubra et al (2004), but worked out in our setting of abstractmixture spaces.Given a mixture space M , we let M ∗ be the vector space of all real-valued mixture-preservingfunctions on M . Let C ⊂ M ∗ be the subspace of constant functions. We give M ∗ the topologyof pointwise convergence: the coarsest topology such that for each x ∈ M , the function M ∗ → R given by f f ( x ) is continuous. We write S for the closure of a subset S of M ∗ . Proposition 5.5.
Let M be a mixture space. Two nonempty sets U , U ′ ⊂ M ∗ represent thesame preorder on M if and only if cone ( U ∪ C ) = cone ( U ′ ∪ C ) . It is easy to check that if U represents % , then the subset of functions in M ∗ that areincreasing with respect to % is the unique maximal mixture-preserving multi-representation of % . Proposition 5.5 is equivalent to the claim that the closure of the convex cone containing U and the constant functions is this maximal multi-representation. A Independence and weak continuity
In this appendix, we clarify how our basic axioms, SI and MC, are related to others common inthe literature on expected utility without completeness, as mentioned in section 2.1.Let M be a mixture space, and consider the following axioms for a preorder % on M . Ind
For x , y , z ∈ M , and α ∈ (0 , x % y = ⇒ xαz % yαz . WCon
For x , y , z , w ∈ M , { α ∈ [0 ,
1] : xαy % zαw } is closed.The first is the independence axiom of expected utility theory. The second is axiom P4 ofShapley and Baucells (1998), and is called ‘weak continuity’ by Dubra et al (2004). Some rela-tionships are clarified by the following lemma. Lemma A.1.
Let % be a preorder on a mixture space M .(i) WCon = ⇒ MC;(ii) MC & Ind = ⇒ WCon;(iii) WCon & Ind ⇐⇒ MC & SI.
Thus MC is weaker than WCon, SI is stronger than Ind, and following Shapley and Baucells(1998), we could have focused on the package of Ind and WCon instead of SI and MC. Wehave emphasized the latter combination partly because MC seems simpler and more intuitivethan WCon, and partly because SI is arguably the central idea of expected utility: if M is aconvex set of probability measures, SI is necessary and sufficient for a preorder on M to have anvector-valued expectational representation (McCarthy, Mikkola, and Thomas, 2020, Lem. 4.3).18 emark A.2. Intermediate between MC and WCon is the Herstein-Milnor axiom HM For x , y , z ∈ M , { α ∈ [0 ,
1] : xαy % z } and { α ∈ [0 ,
1] : z % xαy } are closed.Since it is clear that WCon = ⇒ HM = ⇒ MC, Lemma A.1(iii) shows that all three of theseconditions are equivalent for mixture preorders (i.e. assuming SI). Such an equivalence betweenMC and HM was already noted by Aumann (1962), without proof.
Proof of Lemma A.1. (i) Take w = z in the statement of WCon.(ii) Take M = [0 ,
1] and define % by 1 ≻ x ∼ y for all x , y ∈ [0 , % is easily seento satisfy MC and Ind, but not SI. Shapley and Baucells (1998, Lem. 1.2) is that WCon & Ind= ⇒ SI. It follows that % violates WCon (as one can check with x = y = z = 0, w = 1).(iii) The left-to-right direction is immediate from (i) and the result by Shapley and Baucellsjust mentioned. For the right-to-left direction, assume MC and SI. SI obviously entails Ind, soit remains to derive WCon. It is possible to give a direct proof, using only the mixture spaceaxioms. However, a shorter proof is available in terms of an efficient embedding. We emphasizethat this involves no circularity, as Shapley and Baucells (1998) derived the results concerningefficient embeddings that we presented in section 4.1 using only SI, having first derived it fromWCon and Ind; see note 18.Assume, then, that M ⊂ V is an efficient embedding, with C the positive cone. By Propo-sition 4.1(ii), whose proof does not depend on the present result, C is algebraically closed.Consider the set I = { α ∈ [0 ,
1] : αx + (1 − α ) y % αz + (1 − α ) w } , as in the statement ofWCon. Define f ( α ) = α ( x − z ) + (1 − α )( y − w ). Thus f maps [0 ,
1] onto the line segment I ′ = [ y − w, x − z ] = { α ( x − z ) + (1 − α )( y − w ) : α ∈ [0 , } . Since C is convex, I ′ ∩ C is a(possibly empty) line segment; since C is algebraically closed, this line segment, if not empty,contains its end points. But by (4.2), α ∈ I ⇐⇒ f ( α ) ∈ C , so I = f − ( I ′ ∩ C ). It follows that I is a closed interval, implying WCon. B Weak dominance and Archimedean structures
Let ( M, % ) be a preordered mixture space. In this appendix we prove some general facts aboutweak dominance that we used in section 3. Primarily, we show that weak dominance is a preorderon Γ % . This enables us to define the Archimedean structure (Π % , ≥ ) as in section 3.1: Π % consistsof equivalence classes in Γ % under the symmetric part of the weak dominance preorder. Whileit is not difficult to check the preordering property directly, we proceed in a way that highlightsa geometrical interpretation of the Archimedean structure: it is closely related to the lattice offaces of the positive cone C defined by an efficient embedding M ⊂ V (cf. section 4.1). This wasillustrated in Example 3.2.Recall that a non-empty convex subcone F ⊂ C is called a face of C if, for all x, y ∈ C , x + y ∈ F = ⇒ x, y ∈ F . The set F of faces is partially ordered by inclusion, and indeed itis a complete lattice. This means in particular that, for any v ∈ C , there is a smallest faceΦ( v ) containing v . Let us say that F ∈ F is regular if F is not the union of its proper subfaces:equivalently, F = Φ( v ) for some v ∈ C . Let F r ⊂ F be the set of regular faces. Proposition B.1. (i) For any ( x, y ) , ( s, t ) ∈ Γ % , ( x, y ) weakly dominates ( s, t ) if and only if Φ( x − y ) ⊃ Φ( s − t ) .(ii) Weak dominance is a preorder on Γ % .(iii) (Π % , ≥ ) is isomorphic to ( F r , ⊂ ) as a partially ordered set. See Barker (1973), from which we take our simple definition of a face of a convex cone; it is compatible withthe standard definition of the face of a convex set. iv) Any [( x, y )] ∈ Π % is minimal if and only if Φ( x − y ) = C . In particular, Π % contains atmost one minimal element.Proof. For (i), suppose that ( x, y ) weakly dominates ( s, t ). Then there exists α ∈ (0 ,
1) such that αx + (1 − α ) t % αy + (1 − α ) s . Let λ = − αα . It follows from (4.3) that ( x − y ) − λ ( s − t ) ∈ C .At this point we appeal to Barker (1973, Lemma 2.8): w ∈ Φ( v ) if and only if there exists λ > v − λw ∈ C . (We note that Barker’s lemma does not use his standing assumption offinite-dimensionality.) In our case, we find s − t ∈ Φ( x − y ), and therefore Φ( s − t ) ⊂ Φ( x − y ).The argument is reversible.Part (ii) now follows from the fact that ‘ ⊃ ’ is a preorder on F .Now for part (iii). It follows from part (i) that [( x, y )] Φ( x − y ) is a well-defined, order-preserving, injective function Π % → F r , and we just have to show it is surjective, i.e. that everyregular face is of the form Φ( x − y ) with ( x, y ) ∈ Γ % . Every regular face is of the form Φ( v ), with v ∈ C , and, by (4.3), every such v is of the form λ ( x − y ) with λ > x, y ) ∈ Γ % . Sinceevery face containing v contains λ v , and vice versa, we find that Φ( v ) = Φ( x − y ).For (iv), C is the minimal face of C with respect to the preorder ‘ ⊂ ’ (i.e. it is set-theoreticallythe largest face). So, if Φ( x − y ) = C , then certainly C is a minimal regular face, and therefore[( x, y )] is minimal. Conversely, if [( x, y )] is minimal, then Φ( x − y ) is a minimal regular face. Itremains to show that, if there is a minimal regular face, then it is C . Suppose Φ( v ) is a minimalregular face. Note that, for any w ∈ C , any face containing Φ( v + w ) contains v + w , andtherefore contains both v and w . Therefore Φ( v ) ⊂ Φ( v + w ). Since Φ( v ) is minimal regular,Φ( v ) = Φ( v + w ) ∋ w . That is, Φ( v ) contains every w ∈ C ; so Φ( v ) = C . C Proofs of auxiliary results
Proof of Proposition 3.5.
For the first claim, suppose that ( M, % ) is a preordered mixturespace of countable dimension. We appeal to some results from section 4, the proofs of which donot depend on this one. In the terminology of section 4.1, let M ⊂ V be an efficient embedding,with C the positive cone. Proposition 4.1(i) shows that V has countable dimension. Thereforeits subspace span( C ) has countable dimension. Corollary 4.9 then tells us that % satisfies CD(note that span( C ) is a cofinal subspace of itself).The second claim, that the converse does not hold, even for mixture preorders that satisfyMC, is illustrated by Examples 3.6 and 3.7. Proof of Proposition 4.1.
For (i), let A ⊂ M be nonempty. Fix any a ∈ A and let A ′ = { a − a : a ∈ A \ { a }} . Since M ⊂ V is an efficient embedding, V = span( M − M ) = span( M − { a } ).Thus, A ′ is a basis for V if and only if it is linearly independent and maximal among linearlyindependent subsets of M − { a } . We claim that A ′ is linearly independent if and only if A ismixture independent. It follows that A ′ is a basis for V if and only if A is a maximal mixture-independent subset of M . Since | A ′ | = | A | −
1, it follows that the vector-space dimension of V equals the mixture-space dimension of M .To prove the claim, first suppose that A is not mixture independent. There must be nonempty A , A ⊂ A such that A ∩ A = ∅ but M ( A ) ∩ M ( A ) = ∅ . Given the embedding of M into V , M ( A ) equals the convex hull of A ; it consists of all convex combinations of elements of A .Since M ( A ) ∩ M ( A ) = ∅ , there is an equality between two convex combinations of the form P i α i x i = P i β i y i with α i , β i ∈ [0 , x i ∈ A , y i ∈ A , and P α i = P β i = 1. But then we also have P i α i ( x i − a ) = P i β i ( y i − a )20howing that A ′ is linearly dependent. Conversely, suppose that A ′ is linearly dependent. Thenthere are disjoint, finite A , A ⊂ A \ { a } and an equation of the form P i λ i ( a i − a ) = P i µ i ( b i − a )where at most one of the sums is empty (in which case it is zero), with all λ i , µ i > a i ∈ A , b i ∈ A . Without loss of generality, we can assume that λ := P i λ i ≥ P i µ i =: µ , so that A isnonempty. Moving all terms involving a to the right-hand side, and dividing by λ , we have P i λ i λ a i = P i µ i λ b i + λ − µλ a . This shows that M ( A ) ∩ M ( A ∪ { a } ) = ∅ . Therefore A is not mixture independent.Now for part (ii). Suppose first that % satisfies MC. Let ( v, w ] ⊂ C , so that, by (4.3), z % V z ∈ ( v, w ]. To show that C is algebraically closed, we have to show v ∈ C .Suppose first that z ∼ V z ∈ ( v, w ]. Then − z ∼ V
0, so, by (4.3) again, − z ∈ C . Let z ′ = v + z ∈ ( v, w ]. We have v = 2 z ′ − z . Since both z ′ and − z are in C , and C is a convexcone, it follows that v ∈ C , as desired. We are thus reduced to the case where z ≻ V z ∈ ( v, w ].Now we claim that there exists λ > x , x , x ∈ M such that v = λ ( x − x ) and w = λ ( x − x ). Since M ⊂ V is an efficient embedding, using (4.1) we can write v = λ ( x − y )and w = µ ( s − t ) for some λ, µ > x, y, s, t ∈ M . Set β = λ/ ( λ + µ ), so 1 − β = µ/ ( λ + µ ).The claim is easily verified with λ = λ + µ, x = βy + (1 − β ) t, x = βx + (1 − β ) t, x = βy + (1 − β ) s. Any z ∈ ( v, w ] can be written as z = (1 − α ) v + αw , with α ∈ (0 , z = (1 − α ) λ ( x − x ) + αλ ( x − x ) = λ ((1 − α ) x + αx − x ) . Since, as in the first step, z ≻ V
0, it follows that (1 − α ) x + αx ≻ V x . Then, by (4.2),(1 − α ) x + αx ≻ x . This holds for all α ∈ (0 , x % x . Therefore, by (4.3), v = λ ( x − x ) ∈ C .Conversely, suppose that C is algebraically closed. To show that % satisfies MC, supposethat αx + (1 − α ) y ≻ z for all α ∈ (0 , α ( x − z ) + (1 − α )( y − z ) ∈ C for all such α . Since C is algebraically closed, it follows that y − z ∈ C . By (4.2), y % z , validating MC.For (iii), let V have the weak topology. Suppose first that % has a mixture-preserving multi-representation U . Then (4.4) presents C as the intersection of closed sets, so it is closed.Conversely, suppose that C is closed. If C = V , then by (4.3) % is the indifference relation,which has a mixture-preserving multi-representation consisting of a single constant function.Assume then C = V . The weak topology on V is locally convex, so by the strong separatinghyperplane theorem (Aliprantis and Border, 2006, Cor. 5.84), for any v / ∈ C , there exists a linearfunctional L v : V → R such that L v ( C ) ⊂ [0 , ∞ ) and L v ( v ) <
0. Let L = { L v : v / ∈ C } . Thenby (4.2), x % y ⇐⇒ x − y ∈ C ⇐⇒ L ( x ) ≥ L ( y ) for all L ∈ L . It follows that the restriction of L to M is a mixture-preserving multi-representation of % . Proof of Proposition 4.2.
We show that the cone K defined in Example 4.3 is algebraicallyclosed but not closed (recall that V has the weak topology).As a first step, we show that, for any finite, non-empty A ⊂ B , the subcone K ∩ span( A ∪{ b } )of K is algebraically closed. Any convex cone generated by finitely many elements is algebraicallyclosed (see e.g. Ok, 2007, G.1.6, Thm. 1), so it suffices to prove K ∩ span( A ∪ { b } ) = cone { y A ′ + b : A ′ = ∅ , A ′ ⊂ A } . (C.1)21he inclusion of the right-hand side in the left is obvious. Conversely, suppose v is a member ofthe left-hand side. We may assume v = 0. Since v ∈ K , it may be written v = n X k =1 λ k ( y A k + b ) (C.2)where n is a positive integer, each coefficient λ k is strictly positive, and each A k is a finite,nonempty subset of B . It follows that v is a linear combination, with all coefficients strictlypositive, of every member of S nk =1 A k ∪ { b } . Since v ∈ span( A ∪ { b } ), this is only possible if A k ⊂ A for each k . Therefore (C.2) presents v as a member of the right-hand side of (C.1).We can now show that K itself is algebraically closed. Suppose given a half-open line segment( v , v ] ⊂ K ; we have to show v ∈ K . We can find a finite set of basis elements A ⊂ B such that v , v ∈ span( A ∪{ b } ), and therefore such that ( v , v ] ⊂ span( A ∪{ b } ). Since K ∩ span( A ∪{ b } )is algebraically closed, it contains v ; therefore v ∈ K , as desired.Finally, we show that K is not closed. In this proof, let K denote the closure of K . Notethat b / ∈ K ; we show that b is nonetheless in K . Suppose for a contradiction b / ∈ K . Bythe strong separating hyperplane theorem there exists a linear functional f : V → R such that f ( b ) < f ( K ) ⊂ [0 , ∞ ). Now, since B is uncountable, there exists some n ∈ N for whichthere are infinitely many b ∈ B with f ( b ) < n . Let A be a nonempty, finite set of such b . Then f ( y A ) < | A | − P b ∈ A n = n/ | A | . Therefore f ( y A + b ) < f ( b ) + n/ | A | . Since | A | may be chosento be arbitrarily large, and f ( b ) <
0, we can find some y A such that f ( y A + b ) <
0, contraryto f ( K ) ⊂ [0 , ∞ ). We conclude that b ∈ K . Proof of Proposition 4.4.
Let C be an algebraically closed convex subset of a vector space V . We may assume C is nonempty; we want to show it is closed when V is endowed with theweak topology.First consider the case when dim V is finite. The weak topology on V is then the same asthe Euclidean topology. The following argument is based on Holmes (1975, § C , like any convex subset in a finite-dimensional vector space, has a non-emptyrelative interior ri C (Aliprantis and Border, 2006, Lemma 7.33). This is an open subset of aff C .Translating C , we can assume that 0 ∈ ri C , in which case aff C = span C . Let x be in theclosure of C , which is contained in aff C . For any α ∈ (0 , X = − − αα ri C is open in aff C , so x + X contains a point x ′ ∈ C . Then αx ∈ α ( x ′ − X ) = αx ′ + (1 − α ) ri C ⊂ C. Thus ( x, ⊂ C . Since C is algebraically closed, x ∈ C ; thus C is closed.Now suppose V has countable dimension. By definition, a subset X of V is closed in the finite topology on V if and only if X ∩ W is closed in the Euclidean topology in every finite-dimensionalsubspace W of V . Since, for each finite-dimensional W ⊂ V , C ∩ W is algebraically closed, thepreceding argument shows that C is closed in the finite topology. By a result due to Klee(1953), but stated more fully in Kakutani and Klee (1963), the finite topology on a countabledimensional vector space makes it a locally convex topological vector space. By another versionof the strong separating hyperplane theorem (Aliprantis and Border, 2006, Cor. 5.80), C is theintersection of half-spaces that are closed in the weak topology. C itself is therefore closed in theweak topology.The proof of Proposition 4.5 will use the following observation, given an efficient embedding M ⊂ V of a preordered mixture space. Lemma C.1.
Suppose given ( s, t ) ∈ Γ % , x, y ∈ M , and µ > . The following are equivalent: i) There exists λ > such that λ ( x − y ) % V µ ( s − t ) .(ii) We have ( x, y ) ∈ Γ % , and ( x, y ) weakly dominates ( s, t ) .Proof. We repeatedly use facts (4.2) and (4.3) about efficient embeddings. Suppose (i) holds.We have ( s, t ) ∈ Γ % = ⇒ s % t = ⇒ µ ( s − t ) % V ⇒ λ ( x − y ) % V ⇒ x % y = ⇒ ( x, y ) ∈ Γ % . Rearranging the inequality in (i), and setting α = λ/ ( λ + µ ), we find αx + (1 − α ) t % αy + (1 − α ) s . Therefore ( x, y ) weakly dominates ( s, t ). Thus (ii) holds.Conversely, given (ii), we have αx + (1 − α ) t % αy + (1 − α ) s for some α ∈ (0 , λ ( x − y ) % V µ ( s − t ) with λ = αµ/ (1 − α ). Thus (i) holds. Proof of Proposition 4.5.
For (i), it is a standard result that the algebraic interior of a convexcone consists of its order units; see e.g. Aliprantis and Tourky (2007, Lemma 1.7). The proofof (i) essentially translates this fact into a result about M itself. We will rely on the basic facts(4.2) and (4.3) about efficient embeddings without further comment.Suppose SD holds with respect to some ( x, y ) ∈ Γ % . Let v = x − y ∈ C . We note that,since C is a convex cone, aff( C ) = span( C ) = C − C . Thus, given w ∈ aff( C ), we can write w = w − w with w , w ∈ C . Since w ∈ C , we also have w = µ ( s − t ) for some µ > s, t ) ∈ Γ % . By SD, ( x, y ) weakly dominates ( s, t ). So there exists, by Lemma C.1, some λ > λv = λ ( x − y ) % V µ ( s − t ) = w . Therefore v − λ w ∈ C . Since also λ w ∈ C , wefind that v + λ w − λ w = v + λ w ∈ C . Since C is convex, we deduce [ v, v + λ w ] ⊂ C . Since w ∈ aff( C ) was arbitary, this shows v is in the relative algebraic interior rai( C ).Conversely, suppose that rai( C ) is nonempty. Fix v ∈ rai( C ); then v = λ ( x − y ) for some λ > x % y . Given any ( s, t ) ∈ Γ % , we have t − s ∈ − C ⊂ aff( C ). For some ǫ >
0, we musthave v + ǫ ( t − s ) ∈ C , so λ ( x − y ) % V ǫ ( s − t ). By Lemma C.1, we have ( x, y ) ∈ Γ % and ( x, y )weakly dominates ( s, t ). Therefore this ( x, y ) weakly dominates every ( s, t ) ∈ Γ % , so SD holds.For (ii), suppose CD holds, so that every ( s, t ) ∈ Γ % is weakly dominated by an element ofsome countable set D ⊂ Γ % . Let S = { n ( x − y ) : n ∈ N , ( x, y ) ∈ D } ⊂ C . Since D is countable,so is S . We claim S is cofinal in C . Let w ∈ C . We can write w = µ ( s − t ) with µ > s % t .Some ( x, y ) ∈ D weakly dominates ( s, t ). Therefore, by Lemma C.1, there exists λ > λ ( x − y ) % V µ ( s − t ) = w . Choose an integer n > λ . Then n ( x − y ) % V λ ( x − y ) % V w . Since n ( x − y ) ∈ S , S is cofinal in C .Conversely, suppose that S is a countable set, cofinal in C . For each v ∈ S , we can choose λ v > x v , y v ∈ M with x v % y v such that v = λ v ( x v − y v ). Let D = { ( x v , y v ) : v ∈ S } .Since S is countable, so is D . To prove CD, we show that every ( s, t ) ∈ Γ % is weakly dominatedby an element of D . Since s % t , we have s − t ∈ C . Since S is cofinal, there exists v ∈ S suchthat v % V s − t . It follows from Lemma C.1 that ( x v , y v ) weakly dominates ( s, t ). Proof of Corollary 4.9.
By Proposition 4.5(ii), it suffices to show that there is a countableset cofinal in C if and only if there is a countable-dimensional subspace cofinal in span( C ).Suppose S ⊂ C is countable and cofinal. Let Z = span( S ). Because C is a convex cone, any v ∈ span( C ) can be written in the form v = x − y with x, y ∈ C . There is some s ∈ S such that s % V x ; but then s % V v . Since s ∈ Z , Z is cofinal in span( C ). It has countable dimension since S is countable.Conversely, suppose a countable-dimensional subspace Z is cofinal in span( C ). Let b , b , . . . be a countable (finite or infinite) basis for Z . Since b i ∈ span( C ), it can be written as x i − y i with x i , y i ∈ C . Note that x i % V b i . Let S consist of all linear combinations of the x i withnon-negative integer coefficients; it is a countable subset of C . Let v ∈ C . There exists z ∈ Z such that z % V v . We can write z as a finite sum z = P i λ i b i , for some λ i ∈ R . If λ is a positiveinteger greater than all the λ i , then S ∋ P i λx i % V z % V v . Therefore S is cofinal in C .23 roof of Lemma 4.10. The first claim, at least, is well-known; Bosi and Herden (2016), forexample, provide two proofs of the first implication. But we give the short proofs for convenience.To show CMR = ⇒ Cl, suppose U is a continuous mixture-preserving multi-representationof % . For each u ∈ U , define ˜ u : M → R by ˜ u ( x, y ) = u ( x ) − u ( y ). This ˜ u is continuous, andΓ % = T u ∈U ˜ u − ([0 , ∞ )). Thus Γ % is the intersection of closed sets, so Cl holds.To show Cl = ⇒ Con, assume that Γ % is closed. Let x ∈ M . The map f x : M → M givenby f x ( y ) = ( y, x ) is continuous. Therefore, { y : y % x } = f − x (Γ % ) is closed. A similar argumentshows that { y : x % y } is closed. Hence Con holds.For the second claim of the lemma, suppose M is a mixture space and the maps f x,y arecontinuous. To show Con = ⇒ MC, suppose that % is continuous. Suppose that xαy ≻ z for all α ∈ (0 , { w : w % z } is closed, so is f − x,y ( { w : w % z } ). The latter contains (0 , y % z , establishing MC.The next lemma records some basic facts about the weak topology that will be used in theproof of Proposition 4.11. Lemma C.2.
Let M and M be mixture spaces, each with the weak topology.(i) Suppose f : M → M is mixture-preserving. Then f is continuous.(ii) The weak topology on M × M equals the product topology. (iii) If M is a mixture subspace of M , then it is a topological subspace.(iv) If M is a vector space and M ⊂ M is a linear subspace, then M is closed in M .Proof. (i) By definition of the weak topology on M , a function f : X → M from an arbitrarytopological space X is continuous if and only if g ◦ f is continuous for every mixture-preserving g : M → R . Our f : M → M is mixture preserving, so g ◦ f is mixture-preserving, and thereforecontinuous on M .(ii) The weak topology on M × M is the coarsest one such that every mixture-preserving f : M × M → R is continuous. The product topology is the coarsest one such that the projections π i of M × M onto M i are continuous. Equivalently, it is the coarsest one such that for allmixture-preserving f : M → R and f : M → R , the function f ◦ π + f ◦ π : M × M → R is continuous. Since the latter function is clearly mixture-preserving, it suffices to show that(conversely) every mixture-preserving f is of this form.Fix z ∈ M and z ∈ M . For x i ∈ M i define f ( x ) = f ( x , z ) and f ( x ) = f ( z , x ) − f ( z , z ). It is easy to check that f , f so defined are mixture preserving. Moreover, using themixture-preservation property of f , f ( x ) + f ( x ) − f ( x , x ) = f ( x , z ) + f ( z , x ) − ( f ( z , z ) + f ( x , x ))= 2 f ( x z , z x ) − f ( z x , z x ))= 0 . Therefore f ◦ π + f ◦ π = f , as desired.(iii) The claim is that the weak topology on M coincides with the subspace topology inheritedfrom M . The restriction to M of a mixture-preserving function on M is mixture preserving; itfollows that the subspace topology on M is contained in its weak topology. To show the converse,it suffices to show that any mixture-preserving M → R extends to a mixture-preserving function M → R . To prove this using standard facts from linear algebra, we can first embed M as aconvex set in a vector space V (see section 4.1); thus M is also a convex subset of V . Any Here M × M is a mixture space with respect to the component-wise mixing operation: ( x , x ) α ( y , y ) =( x αy , x αy ). f : M → R extends to an affine (i.e. linear plus constant) functionon V ; the restriction of this affine function to M is a mixture-preserving extension of f .(iv) For any x ∈ M \ M , there is a linear (hence mixture-preserving) function g : M → R such that g ( M ) = { } and g ( x ) = 1. Then g − ((0 , ∞ )) is an open neighbourhood of x disjointfrom M . Thus M \ M is open and M is closed in M . Proof of Proposition 4.11.
We first show that the mixture preorder % defined in Exam-ple 4.12 is continuous. Fix ( v, w ) ∈ M . Let U = { ( x, y ) : ( x, y ) % ( v, w ) } and L = { ( x, y ) :( v, w ) % ( x, y ) } . We need to show that U and L are closed in M , which has the weak topology.The two cases are similar, so we consider the former.Let K v = K ∩ V v . Define a function f : M → V × V by ( x, y ) ( x, y ) − ( v, w ). It follows from(4.5) that U = f − ( { } × K v ). Give V × V the weak topology. Since f is mixture-preserving,Lemma C.2(i) tells us that f is continuous. So, to show that U is closed, it suffices to show that { } × K v is closed in V × V .In the first step of proving Proposition 4.2 we showed that K v , that is, K ∩ span( A v ∪ { b } ),is an algebraically closed convex cone. Thus { } × K v is an algebraically closed convex subsetof { } × V v . Since V v , and hence { } × V v , is a finite-dimensional vector space, Proposition 4.4implies that { } × K v is closed in the weak topology on { } × V v .By Lemma C.2(iii), { } × V v , with the weak topology, is a topological subspace of V × V .Moreover, it is a closed subspace, by Lemma C.2(iv). In summary, { } × K v is closed in a closedsubspace of V × V ; therefore it is closed in V × V .We now show that Γ % is not closed in M × M . Note that z = (0 , b ; 0 ,
0) is an element of M × M , but not of Γ % . It suffices to show that z is in the closure of Γ % in M × M . Therefore, itsuffices to find a net ( z α ) in Γ % converging to z in M × M . Here M has the weak topology and M × M has the resulting product topology. Similarly, give V the weak topology, and V × V the product topology. By Lemma C.2(ii), both these product topologies are again the weaktopologies; Lemma C.2(iii) then implies that M × M is a topological subspace of V × V . So itwill suffice that ( z α ) converges to z in V × V .Recall that b is in the closure K of K in V , as proved as the last step in the proof ofProposition 4.2. Let ( y α ) be a net in K converging to b . Note that, by definition, K ⊂ cone( B ) = V + . Therefore each y α can be written as y α = x α + λ α b , with x α ∈ cone( B ) and λ α ≥
0. Note x α ∈ V + and y α ∈ V x α , so ( x α , y α ) is in M . Moreover, by (4.5), ( x α , y α ) % ( x α , z α := ( x α , y α ; x α ,
0) is in Γ % .Now, any element of V can be written uniquely in the form y = x + λb with x ∈ span( B )and λ ∈ R . Define a linear map f : V → V × V by f ( y ) = ( x, y ; x, z α = f ( y α ). Since,by Lemma C.2(i), f is continuous, we have lim α z α = f ( b ) = z . Proof of Proposition 5.1.
It is obvious that a preorder satisfying SMR satisfies MR and ad-mits a strictly increasing mixture-preserving function. (Note that we require multi-representationsto be nonempty.) Conversely, let u ′ : M → R be mixture-preserving and strictly increasing,and U be a mixture-preserving multi-representation. Let U ′ = { u ′ + nu : n ∈ N , u ∈ U} .First, note that for any n ∈ N and u ∈ U , u ′ + nu is strictly increasing. Now suppose that u ′ ( x ) + nu ( x ) ≥ u ′ ( y ) + nu ( y ) for all n ∈ N , u ∈ U . Since, for each u , n can be arbitrarilylarge, we must have u ( x ) ≥ u ( y ). Since U is a multi-representation, we find x % y , so U ′ is amixture-preserving multi-representation containing only strictly increasing functions. Proof of Lemma 5.3.
Let M ⊂ V be an efficient embedding, with positive cone C ⊂ V .Suppose given a mixture-preserving multi-representation U . For each u ∈ U , let ˜ u be its extension25o an affine function V → R , and let A u be the open half-space A u = { v ∈ V : ˜ u ( v ) < ˜ u (0) } . Itfollows from (4.4) that A = { A u : u ∈ U} is an open cover of V \ C , in the weak topology on V .Consider first the case where dim M is finite, and hence, by Proposition 4.1(i), dim V isfinite. Then the weak topology on V coincides with the Euclidean topology, and V is a second-countable topological space, as is its topological subspace V \ C . By Lindel¨of’s lemma, A containsa countable subcover A ′ . We can write A ′ = { A u : u ∈ U ′ } for some countable subset U ′ ⊂ U .Then C = \ u ∈U ′ { v ∈ V : ˜ u ( v ) ≥ ˜ u (0) } . (C.3)It follows from (4.4) that U ′ is a mixture-preserving multi-representation of % . Finally we notethat |U ′ | = ℵ ≤ max( ℵ , dim M ).Now suppose dim M = dim V = κ for some infinite cardinal κ . Let B be a basis of V , and let P be the set of finite subsets of B ; note that |P| = κ . For each P ∈ P , A P := { A u ∩ span P : u ∈ U} is an open cover of span P \ C in the weak topology on span P . As in the previous paragraph, itcontains a countable subcover A ′ P , which we can write in the form A ′ P = { A u ∩ span P : u ∈ U ′ P } ,with U ′ P ⊂ U countable. Let U ′ = S P ∈P U ′ P . Choose any v ∈ V \ C . It is in span P for some P ,and therefore it is in A u for some u ∈ U ′ . So U ′ = { A u : u ∈ U ′ } is an open cover of V \ C . Forthe same reason as before, U ′ is a mixture-preserving multi-representation of % . Finally, since |P| = κ and each U ′ P is countable, |U ′ | = κ ≤ max( ℵ , dim M ). Proof of Proposition 5.2.
For (i), assume that dim M is countable and let % be a mixturepreorder on M that has a mixture-preserving multi-representation; we have to show that it hasone using only strictly increasing functions. Let M ⊂ V be an efficient embedding, so, by Propo-sition 4.1(i), dim V is countable. Since V = span M , we can pick a (finite or countably infinite)basis B = { v , v , . . . } ⊂ M of V . By Lemma 5.3, % has a finite or countably infinite mixture-preserving multi-representation U = { u , u , . . . } . Let ˜ u i : V → R be the unique extension of u i to an affine function; thus L i := ˜ u i − ˜ u i (0) is a linear functional on V . Rescaling the u i asnecessary, we can assume | L i ( v j ) | ≤ j ≤ i . We define a mixture-preserving function u on M by u ( x ) = |U| X i =1 − i L i ( x ) . This is clearly well-defined when |U| is finite. If |U| is infinite, note that every x ∈ M canbe written in the form x = P | B | j =1 c j v j , with finitely many nonzero c j ∈ R . It follows that | L i ( x ) | ≤ P | B | j =1 | c j || L i ( v j ) | ≤ P | B | j =1 | c j | , for all sufficiently large i . Therefore the sum defining u ( x ) is absolutely convergent, making u a well-defined mixture-preserving function. It is alsostrictly increasing. By Proposition 5.1, % has a mixture-preserving multi-representation usingonly strictly increasing functions.For part (ii), we show that the mixture preorder defined in Example 5.4 satisfies MR but notSMR.That preorder was defined by a mixture-preserving multi-representation, so it satisfies MR.We show that it does not admit any strictly-increasing function M → R . Suppose for con-tradiction that u is such a function. In the notation of the example, for each α < κ , define f ( α ) = − u ( v α ). Given α < β < κ , we have v α ≻ v β , and hence u ( v α ) > u ( v β ). This showsthat f is a strictly increasing function of α , and hence there are uncountably many intervals( f ( α ) , f ( α + 1)) ⊂ R that are nonempty, pairwise disjoint, and open. But that is impossible:each open interval must contain a rational number, of which there are countably many.26 roof of Proposition 5.5. Suppose a preorder % on M is represented by U ⊂ M ∗ . Let( M ∗ ) + ⊂ M ∗ consist of the functions in M ∗ that are increasing with respect to % . Write K = cone ( U ∪ C ). To prove the Proposition, it is sufficient to show that K = ( M ∗ ) + .We first verify K ⊂ ( M ∗ ) + . It is obvious that K ⊂ ( M ∗ ) + . Suppose ( f α ) is a net in K converging to f , and suppose x % y . Then f α ( x ) ≥ f α ( y ) for all α . Since M ∗ has the topologyof pointwise convergence, lim α f α ( x ) = f ( x ) and lim α f α ( y ) = f ( y ); therefore f ( x ) ≥ f ( y ). Thus f is increasing, i.e. f ∈ ( M ∗ ) + .Conversely, to show ( M ∗ ) + ⊂ K , we first embed M in M ∗∗ , the algebraic dual of M ∗ ,via the mapping φ : M → M ∗∗ given by φ ( x )( f ) = f ( x ). It is easy to check that φ is mixture-preserving (it is also injective, as shown in Mongin (2001), but we do not use this). The subspacespan( φ ( M )) ⊂ M ∗∗ separates the points of M ∗ , so ( M ∗ , span( φ ( M )) is a dual pair of vectorspaces. Moreover, the topology on M ∗ is the weak topology with respect to this pairing, so itfollows from the fundamental theorem of duality (Aliprantis and Border, 2006, Thm. 5.93) thatspan( φ ( M )) is the continuous dual of M ∗ .Suppose for a contradiction that f ∈ ( M ∗ ) + but f / ∈ K . The vector space M ∗ is locallyconvex, and since K is a convex cone, we may use the strong separating hyperplane theorem(Aliprantis and Border, 2006, Cor. 5.80) to obtain F ∈ span( φ ( M )) such that F ( K ) ⊂ [0 , ∞ )and F ( f ) <
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0. Since f is increasing,it follows that x % y . Thus for some g ∈ U , g ( x ) < g ( y ), implying that F ( g ) <
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