Conditional Preference Orders and their Numerical Representations
CConditional Preference Orders and their Numeri-cal Representations
Samuel Drapeau a,1 , Asgar Jamneshan b,2
August 29, 2018 A BSTRACT
We provide an axiomatic system modeling conditional preference orders whichis based on conditional set theory. Conditional numerical representations areintroduced, and a conditional version of the theorems of Debreu on the ex-istence of numerical representations is proved. The conditionally continuousrepresentations follow from a conditional version of Debreu’s Gap Lemma theproof of which relies on a conditional version of the axiom of choice, freeof any measurable selection argument. We give a conditional version of thevon Neumann and Morgenstern representation as well as automatic conditionalcontinuity results, and illustrate them by examples.K
EYWORDS : Conditional Preferences, Utility Theory, Gap Lemma, von Neu-mann and Morgenstern A
UTHORS I NFO a SAIF/(CAFR) and Mathematics Departement,Shanghai Jiao Tong University, 211 Huaihai Road,Shanghai, China 200030 b Konstanz University, Universitätsstraße 10, 78464Konstanz, Germany [email protected] [email protected] P APER I NFO
AMS C
LASSIFICATION : JEL C
LASSIFICATION : C60, D81
1. Introduction
In decision theory, the normative framework of preference ordering classically requires the completenessaxiom. Yet, there are good reasons to question completeness as famously pointed out by Aumann [2]:Of all the axioms of utility theory, the completeness axiom is perhaps the most question-able. [. . . ] For example, certain decisions that an individual is asked to make might involvehighly hypothetical situations, which he will never face in real life. He might feel that he can-not reach an “honest” decision in such cases. Other decision problems might be extremelycomplex, too complex for intuitive “insight”, and our individual might prefer to make nodecision at all in these problems. Is it “rational” to force decision in such cases?Aumann’s remark, supported by empirical evidence, triggered intensive research in terms of interpreta-tion, axiomatization and representation of general incomplete preferences, see [3, 15, 16, 18, 20, 34, 37]and the references therein. These authors consider incompleteness either as a result of status quo, seeBewley [3], or procedural decision making, see Dubra and Ok [15], and the numerical representations arein terms of multi-utilities. However, Aumann’s quote and a correspondence with Savage [38], where heexposes the idea of state-dependent preferences, suggest that the lack of information underlying a deci-sion making is a natural source of incompleteness. For instance, consider the simple situation where aperson has to decide between visiting a museum or going for a walk on Sunday in one month from now.She cannot express an unequivocal preference between these two prospective situations since it dependson the knowledge of uncertain factors like the weather, availability of an accompanying person, etc. Thisinformation-based incompleteness suggests a contingent form of completeness. For instance, conditionedon the event “sunny and warm day” she prefers a walk. In this way, a complex decision problem, provided1 a r X i v : . [ q -f i n . E C ] J a n ufficient information, leads to an “honest” decision. The present work suggests a framework formalizingthis idea of a contingent decision making and its quantification.Numerous quantification instruments in finance and economics entail a conditional dimension by map-ping prospective outcomes to random variables such as for instance conditional and dynamic monetaryrisk measures [1, 6, 7, 11], conditional expected utilities and certainty equivalents, dynamic assessmentindices [4, 24] or recursive utilities [17, 19]. However, few papers address the axiomatization of condi-tional preferences underlying these conditional quantitative instruments. In this direction is the work ofLuce and Krantz [31] where an event-dependent preference ordering is considered and studied. Their ap-proach is further refined and extended in Wakker [42] and Karni [27, 28]. State-wise dependency is usedin Kreps and Porteus [29, 30] and Maccheroni et al. [32] to study intertemporal preferences and a dy-namic version of preferences, respectively. Remarkable is the abstract approach by Skiadas [39, 40]. Heprovides a set of axioms modeling conditional preferences on random variables which admit a conditionalSavage representation of the form U ( x ) = E Q [ u ( x ) | A ] where A is an algebra of events representing the information, Q is a subjective probability measure and u is a utility index. As in the previous works, its decision-theoretical foundation consists of a whole familyof total pre-orders (cid:60) A , one for each event A ∈ A , and a consistent aggregation property in order to obtainthe conditional representation. However, the decision maker is assumed to implicitly take into account alarge number of complete pre-orders.Our axiomatic approach differs in so far as it considers a single but possibly incomplete preferenceorder (cid:60) instead of a whole family of complete preference orders. Even if one cannot a priori decidewhether x (cid:60) y or y (cid:60) x for any two prospective outcomes, or acts, there may exist a contingent infor-mation A conditioned on which x is preferable to y . In this case we formally write x | A (cid:60) y | A . The setof contingent information is modeled as an algebra of events A = ( A , ∩ , ∪ , c , ∅ , Ω) of a state space Ω . In order to describe the conditional nature of the preference, we require that (cid:60) interacts consistently withthe information, that is,• consistency: if x | A (cid:60) y | A and B ⊆ A , then x | B (cid:60) y | B ;• stability: if x | A (cid:60) y | A and x | B (cid:60) y | B , then x | A ∪ B (cid:60) y | A ∪ B ;• local completeness: for every two acts x and y there exists a non-empty event A such that either x | A (cid:60) y | A or x | A (cid:60) y | A .These assumptions bear a certain normative appeal in view of the conditional approach that we are aimingat. In the context of the previous example, consistency says that if the person prefers a walk over a visit toa museum whenever it is “sunny” or “warm”, then a fortiori she prefers a walk if it is “sunny”. Stabilitytells that if she prefers a walk whenever it is “sunny” or “rainy”, then on any day where at least one ofthese conditions is met she will go for a walk. In contrast to classical preferences, we only assume a localcompleteness: For any two situations she is able to meet a decision provided enough – possibly extremelyprecise – information. In our example, there exists a rather unlikely, but still non-trivial, coincidence ofthe conditions ‘sunny’ , ‘humidity between 15 and 20%’ and ‘wind between 0 and 10km/h’ under whichshe prefers a walk to the museum. Unlike classical completeness, the information necessary to decide In a five steps binary tree, . . . is the cardinality of the family of total pre-orders (cid:60) A . Conditional set theory [14] allows the contingent information to be any complete Boolean algebra. Indeed, the smaller the event, the more precise in which state of the world this event may occur. The most precise event beingthe singleton. x and y depends on the pair ( x, y ) . Note that if the set of contingent information reducesto the trivial information A = {∅ , Ω } , then, as expected, a conditional preference is a classical completepreference order. In particular, classical decision theory is a special case of the conditional one.Observe that our approach, as [29, 31, 39, 40, 42], considers an exogenously given set of informationsor events as the source of incomplete decision making. Whereas in [3, 15] and the related subsequentliterature on incomplete preference, the incompleteness and the resulting multi-valued representationsyield an endogenous information about the nature of the incompleteness. Incompleteness there is howevernot in terms of an algebra of events, and therefore not specifically related to a contingent decision making.Our approach is also not a priori dynamic in the sense that a single algebra of available informationis given for the contingent decision making. We do not address the question of progressive learningover time as new information reveals, resulting in an update of decisions. This incremental learningapproach in decision making is investigated by Kreps and Porteus [29], and recently by Dillenbergeret al. [12] as well as Piermont et al. [35]. In these articles, the agent learns over time and may modifyher behavior according to the new information as well as her previous choice making. However, theunderlying information structure is exogenously given – either by a fixed dynamic structure by meansof a filtration or a random tree, or by the filtration generated by the consumption paths, or even bythe filtration generated by the previous preference orders. Our approach may help in these cases byconsidering a sequence of conditional preference orders (cid:60) , (cid:60) , . . . , (cid:60) t , . . . with respect to an increasingsequence of algebra of events A ⊆ A ⊆ · · · ⊆ A t ⊆ · · · each of which for every point in time. Wecan provide an axiomatic system to describe these conditional preference orders (cid:60) t for each given time t , and derive a sequence of conditional numerical representations U t . Since we only address the case ofa single information structure, that is, at a fixed given time t , we intentionally left out the following twoquestions in the dynamic context. First, whether the decision making at time t is influenced by the pastinformation, that is a Markovian versus non-Markovian decision making. Second, the impact at time t of past and eventually future decisions. In other terms, the interdependence structure over time of thesepreferences and the consequences for the dynamic utility representation in terms of time consistency. Although being intuitive, it is mathematically not obvious what is meant by a contingent prospectiveact x | A . The formalization of which corresponds to the notion of a conditional set, introduced recently byDrapeau et al. [14]. An heuristic introduction to conditional sets is given in Section 2. For an exhaustivemathematical presentation we refer to [14]. The formalization and properties of conditional preferencesare given in Section 3. In Section 4, we address the notion of conditional numerical representation andprove a conditional version of Debreu’s existence result of continuous numerical representations. Whilethe proof technique differs, the classical statements in decision theory translate into the conditional frame-work. For instance, a conditional version of the classical representation of von Neumann and Morgenstern[41] is presented in Section 5. The representation of Debreu requires topological assumptions that oftenare not met in practice. In Section 6, we provide conditional results that allow to extend Debreu andRader’s theorem in a more general framework, and present automatic continuity results which allow tobypass topological assumptions. We illustrate each of these cases by examples. These results in theirclassical form rely on the Gap Lemma of Debreu [9, 10] the conditional adaptation of which does not in-volve any measurable selection arguments but derives from a conditional version of the axiom of choice.Section 7 is dedicated to the formulation and the proof of this conditional Gap Lemma. In Appendix A,we gather some technical results and most of the proofs. Though, Dillenberger et al. [12] consider a static approach resulting in dynamic utility valuations that are deterministic. For instance, time consistency, Bellman principle, weaker time consistency, etc. A topic of intensive study in mathematical finance, see [1, 4, 6–8] among others. . Conditional Sets As mentioned in the introduction, we model the contingent information, conditioned on which a decisionmaker ranks prospective outcomes, by an algebra of events A . For technical reasons, we assume that it isa σ -algebra with a probability measure defined on it. The inclusion of two events is then to be understoodin the almost sure sense. A set X – that in the present context describes acts – is a conditional set of A if it allows for conditioning actions A : X → X | A for each event A ∈ A which satisfy a consistency andan aggregation property: Consistency:
For any two acts x, y ∈ X and events A ⊆ B , if the acts x and y coincide conditionedon B , that is, x | B = y | B , then they also coincide conditioned on A , that is, x | A = y | A . Stability:
For any two acts x, y ∈ X and event A ∈ A , there exists an act z ∈ X such that z coincides with x conditioned on A and with y otherwise. We denote this element z = x | A + y | A c . Intuitively, the action x (cid:55)→ x | A tells how acts are conditioned on the information A and X | A representsthe acts in x conditioned to A . Example 2.1.
Following the example from the introduction, there are two unconditional alternatives x = ‘going for a walk’ and y = ‘going to the museum’ , and the information is reduced to a single condition A = ‘sunny’ which yields the algebra A = { , A, A c , Ω } = { ‘no information’ , ‘sunny’ , ‘not sunny’ , ‘full information’ } . The corresponding conditional set of acts is then given by X = { x, y, x | A + y | A c , y | A + x | A c } . For instance, the act x | A + y | A c stays for going for a walk provided it is sunny and going to the museumotherwise. ♦ Example 2.2.
The conditional rational numbers are defined as follows: given two rational numbers q , q ∈ Q and an event A ∈ A , let q := q | A + q | A c be the conditional rational number which is q conditioned on A and q otherwise. The set of conditional rational numbers, denoted by Q , is aconditional set where the conditioning action is given by q | B = q | A ∩ B + q | A c ∩ B ∈ Q | B . The con-ditional natural numbers N are defined analogously. In analogy to the next example, N and Q correspondto the set of random variables with rational values and natural values, respectively. ♦ In the theory of conditional sets, any complete algebra can be considered as a source of information and what follows also holdsin this slightly more general framework. Though, from an economical point of view, most standard frameworks consider finitealgebras of events or σ -algebras with a probability measure on it that describe the events of null measure, that is, those eventsthat are considered as never occurring. A σ -algebra on which sets are identified if they coincide almost surely is complete, see[14, 25]. If one does not want to consider probability spaces, the Borel sets of a Polish space factorised by the sets of categoryone is a complete Boolean algebra. Since we assume that A can be made complete, the concatenation property is required for any partition of events ( A i ) ⊆ A andfamily of acts ( x i ) ⊆ X , and we denote x = (cid:80) x i | A i the unique element such that x coincides with x i conditioned on A i . More generally, given a partition of events ( A n ) ⊆ A and a corresponding family of rationals ( q n ) ⊆ Q , define the conditionalrational number q := (cid:80) q n | A n as the conditional element which has the value q n conditioned on A n . xample 2.3. Another example is the collection L ( A ) = { x : Ω → R such that x is A -measurable } of random variables. Given a random variable x and an event A , the conditioning of x on A is therestriction x | A : A → R , ω (cid:55)→ x | A ( ω ) := x ( ω ) for ω ∈ A . For any two random variables x, y and anevent A , then z = x | A + y | A c corresponds to the random variable x A + y A c where A is the indicatorfunction of the event A .In many cases, the information algebra A describes the exogenous information which is however onlypartially available to the agent for a decision making. For instance, the information available tomorrowto decide about random outcomes that are due in a year from now and depend on the whole informationduring that year. This can be modelled as follows. Given another algebra B with a probability measureon it and such that A ⊆ B , we can define L ( B ) := { x : Ω → R such that x is B -measurable with E [ | x | | A ] < ∞} . as the set of B -measurable random variables with finite conditional expectation with respect to tomorrow’sinformation A . Inspection shows that it defines a conditional set of A when considering the restrictions x | A for events A which are in the smaller algebra A . ♦ Example 2.4.
As it concerns decision theory, lotteries – or probability distributions – are often used asobjects for decision making. We define P ( A ) := { µ : Ω → P such that µ is A -measurable } where P is a set of lotteries. A conditional lottery can be seen as a state-dependent lottery providing foreach state ω a lottery µ ( ω, dx ) . Throughout, we denote by P ( A ) the set of state-dependent measurablelotteries. Likewise random variables, it defines a conditional set where µ | A is the conditional lotteryrestricted to the event A and for every two conditional lotteries µ, ν and event A , the conditional lottery η = µ | A + ν | A c corresponds to the conditional lottery µ A + ν A c .Another typical object are Anscombe-Aumann acts which are extensions of lotteries. Actually theconditional set of conditional lotteries P ( A ) already represents, strictly speaking, Anscombe-Aumannacts. However, in our context, the decision making is contingent and therefore realised with respect tothe available information A . In the present form, an Anscombe-Aumann act is a state-dependent lotterybut measurable with respect to a larger algebra of events B on which the decision maker cannot make anhonest decision. Just as L ( B ) , the conditional set of conditional Anscombe-Aumann acts is defined as P ( B ) := { µ : Ω → P such that µ is B -measurable } . ♦ The relation between conditional sets is described by the conditional inclusion which is characterizedby two dimensions, a classical inclusion and a conditioning:• On the one hand, every non-empty set Y ⊆ X which is stable , that is, x | A + y | A c ∈ Y for every x, y ∈ Y and A ∈ A , is a conditional subset of X .• On the other hand, X | A is a conditional set but on the relative algebra A A := { B ∩ A : B ∈ A} and a subset of X conditioned on A . It is in fact an L -module as studied and introduced in [21, 22] Y is said to be conditionally included in X , denoted Y (cid:118) X , if Y = Z | A for some stable Z ⊆ X and a condition A ∈ A . In that case, we say that Y isa conditional set “living” on A and if we want to emphasize the condition on which this conditional setlives, we denote it Y | A . The conditional inclusion is illustrated in Figure 1a. If A = ∅ , then X |∅ livesnowhere and in particular is conditionally contained in any conditional set, and thus is conditionally theemptyset. The conditional powerset P ( X ) := { Y : Y (cid:118) X } = { Y : Y = Z | A for some event A ∈ A and a stable set Z ⊆ X } consists of the collection of all conditional subsets of X . Example 2.5.
In Example 2.1, the set { x, y } ⊆ X is not stable since x | A + y | A c (cid:54)∈ { x, y } . Hence { x, y } is not a conditional set whereas Z := { x ; y | A + x | A c } is stable, and therefore a conditional subset of X living on Ω . However, Y := { x | A, y | A } is a conditional subset of X living on A . Indeed, Y | A = Z | A . ♦ The conditional intersection of two conditional sets
Y, Z is the intersection on the largest condition A ∗ onwhich Y and Z have a non-empty classical intersection as illustrated in Figure 1b. The conditional union (a) Illustration of the conditional inclusion. (b) Illustration of the conditional intersection. of two conditional subsets Y, Z is the collection of all elements which can be concatenated such that eachpiece of the concatenation conditionally falls either in Y or in Z . The conditional union is defined by Y (cid:116) Z := { y | A + z | B : y | A ∈ Y | A, z | B ∈ Z | B and A ∩ B = ∅} and illustrated in Figure 1c. Finally, the conditional complement Y (cid:64) of a conditional subset Y is thecollection of all those elements y which nowhere fall into Y , as illustrated in Figure 1d.A main result in [14] is that the conditional powerset together with these operations forms a completeBoolean algebra (cid:0) P ( X ) , (cid:116) , (cid:117) , (cid:64) , X |∅ , X (cid:1) . Following the classical constructions, the conditional powerset allows to define conditional relations, andfunctions, and topologies, and other conditional structures, see [14].6 c) Illustration of the conditional union. (d) Illustration of the conditional complement.
3. Conditional Preference Orders
For the remainder of the paper X denotes a conditional set. A conditional binary relation (cid:60) is a condi-tional subset G (cid:118) X × X living on Ω and we write x (cid:60) y if and only if ( x, y ) ∈ G . In particular, aconditional binary relation is at first a classical binary relation. However, due to the fact that the graph G is a conditional set and writing x | A (cid:60) y | A for ( x | A, y | A ) ∈ G | A , the following additional propertieshold• consistency: if x | A (cid:60) y | A and B ⊆ A , then x | B (cid:60) y | B ;• stability: if x | A (cid:60) y | A and x | B (cid:60) y | B , then x | A ∪ B (cid:60) y | A ∪ B ;corresponding to two of the normative properties mentioned in the introduction. Given a conditionalbinary relation, ∼ denotes the symmetric part of the binary relation and we use the notation x (cid:31) y if and only if x (cid:60) y and y | A (cid:54) (cid:60) x | A for every non-empty event A ∈ A . In other words, x (cid:31) y means that x is strictly preferred to y on any non-empty condition. Both ∼ and (cid:31) are conditional binary relations. Definition 3.1.
A conditional binary relation (cid:60) on X is called a conditional preference order if (cid:60) is• reflexive: x (cid:60) x for every x ;• transitive: From x (cid:60) y and y (cid:60) z it follows that x (cid:60) z ;• locally complete: for every x (cid:54)∼ y there exists a non-empty event A such that either x | A (cid:31) y | A or y | A (cid:31) x | A .Although a conditional preference is not total, the following lemma shows that local completeness allowsto derive for every two elements a partition on which a comparison can be achieved. Lemma 3.2.
Let (cid:60) be a conditional preference order on X and x, y ∈ X . There is a pairewise disjointfamily of conditions A, B, C such that A ∪ B ∪ C = Ω and x | A ∼ y | A, x | B (cid:31) y | B and y | C (cid:31) x | C. roof. Let x, y ∈ X and define A = ∪{ ˜ A ∈ A : x | ˜ A ∼ y | ˜ A } , B = ∪{ ˜ B ∈ A : x | ˜ b (cid:31) y | ˜ B } and C = ∪{ ˜ C ∈ A : ˜ Cy (cid:31) ˜ Cx } which are the largest conditions on which x is conditionally equivalent, strictly better or worse than y ,respectively. Due to the consistency property of conditional relations, it follows that these conditions aremutually disjoint. For the sake of contradiction, suppose that D := A ∪ B ∪ C (cid:54) = Ω . It follows that outside D , that is, conditioned on D c , the element x is nowhere either equivalent, strictly better or worse than y .Otherwise, this contradicts the definition of A , B and C . Define ˜ y = x | D + y | D c being x conditionedon D and y conditioned on D c . Since x (cid:54)∼ y and ∼ is a conditional equivalence relation, it follows that x (cid:54)∼ ˜ y , otherwise x | D c ∼ y | D c contradicting the definition of A . By local completeness, there exists anon-empty condition E such that either x | E (cid:31) ˜ y | E or ˜ y | E (cid:31) x | E . Without loss of generality, supposethat x | E (cid:31) ˜ y | E . Since ˜ y | D = x | D ∼ x | D by reflexivity and consistency, it follows that E is disjointfrom D , in other words E ⊆ D c . In particular, ˜ y | E = y | E implying that x | E (cid:31) y | E , which togetherwith ∅ (cid:54) = E ⊆ D c contradict the maximality of B . Thus D = Ω which ends the proof. (cid:3) Example 3.3.
Let us give a complete formal description of the example in the introduction. Recallfrom Example 2.1 that A = { ‘no information’ , ‘sunny’ , ‘not sunny’ , ‘full information’ } = {∅ , A, A c , Ω } ,and that the conditional set generated from the two unconditional choices x = ‘going for a walk’ and y = ‘going to the museum’ is given by X = { x, y, x | A + y | A c , y | A + x | A c } . The conditional preferencebeing reflexive, it trivially holds x (cid:60) x, y (cid:60) y, x | A + y | A c (cid:60) x | A + y | A c , y | A + x | A c (cid:60) y | A + x | A c . Further, the individual prefers going for a walk if it is sunny and to the museum otherwise. This translatesinto x | A (cid:31) y | A and y | A c (cid:31) x | A c . Since the preference is assumed to be conditional, it also holds x | A + y | A c (cid:60) y | A + x | A c , x (cid:60) y | A + x | A c , x | A + y | A c (cid:60) y. For instance, the relation x (cid:60) y | A + x | A c states that going for a walk is in any case better than going to themuseum if it is sunny and going for a walk otherwise. Inspection shows that the conditional preferenceis indeed a transitive and reflexive conditional relation. As mentioned, this relation does not tell whether x is preferred to y , that is, whether she wants to go for a walk or to the museum. There exists howevera condition A = ‘sunny’ , such that x | A (cid:60) y | A which shows that it is locally complete. In particular, thepartitioning given in Lemma 3.2 corresponds to x |∅ ∼ y |∅ , x | A (cid:31) y | A, y | A c (cid:31) x | A c . Note also that conditional sets allow to solve the following puzzle: Define Y = { z (cid:31) y } , the set ofelements which are strictly preferred to y . In a classical setting this set is empty. Indeed, there existsno alternative which is strictly preferred to going to the museum since conditioned on A c , y is maximalfor the preference order. However, as our intuition suggests, this set should not be empty and indeed itis conditionally non-empty since Y = { x | A } . The importance of this fact is observed in the proof ofDebreu’s Theorem 4.6 with the definitions of Z ± towards the construction of a conditional numericalrepresentation. ♦ Recall that we assume that A is an algebra that can be factorized in such a way that it is complete with respect to the formationsof unions and intersections. xample 3.4. In the framework of Example 2.3 of A -measurable random variables, the natural partialorder on L ( A ) given by x (cid:62) y if and only if x ( ω ) ≥ y ( ω ) for almost all ω is an example of a conditionalpreference order. Indeed, it is consistent since if the random variable x is greater than the random variable y on an event A , then it is also the case on any event B ⊆ A . Likewise it is stable, reflexive andtransitive, even asymmetric and therefore a conditional partial order. However, it is also locally complete.Indeed, if x (cid:54) = y , then it follows immediately that either the event B := { ω : x ( ω ) > y ( ω ) } or theevent C := { ω : x ( ω ) < y ( ω ) } is non-empty. Actually, defining the events A = { ω : x ( ω ) = y ( ω ) } , B = { ω : x ( ω ) > y ( ω ) } and C := { ω : x ( ω ) < y ( ω ) } provide the partition of Lemma 3.2.Since the conditional rational numbers Q coincide with a subset of A -measurable random variables,the same conditional total ordering in the almost sure sense can be defined on them. ♦ Remark 3.5.
In general, a conditional preference can be an equivalence relation conditioned on A andstrictly non-trivial on A c . However, the case of interest lives on A c . Therefore, throughout this paper, weassume that a conditional preference is conditionally non-trivial, that is, there exists a pair x, y ∈ X suchthat x (cid:31) y (cid:7)
4. Conditional Numerical Representations
Next we address the quantification of such a conditional ranking. First, we need the notion of a conditionalfunction . A conditional function f : X → Y between two conditional sets is a classical function with theadditional property of stability: f ( x | A + y | A c ) = f ( x ) | A + f ( y ) | A c . Example 4.1.
The A -conditional expectation of elements of L ( B ) introduced in Example 2.3, is a con-ditional function. Indeed, for every x, y ∈ L ( B ) and each A ∈ A it holds f ( x | A + y | A ) := E [1 A x + 1 A c y | A ] = 1 A E [ x | A ] + 1 A c E [ y | A ] = f ( x ) | A + f ( y ) | A c since A is A -measurable. ♦ Example 4.2.
For q = q | A + q | A c and r = r | B + r | B c , define the conditional addition and condi-tional absolute value on Q as q + r := q + r on A ∩ Bq + r on A ∩ B c q + r on A c ∩ Bq + r on A c ∩ B c and | q | := | q | | A + | q | | A c . Together with an analogous definition for conditional multiplication these operations make Q a condi-tional totally ordered field as defined in [14].In particular, this allows to define on Q the conditional variant of the Euclidean topology on Q by theconditional balls B r ( q ) := { p ∈ Q : | q − p | (cid:54) r } , q ∈ Q and r ∈ Q ++ := { p ∈ Q : p > } . It behaves like the standard topology on Q with theadditional local property: B r ( q ) | A + B r ( p ) | A c := B r | A + r | A c ( q | A + p | A ) , ♦ for every r , r ∈ Q ++ and q, p ∈ Q ++ . In other words, a conditional neighborhood of conditionedon A and / on A c is itself a conditional neighborhood of the conditional rational | A + 2 / | A c .For the quantification, we secondly need a conditional analogue of the real line which allows to rep-resent the conditional preferences. The conditional real numbers, denoted by R , are obtained from theconditional rational numbers by adapting Cantor’s construction, that is, identifying conditional Cauchysequences in Q . As in the standard theory, the conditional real numbers can be characterized as a con-ditional field where every bounded subset has an infimum and a supremum and which is topologicallyconditionally separable. In particular, Q is conditionally dense in R . Remark 4.3.
In our context of an algebra of events, the conditional real line R corresponds exactly to theconditional set of random variables L ( A ) endowed with the L -topology introduced in [21], as shown in[14]. Therefore, in the following, the reader may always think of the conditional real line as being the setof A -measurable random variables. In particular, conditional numerical representations map conditionalpreferences to the almost sure order between A -measurable random variables. (cid:7) Definition 4.4. A conditional numerical representation of a conditional preference order (cid:60) on X is aconditional function U : X → R such that x (cid:60) y if and only if U ( x ) (cid:62) U ( y ) . (4.1)Note that every conditional function U : X → R defines a conditional preference order by means of(4.1). Furthermore, if U : X → R is a conditional numerical representation, then ϕ ◦ U is a conditionalnumerical representation for every conditionally strictly increasing function ϕ : R → R . Remark 4.5.
The conditional entropic monetary utility function studied for instance in [24] as a specialcase of a conditional certainty equivalent and given by U ( x ) = ln (cid:0) E (cid:2) e x (cid:12)(cid:12) A (cid:3)(cid:1) , x ∈ L ( B ) , is a representation of a conditional preference. Indeed, this function is local since for every A ∈ A itholds U ( x | A + y | A c ) = ln (cid:0) E (cid:2) e A x +1 Ac y (cid:12)(cid:12) A (cid:3)(cid:1) = ln (cid:0) A E (cid:2) e x (cid:12)(cid:12) A (cid:3) + 1 A c E (cid:2) e y (cid:12)(cid:12) A (cid:3)(cid:1) = 1 A ln (cid:0) E (cid:2) e x (cid:12)(cid:12) A (cid:3)(cid:1) + 1 A c ln (cid:0) E (cid:2) e y (cid:12)(cid:12) A (cid:3)(cid:1) = U ( x ) | A + U ( y ) | A c . The same argumentation holds for all conditional certainty equivalents, conditional/dynamic risk mea-sures or acceptability indices mentioned in the introduction. (cid:7)
Given a conditional preference order, we address the necessary and sufficient conditions under which aconditional numerical representation exists. The first result is a conditional version of Debreu’s statementin [9] and necessitates the notion of conditionally order dense. A conditional subset Z (cid:118) X is condition-ally order dense if for every x, y ∈ X with x (cid:31) y , there exists z ∈ Z such that x (cid:60) z (cid:60) y . The caseof interest is when Z is conditionally countable, that is, there exists a conditional injection ϕ : Z → Q .10quivalently, Z is conditionally countable if it is a conditional sequence Z = ( z n ) n ∈ N where N is theconditional natural numbers. There exists a difference between a conditional sequence and a standardsequence: Analogous to the classical case, a conditional sequence ( z n ) in Z is a conditional function f : N → Z , n (cid:55)→ f ( n ) = z n . However stability yields z n | A + z m | A c = f ( n ) | A + f ( m ) | A c = f ( n | A + m | A c ) = z n | A + m | A c . In other words, the sequence step n conditioned on A and the sequencestep m conditioned on A c result into the sequence step n | A + m | A c . Theorem 4.6.
A conditional preference order (cid:60) on X admits a conditional numerical representation ifand only if X has a conditionally countable order dense subset.Proof. The if-part:
Without loss of generality, assume Z = { z n : n ∈ N } is a conditionally countableorder dense subset of X which is not conditionally finite. Consider now Z + ( x ) := { z ∈ Z : z (cid:31) x } and Z − ( x ) := { z ∈ Z : x (cid:31) z } . Since (cid:60) is a conditional binary relation, Z + ( x ) and Z − ( x ) are conditional subsets of Z for every x ∈ X .However, as mentioned in Example 3.3, Z ± ( x ) may both live on some event smaller than Ω . Further, ( Z ± ( x )) x ∈ X is a conditional family in the conditional powerset P ( Z ) , that is, Z ± ( x ) = Z ± ( x ) | A + Z ± ( x ) | A c for every x = x | A + x | A c ∈ X . Due to transitivity, x (cid:60) y implies Z + ( x ) (cid:118) Z + ( y ) and Z − ( y ) (cid:118) Z − ( x ) . (4.2)It follows from conditional order denseness that x (cid:31) y implies that there is z ∈ Z such that x (cid:31) z (cid:60) y on some event A and x (cid:60) z (cid:31) y on A c . Thus z | A ∈ (cid:2) Z − ( x ) \ Z − ( y ) (cid:3) | A and z | A c ∈ (cid:2) Z + ( y ) \ Z + ( x ) (cid:3) | A c for some z ∈ Z and A ∈ A . (4.3)Let now µ be a strictly positive conditional measure on Z , that is, µ ( { z n } ) > for every n ∈ N .Define then U ( x ) = µ ( Z − ( x )) − µ ( Z + ( x )) for every x ∈ X . Then U is a conditional function since ( Z ± ( x )) x ∈ X is a conditional family and µ is a conditional function. On the one hand, from (4.2) and µ being conditionally increasing it follows that x (cid:60) y implies U ( x ) (cid:62) U ( y ) . On the other hand, assumethat x (cid:31) y on some non-empty event A . Without loss of generality, A = Ω . Then from (4.3) and µ beingstrictly positive it follows that x (cid:31) y yields U ( x ) = µ ( Z − ( x )) − µ ( Z + ( x )) (cid:62) (cid:2) µ ( { z } ) + µ ( Z − ( y )) (cid:3) | A − (cid:2) µ ( Z + ( y )) − µ ( { z } ) (cid:3) | A c = µ ( { z } ) + U ( y ) > U ( y ) . From the conditional completeness of (cid:60) it follows that U is a conditional numerical representation. The only if-part:
A conditional preference order which admits a conditional numerical representationis conditionally complete since the conditional reals are so. It holds that Y := Im ( U ) is a conditional sub-set of R . Choose a conditionally countable order dense subset I (cid:118) Y by Lemma 7.1. Then Z := U − ( I ) is conditionally countable and since U is a conditional numerical representation, it is conditionally orderdense. (cid:3) Let A be the event on which Z + ( x ) lives. It means that there exists no z ∈ Z such that z is strictly preferred to x conditionedon A c . Since Z is conditionally order dense, it follows that x is a maximal element conditioned on A c . For instance, define µ ( { z n } ) = 2 − n := (cid:80) − n k | A k for every n = (cid:80) n k | A k ∈ N . Definition 4.7.
Let (cid:60) be a conditional preference order on a conditional topological space X . We saythat (cid:60) is conditionally upper semi-continuous if U ( x ) := { y ∈ X : y (cid:60) x } is conditionally closed forevery x ∈ X . A conditional numerical representation U : X → R is said to be conditionally uppersemi-continuous if { x ∈ X : U ( x ) (cid:62) m } is conditionally closed for every m ∈ R . Theorem 4.8.
Let (cid:60) be a conditionally upper semi-continuous preference order on a conditionally sec-ond countable topological space X . Then (cid:60) admits a conditionally upper semi-continuous numericalrepresentation. In particular, if (cid:60) is conditionally continuous , then it admits a conditionally continuousnumerical representation. Example 4.9.
The conditional set L ( B ) of Example 2.3 is a typical framework in which Rader’s The-orem applies. Indeed, as soon as B is regular enough , then it follows that L ( B ) is a conditionallysecond countable Banach space for the conditional norm (cid:107) x (cid:107) = E [ | x | |A ] . Therefore, any conditionallyupper semi-continuous conditional preference order (cid:60) on the set of random outcomes in a year from nowconditioned on the information tomorrow admits a numerical representation U such that x (cid:60) y if and only if U ( x ) (cid:62) U ( y ) almost surely. ♦
5. A Conditional von Neumann-Morgenstern Representation
A classical class of preferences are the affine ones and the resulting affine numerical representation dueto von Neumann and Morgenstern [41]. This representation can be carried over to the conditional caseas follows. Let X be a conditionally convex subset living on of some conditional vector space. We saythat a conditional preference order on X satisfies the• conditional independence axiom: if x (cid:31) y then αx + (1 − α ) z (cid:31) αy + (1 − α ) z for every z ∈ X and each α ∈ ]0 , ;• conditional Archimedean axiom: if x (cid:31) y (cid:31) z then αx + (1 − α ) z (cid:31) y (cid:31) βx + (1 − β ) z forsome α, β ∈ ]0 , .A conditional real-valued numerical representation U of (cid:60) is conditionally affine , if U ( αx + (1 − α ) y ) = αU ( x ) + (1 − α ) U ( y ) , for every x, y ∈ X and each α ∈ [0 , . A conditional topology is the counterpart to a classical topology but with respect to the conditional operations of union andintersection, see [14]. The conditional topology of which is generated by a conditionally countable neighborhood base. That is, U ( x ) = { y ∈ X : y (cid:60) x } and U ( y ) = { y ∈ X : x (cid:60) y } are conditionally closed for every x ∈ X . That is, a separable σ -algebra. heorem 5.1. Let (cid:60) be a conditional preference order satisfying both the conditional Archimedean andindependence axioms. Then (cid:60) admits a conditionally affine representation U . Moreover, if ˆ U is anotherconditionally affine representation, then ˆ U = αU + β , where α > and β ∈ R . The result of Neumann and Morgenstern goes a step forward by providing a utility index against whichlotteries are ranked according to expectation. In our context, following Example 2.4, conditional lotterieson the real line is the conditional set P ( A ) := { µ : Ω → P such that µ is measurable } where P denotes a set of deterministic lotteries on the real line. We endow this conditional set with theconditional weak ∗ topology generated by the conditional set of bounded functions C b ( A ) := { f : Ω → C b such that f is measurable } where C b is the set of continuous functions from the real line into the real line. In other words, a function u ∈ C b ( A ) is a state-dependent continuous function u ( ω, x ) , that is, a state-dependent utility index. Theconditional scalar product is given by the random variable ω (cid:55)−→ (cid:104) f, µ (cid:105) ( ω ) = (cid:90) R f ( ω, x ) µ ( ω, dx ) , for almost all ω ∈ Ω which is an element of R = L ( A ) . In this framework, the classical representation theorem of vonNeumann and Morgenstern carries over as follows. Theorem 5.2.
Let (cid:60) be a conditional preference order on the conditional convex set of lotteries P ( A ) .Suppose that (cid:60) fulfills the conditional independence and Archimedean axioms and (cid:60) is weak ∗ -continuous.Then there exists a unique, up to strictly positive conditionally affine transformation, conditional utilityfunction u ∈ C ( A ) such that µ (cid:60) ν if and only if (cid:90) u ( ω, x ) µ ( ω, dx ) ≥ (cid:90) u ( ω, x ) ν ( ω, dx ) for almost all ω ∈ Ω . Proof.
This is a consequence of Theorem 5.1 together with a conditional version of the Riez theorem, see[26]. (cid:3)
6. Further Conditional Representations
As in the classical case, the assumptions of Theorem 4.8 are empirically as well as mathematically prob-lematic for the following reasons• Many of the topologies of interest for practical representations are not second countable, not evenmetrizable;• Requiring upper semi-continuity is an empirical issue in particular for non-metrizable topologiessince it is not practically falsifiable. These are related to conditional distribution on the conditional real line R , see Jamneshan et al. [26] for the construction anddefinition of such conditional probability distributions.
13n answer to the first point is the following proposition that relies on a conditional version of Banach-Alaoglu based on the notion of conditional compactness introduced in [14].
Proposition 6.1.
Let Y be a conditionally separable locally convex topological vector space admitting aconditionally countable neighborhood base of , and denote by X its topological dual endowed with theconditional weak ∗ topology σ ( X, Y ) . Then every conditionally upper semi-continuous preference order (cid:60) on X admits an upper semi-continuous numerical representation. Example 6.2.
Suppose that we are interested into the decision making of an agent according to tomor-row’s information A between random outcomes in a year from now, that is measurable with respect tosome wider information algebra B ⊇ A . Suppose further that these random outcomes are bounded, thatis, belong to the following conditional set L ∞ ( B ) := { x : Ω → R such that x is measurable and bounded by an A -measurable random variable } . From [22] it is known that it is the conditional dual of L ( B ) introduced in Example 2.3 which is con-ditionally separable provided that B is regular enough. It follows that if (cid:60) is σ ( L ∞ ( B ) , L ( B )) -uppersemi-continuous, then it admits a numerical representation U . If furthermore, (cid:60) is• conditionally convex : x (cid:60) y implies αx + (1 − α ) y (cid:60) y for every conditional real number (cid:54) α (cid:54) which describes a form of preference of diversification;• monotone : x (cid:60) y whenever x (cid:54) y , which describes a preference for almost sure better outcomes;then, it follows that U is conditionally quasiconvex and monotone and by means of [5, 13], and theconditional extension in [4, Theorem 2.12 and Remark 2.13], it admits the following robust representation U ( x ) = inf Q ∈ ∆ R ( Q, E Q [ x |A ]) for a unique conditional risk function R : ∆ × R → [ −∞ , ∞ ] where ∆ is the set of probabilitymeasures absolutely continuous with respect to the reference measure. ♦ As for the second question, we show that automatic continuity results taking advantage of monotonicityalso extends to the continuous case and yields the following result.
Proposition 6.3.
Let (cid:60) be a conditionally complete preference order on a conditional Banach space X . Suppose that (cid:60) is conditionally convex and monotone and satisfies the Upper Archimedean Axiom: If x (cid:60) y (cid:31) z , then there exists α ∈ R with < α < such that y (cid:31) αx + (1 − α ) z .Then (cid:60) is conditionally upper semi-continuous. Let us illustrate this automatic continuity result in the framework presented in [4]. By x t , x t +1 , . . . , x T we denote a future cumulative cash-flow stream starting from a given future time t . We are interestedin the study of an investor’s assessment (cid:60) of these future cash-flows but conditioned on the availableinformation at this time A := A t . The information at time s is denoted by A s and it holds A s ⊆ A s +1 The characterization of which can be found in [4, Theorem 2.12 and Remark 2.13]. It also holds on a Fréchet lattice. s ≥ t . The cash-flow x s is adapted to A s and is square integrable, that is E [ | x s | |A t ] < ∞ .We denote by L ( A s , s ≥ t ) := (cid:8) x = ( x t , . . . , x T ) : x s ∈ L ( A s ) , s = t, . . . , T (cid:9) which is a conditionally reflexive Hilbert space, see [14, 22]. We denote by• ∆ the set of probability measures Q absolutely continuous with respect to the reference measure;• D the set of discounting factors, that is those processes D = ( D t , . . . , D T ) where D t (cid:62) D t +1 (cid:62) . . . (cid:62) D T (cid:62) where D s is A s − adapted.It follows that E Q (cid:34) T (cid:88) k = t D k ( x k − x k − ) (cid:12)(cid:12) A t (cid:35) is the expected discounted value of the cash-flow stream x k − x k − for the discount factor D ∈ D underthe probability model Q ∈ ∆ . For reasons discussed in [4], we denote by Q ⊗ D ∈ ∆ ⊗ D the set ofthose Q, D with some L -integrability conditions. Proposition 6.4.
Let (cid:60) be a conditional preference order on L ( A s , s ≥ t ) . Suppose that it is• convex: x (cid:60) y implies αx + (1 − α ) y (cid:60) y for every conditional real number (cid:54) α (cid:54) ;• monotone: x (cid:60) y whenever x s (cid:62) y s for every s ≥ t ;• upper Archimedean: if x (cid:60) y (cid:31) z , then there exists some conditional real number < α < suchthat y (cid:31) αx + (1 − α ) z .Then, (cid:60) is upper semi-continuous and admits an upper-semi continuous numerical representation U withrobust representation U ( x ) = inf Q ⊗ D ∈ ∆ ⊗D R (cid:32) Q ⊗ D ; E Q (cid:34) T (cid:88) k = t D k ( x k − x k − ) (cid:12)(cid:12) A t (cid:35)(cid:33) (6.1) for a unique minimal risk function R : ∆ ⊗ D × R → [ −∞ , ∞ ] . This representation shows that convex and monotone conditional assessment of future cash flows excerptsa prudent assessment of probability as well as discounting model uncertainty. If we can ensure the ex-istence of an upper semi-continuous, quasiconvex, conditional and monotone numerical representation U , then existence and uniqueness of Representation 6.1 is a consequence of [4, Theorem 3.4]. How-ever, L ( A s , s ≥ t ) being a Banach space, it follows that the assumption of the propositions fulfills theassumptions of Proposition 6.3 and so we obtain the existence of an upper semi-continuous numericalrepresentation.
7. Conditional Gap Lemma
For s, t ∈ R with s (cid:54) t we denote [ s, t ] = { u ∈ R : s (cid:54) u (cid:54) t } and if s < t we denote [ s, t [= { u ∈ R : s (cid:54) u < t } , ] s, t ] = { u ∈ R : s < u (cid:54) t } , and ] s, t [= { u ∈ R : s < u < t } , all conditionallyconvex subsets of R which live on Ω . In the conditional topology of R the conditionally convex subset15 s, t ] is conditionally closed whereas ] s, t [ is conditionally open. A conditionally convex subset I (cid:118) R is an interval . Denoting by s = inf I and t = sup I , an interval is generically denoted by ( s, t ) . Upto conditioning, all conditionally convex subsets of R are characterized as conditional intervals. If weassume that an interval lives on Ω , convexity yields ( s, t ) = [ s, t ] | A + [ s, t [ | B +] s, t ] | C +] s, t [ | D (7.1)where A = E ∩ F , B = E ∩ F c , C = E c ∩ F , D = E c ∪ F c and E = ∪{ ˜ E : s | ˜ E ∈ I | ˜ E } and F = ∪{ ˜ F : t | ˜ F ∈ I | ˜ F } , as illustrated in Figure 1.Figure 1: Illustration of a Gap.Let S (cid:118) R and ( s, t ) be an interval such that ( s, t ) (cid:118) S (cid:64) . Inspection shows that there exists a uniquemaximal interval ( s ∗ , t ∗ ) with respect to conditional inclusion such that ( s, t ) (cid:118) ( s ∗ , t ∗ ) (cid:118) S (cid:64) . Such amaximal interval with inf S (cid:64) < s ∗ (cid:54) t ∗ < sup S (cid:64) on the conditions where s ∗ , t ∗ are living is called a conditional gap of S . Note that any conditional gap ( s, t ) of S can be decomposed into ( s, t ) = { s }| A + ( s, t ) | B where A ∩ B = ∅ and s < t on B , that is the conditional interior of ( s, t ) lives on B . Moreover, thefamily of conditional gaps of S is itself stable and therefore each of the conditional gaps of S lives on thesame condition. Indeed, suppose that two conditional gaps ( s , t ) and ( s , t ) live on events A and B ,respectively, and A ⊆ B with A (cid:54) = B . Then it follows that ( s , t ) = ( s , t ) | A (cid:118) ( s , t ) | A + ( s , t ) | B ∩ A c (cid:118) S (cid:64) contradicting the maximality of ( s , t ) . Hence A = B . It is possible that s, t attain ±∞ on some positive condition. May live on an event A strictly smaller than Ω . emma 7.1. The conditionally complete order (cid:62) restricted to any S (cid:118) R living on Ω admits a condi-tionally countable order dense subset.Proof. Analogous to conditional gaps, we define a predecessor-successor as a maximal interval ( s, t ) (cid:118) S (cid:64) but under the additional requirement that s < t . In other words, these are conditionally maximalnon-trivial conditional gaps. Alike conditional gaps, predecessor-successor pairs of S form a conditionalfamily and therefore all live on the same condition. This condition is per definition smaller than the oneon which the conditional gaps live.Now, up to conditioning, we may assume that the conditional gaps of S are all living on Ω . Since B /m ( q ) for m ∈ N and q ∈ Q is a conditionally countable base of the topology, the family of thoseintersections B /m ( q ) (cid:117) S living on Ω is a conditionally countable family which we denote ( U n ) . Bymeans of the conditional axiom of choice, see [14, Theorem 2.17], there exists a conditionally countablefamily ( u n ) such that u n ∈ U n for all n . Let further A be the condition on which the conditional family ( s i , t i ) of the predecessors-successors of S is living. It follows that U = ( u n ) (cid:116) ( s i ) is a conditionallycountable order dense subset of S living on Ω . Indeed, let s < t for s, t ∈ S and B the condition onwhich ( s, t ) is a predecessor-successor pair, that is, the maximal condition such that s is an element of ( s i ) . It follows that there exists v ∈ S such that s < v < t on B c . Hence, we may find q ∈ Q and n ∈ N such that s < q − /m < v < q + 1 /m < t on B c which ensures the existence of some u n in the family ( u n ) such that s < u n < t on B c . It follows that u = s | B + u n | B c ∈ U and s (cid:54) u (cid:54) t .We are then left to show that U is conditionally countable. Since ( u n ) is conditionally countable,according to [14, Lemma 2.33], it is enough to show that the conditional family of open sets ] s i , t i [=] s i , t i [ i ∈ I , where ( s i , t i ) is the conditional collection of predecessor-successors, is conditionally count-able. Without loss of generality, suppose that this family lives on Ω . For any two ] s i , t i [ and ] s j , t j [ suchthat s i (cid:54) = s j on any non-empty condition, it follows that ] s i , t i [ (cid:117) ] s j , t j [= R |∅ . This provides a condi-tionally pairwise disjoint family of conditionally open sets on Ω . By means of the conditional axiom ofchoice, [14, Theorem 2.17], we choose a conditional family ( q i ) of elements of Q such that q i ∈ ] s i , t i [ for every i . For P = (cid:116){ q i } (cid:118) Q , define f : I → P , i (cid:55)→ q i . This function is a well-defined conditionalfunction. Indeed, for q i = q j , it follows that q i ∈ ] s i , t i [ (cid:117) ] s j , t j [ . Both being conditional gaps of S , thisimplies that ] s i , t i [=] s j , t j [ and therefore i = j . This also shows that f is a conditional injection, thus I is at most conditionally countable. (cid:3) Theorem 7.2 (Debreu’s Gap Lemma).
For every S (cid:118) R there exists a conditionally strictly increasingfunction g : S → R such that all the conditional gaps ( s, t ) of g ( S ) are of the form ( s, t ) = { s }| A +] s, t [ | B. This theorem says that there exists a strictly increasing transformation of S such that any conditional gapwhich is of the form (7.1) is transformed in a gap which conditionally is either empty, a singleton or anopen set. The following argumentation follows the proof idea in [33]. Proof. Step 1:
According to Lemma 7.1, let U = { u n : n ∈ N } be a conditionally countable orderdense subset of S . We construct a conditionally increasing function f : U → [0 , . Let H be the set of On the condition where none of the gaps lives, it holds S = R for which Q is a conditionally countable order dense subset. f : V → [0 , , where V = { u k : 1 (cid:54) k (cid:54) n } , n ∈ N or V = U and such that f ( u ) = 1 / ,f ( u k ) = sup l (cid:54) k − { f ( u l ) : u l < u k } + inf l (cid:54) k − { f ( u l ) : u k < u l } , k (cid:62) . By definition, any f ∈ H is conditionally strictly increasing on its domain and H is a conditional set.Furthermore f : { u } → [0 , with f ( u ) = 1 / is an element of H so that H lives on Ω . We show thatthere exists a function f ∈ H with domain U . For f : V → [0 , and g : W → [0 , in H , define f (cid:52) g if V (cid:118) W and f = g restricted on V . Let now ( f i ) be a chain in H and define f : V := (cid:116) V i → [0 , , u = (cid:80) u j | A j (cid:55)→ f ( u ) = (cid:80) f j ( u j ) | A j where u j ∈ V j for every j is a well-defined conditional functionin H . Indeed, H is a conditional set, the f i are restrictions of each others and V i (cid:118) V j if f i (cid:52) f j . ByZorn’s Lemma, there exists a maximal function f ∈ H , f : V → [0 , . Next we show that V = U . Forthe sake of contradiction, suppose that V = { u k : 1 (cid:54) k (cid:54) n } for some n ∈ N on some non-trivial event A . Without loss of generality, assume that A = Ω . Define g : { u k : 1 (cid:54) k (cid:54) n + 1 } → [0 , by setting g = f on { u k : 1 (cid:54) k (cid:54) n } and g ( u n +1 ) = sup l (cid:54) n { f ( u l ) : u l < u n +1 } + inf l (cid:54) n { f ( u l ) : u n +1 < u l } As for those n (cid:54) m (cid:54) n + 1 , it follows that m = n | A + ( n + 1) | A c and we set g ( u m ) = g ( u n ) | A + g ( u n +1 ) | A c . By construction, g : { u k : 1 (cid:54) k (cid:54) n + 1 } → [0 , is an element of H which coincides on V with f . Since g is defined on V (cid:116) { u n +1 } it contradicts the maximality of f . Thus the domain of themaximal function f ∈ H is U . Step 2:
Let U = { u n : n ∈ N } and f : U → [0 , as defined in the previous step. Suppose that V, W (cid:118) U living on Ω satisfy(a) V (cid:116) W = U ,(b) V (cid:54) W , (c) ∅ is the unique condition on which V and W have both a maximum and a minimum, respectively.Then sup s ∈ V f ( s ) = inf t ∈ W f ( t ) . By (b) it holds sup V f ( s ) (cid:54) inf W f ( t ) . In order to show the reverse inequality, according to (c), it issufficient to suppose that V and W have both nowhere a maximum and a minimum, respectively, sincethen the gap is the largest. For the sake of contradiction, up to conditioning, suppose that sup V f ( s ) + ε < inf W f ( t ) for some ε > . Choose s = u m ∈ V and t = u n ∈ W such that sup V f ( s ) − ε (cid:54) f ( s ) (cid:54) sup V f ( s ) and inf W f ( t ) (cid:54) f ( t ) (cid:54) inf W f ( t ) + ε. (7.2) With the classical convention that the infimum and supremum over emptyset is equal to and , respectively. That is, f ( u k ) =inf l (cid:54) k − f ( u l ) / on the condition where u k < u l for every l (cid:54) k − and f ( u k ) = (sup l (cid:54) k − f ( u l ) + 1) / on thecondition where u l < u k for every k − (cid:54) l . In the sense that for all s ∈ V and for all t ∈ W it holds s (cid:54) t . V has nowhere a maximum, there exists u k ∈ V such that u m < u k < u n and k > n, m . Let k (cid:48) = min { k > n, m : u n < u k < u m } . By construction of f and since (a) holds, it follows that f ( u k (cid:48) ) = f ( u n ) + f ( u m )2 = f ( s ) + f ( t )2 . Adding both inequalities in (7.2) yields sup V f ( s ) + inf W f ( t )2 − ε (cid:54) f ( u k (cid:48) ) (cid:54) sup V f ( s ) + inf W f ( t )2 + ε , and therefore sup V f ( s ) < f ( u k (cid:48) ) < inf W f ( t ) contradicting u k (cid:48) ∈ V . Step 3:
Define g : S → R by g ( s ) = sup u ∈ U,u (cid:54) s f ( u ) . By construction, g is a conditionally strictlyincreasing extension of f since U is a conditionally countable order dense subset of S . Let ( s, t ) be aconditional gap of g ( S ) and A be the maximal event such that s < t , that is ( s, t ) = ( s, t ) | A + { s }| B .Without loss of generality, suppose that A = Ω and B = ∅ . Define V = { u ∈ U : f ( u ) (cid:54) s } and W = { u ∈ U : f ( u ) (cid:62) t } . Since s < t and V , W satisfy (a) and (b) of the previous step, (c) has to beviolated. Hence V and W have both a conditional maximum and minimum respectively on some maximalnon-empty event C , that is, s = f ( u n ) and t = f ( u m ) on C for some n, m . Thus s | C, t | C ∈ g ( S ) | C showing that ( s, t ) | C =] s, t [ | C . If C is non equal to Ω , we follow the same argumentation but conditionedon C c which yields a contradiction with the maximality of C . Therefore, ( s, t ) =] s, t [ which ends theproof. (cid:3) Theorem 7.3.
Any numerically representable conditionally upper semi-continuous preference order ad-mits a conditionally upper semi-continuous numerical representation.Proof.
Let ˜ U be a numerical representation of a conditionally upper semicontinuous preference order (cid:60) .According to Debreu’s Gap Lemma 7.2 there exists a conditional function g : Im ( ˜ U ) → R such that allthe conditional gaps ( s, t ) of g ( Im ( ˜ U )) are of the following form ( s, t ) = s | A +] s, t [ | B, for s < t. Since g is strictly increasing, it follows that U = g ◦ ˜ U is a conditional numerical representation of (cid:60) as well. Clearly Im ( U ) = g ( Im ( ˜ U )) . In order to verify the upper semi-continuity, pick m ∈ R . Wedistinguish between the following cases:• If m = U ( y ) , then { x ∈ X : U ( x ) (cid:62) m } = { x ∈ X : U ( x ) (cid:62) U ( y ) } = { x ∈ X : x (cid:60) y } whichis conditionally closed by assumption.• If m ∈ ] s, t [ where ] s, t [ is a conditional gap of Im ( U ) , then t = U ( y ) for some y ∈ X , and thus { x ∈ X : U ( x ) (cid:62) m } = { x ∈ X : U ( x ) (cid:62) t } = { x ∈ X : U ( x ) (cid:62) U ( y ) } = { x ∈ X : x (cid:60) y } which is also conditionally closed by assumption.• If m = s where { s } is a conditional gap of Im ( U ) , then let ( s n ) = ( U ( y n )) (cid:118) Im ( U ) be aconditional sequence such that s n (cid:37) s . It holds { x ∈ X : U ( x ) (cid:62) s } = (cid:117) n { x ∈ X : U ( x ) (cid:62) s n } = (cid:117) n { x ∈ X : U ( x ) (cid:62) U ( y n ) } = (cid:117) n { x ∈ X : x (cid:60) y n } which is conditionally closed as theconditional intersection of closed sets.Since R = Im ( U ) (cid:116) [ Im ( U )] (cid:64) and [ Im ( U )] (cid:64) is made of gaps of the form ( s, t ) = { s }| A +] s, t [ | B , itfollows that any m ∈ R belongs conditionally to one of the previous three cases. Thus U is conditionallyupper semi-continuous. (cid:3) . Technical Proofs A.1. Proof of Theorem 4.8.
Proof (Proof of Theorem 4.8).
Let O = ( O n ) n ∈ N be a conditionally countable topological base of X and µ be a strictly positive measure on N . We know that Z ( x ) := U ( x ) (cid:64) is conditionally open for every x ∈ X . Fix some x ∈ X and let A be the event on which lives Z ( x ) . Then { n ∈ N : O n | A (cid:118) Z ( x ) } isa conditional subset of N . Next define U ( x ) = µ ( { n ∈ N : O n | A (cid:118) Z ( x ) } ) | A + 0 | A c . If x (cid:60) y , then U ( x ) (cid:62) U ( y ) since Z ( y ) (cid:118) Z ( x ) . Otherwise if x (cid:31) y , then y ∈ Z ( x ) . Since Z ( x ) is conditionally open, there exists a neighborhood O i of y such that O i (cid:118) Z ( x ) . However, since y ∈ Z ( y ) (cid:64) (cid:117) O i , it follows that O i is nowhere a subset of Z ( y ) . Hence, U ( x ) (cid:62) U ( y ) + µ ( { i } ) > U ( y ) .By Theorem 7.3 we can choose U to be conditionally upper semi-continuous which ends the proof. (cid:3) A.2. Proof of Theorem 5.1.
We will follow the classical proof adapted to the conditional setting.
Lemma A.1.
Let (cid:60) be a conditionally complete preference order satisfying both the conditional inde-pendence and Archimedean axioms. Then the following assertions hold:(i) If x (cid:31) y , then βx + (1 − β ) y (cid:31) αx + (1 − α ) y for all (cid:54) α < β (cid:54) .(ii) If x (cid:31) z and x (cid:60) y (cid:60) z , then there exists a unique α ∈ [0 , with y ∼ αx + (1 − α ) z .(iii) If x ∼ y , then αx + (1 − α ) z ∼ αy + (1 − α ) z for all α ∈ [0 , and all z ∈ M .Proof. (i) Strictly analogous to the classical proof, see for instance [23, p. 54].(ii) Up to conditioning, we may assume that x (cid:31) y (cid:31) z . The candidate is α := sup { β ∈ [0 ,
1] : y (cid:60) βx + (1 − β ) z } We obtain a partition
A, B, C of Ω such that y ∼ αx + (1 − α ) z on A , y (cid:31) αx + (1 − α ) z on B and αx +(1 − α ) z (cid:31) y on C . Conditioned on B and C respectively, we may apply the classical argumentation,see for instance [23, p. 54] yielding a contradiction showing that B = C = ∅ and therefore A = Ω whichends the proof. As for the uniqueness, this is a consequence of the first point.(iii) Let α ∈ [0 , and z ∈ X . There exists a partition of A, B, C of Ω such that αx + (1 − α ) z ∼ αy + (1 − α ) z on A , αx + (1 − α ) z (cid:31) αy + (1 − α ) z on B and αx + (1 − α ) z (cid:31) αy + (1 − α ) z on C . The same contradiction argumentation as in the classical case, [23, p. 54–55], conditioned on B and C , respectively, shows that B = C = ∅ and therefore A = Ω . (cid:3) Proof (Theorem 5.1).
Let x, y ∈ X be such that x (cid:31) y and define the conditional convex subset N x,y := { z ∈ X : x (cid:60) z (cid:60) y } . For z ∈ N x,y , part (ii) of Lemma A.1 yields a unique α ∈ [0 , such that z ∼ αx + (1 − α ) y . Setting U ( z ) := α , z ∈ N x,y provides a well-defined conditional function from N x,y to [0 , . Indeed, let [ a i , z i ] ⊆ A × N x,y , and denote α i = U ( z i ) and α = U ( (cid:80) z i | A i ) . There On the condition where x ∼ y set α = 1 and on the condition where y ∼ z , set α = 0 . A, B, C of Ω such that α = (cid:80) α i | A i on A , α > (cid:80) α i | A i on B and (cid:80) α i | A i > α on C . In particular, α > α i on B ∩ A i and α i > α on C ∩ A i for every i . Hence, if either B or C werenon-empty events, this would contradict the uniqueness of some α i on B ∩ A i (cid:54) = ∅ or C ∩ A i (cid:54) = ∅ . Hence B = C = ∅ showing that A = Ω . The extension to X follows exactly the same argumentation as theclassical case, see [23, p. 55]. (cid:3) A.3. Proof of Proposition 6.1.
In a conditional topological space X with conditional topological dual X ∗ , the conditional absolute polarof a set O (cid:118) X living on Ω is given by O • = { x ∗ ∈ X ∗ : |(cid:104) x ∗ , x (cid:105)| (cid:54) for all x ∈ O } Lemma A.2.
Let X be a conditional topological vector space with conditional dual X ∗ and O a condi-tional base of neighborhoods of in X . Then X ∗ = (cid:71) O ∈O O • . Proof.
Let x ∗ ∈ X ∗ , then V = [ x ∗ ] − ([ − , is a conditional neighborhood of . In particular, x ∗ ∈ V • . Choose O ∈ O such that O (cid:118) V . Then V • (cid:118) O • , and thus x ∗ ∈ O • . The reciprocal is immediatesince O • (cid:118) X ∗ . (cid:3) Proposition A.3.
Let X be a locally convex conditional topological vector space X which is condition-ally separable. Relative to any conditionally σ ( X ∗ , X ) -compact subset C (cid:118) X , the σ ( X ∗ , X ) -topologyis conditionally metrizable.Proof. Without loss of generality, by the conditional version of the Banach-Alaoglu Theorem [14] andthe previous lemma, we may assume that C = O • for some conditional neighborhood O of in X .First, we construct a conditional distance on O • as follows. Let ( x n ) (cid:118) O be a conditionally densesequence in O and define d : O • × O • → R + by d ( x ∗ , y ∗ ) = (cid:88) n ∈ N n |(cid:104) x ∗ − y ∗ , x n (cid:105)| |(cid:104) x ∗ − y ∗ , x n (cid:105)| , x ∗ , y ∗ ∈ O • . (A.1)Straightforward inspection shows that it is a well-defined conditional function and a translation invariantdistance on O • . Indeed, as a locally convex conditional topological vector space, X separates the pointsof X ∗ and a fortiori those of O • . Furthermore, { x ∗ ∈ X ∗ : |(cid:104) x ∗ , x k (cid:105)| < r, (cid:54) k (cid:54) n } (cid:118) (cid:8) x ∗ ∈ X ∗ : d (0 , x ∗ ) < r + 2 − n +1 (cid:9) = B r +2 − n +1 (0) , for every n ∈ N and r > . This shows that the conditional topology generated by d on O • is weakerthan σ ( X ∗ , X ) , that is, τ d (cid:118) σ ( X ∗ , X ) . Second, we show that these topologies coincide. To this end, we consider the identity map Id :( O • , σ ( X ∗ , X )) → ( O • , d ) which is a bijection. Let ( x ∗ α ) (cid:118) O • be a conditional net converging in Note that the construction of such a metric can be done similarly on X ∗ by considering a conditionally dense sequence ( x n ) ofelements in X , and the topology induced by d on X ∗ is therefore weaker that σ ( X ∗ , X ) . ( X ∗ , X ) to x ∗ ∈ O • . For r > , choose k ∈ N such that (cid:80) n>k − n < r . Since x ∗ α , x ∗ ∈ O • and x n ∈ O , it follows that |(cid:104) x ∗ α − x ∗ , x n (cid:105)| (cid:54) for every n ∈ N . Hence, d ( x α , x ) (cid:54) (cid:88) (cid:54) n (cid:54) k |(cid:104) x ∗ α − x ∗ , x n (cid:105)| + 2 r (A.2)Since |(cid:104) x ∗ α − x ∗ , x n (cid:105)| → for every n ∈ N , it follows that lim sup d ( x α , x ) (cid:54) r for every r > .This shows that Id is continuous. Now, ( O • , σ ( X ∗ , X )) is conditionally compact due to the conditionalversion of Banach-Alaoglu and ( C, d ) is conditionally Hausdorff, it follows that Id is conditionally bi-continuous. Hence, V ∈ τ d for every V ∈ σ ( X ∗ , X ) showing that τ d = σ ( X ∗ , X ) relative to O • . (cid:3) Proof (of Proposition 6.1).
Denoting by ( O n ) the conditional countable neighborhood of in X , de-fine (cid:60) n as the conditional restriction to O • n of (cid:60) . Clearly, (cid:60) n is conditionally σ ( X ∗ , X ) -upper semi-continuous for every n . Furthermore, by the conditional version of Banach-Alaoglu, O • n is σ ( X ∗ , X ) -compact. By Proposition A.3, it follows that O • n is conditionally metrizable and compact, hence condi-tionally second countable. Theorem 4.8 implies that (cid:60) n is representable and by Theorem 4.6, it admitsa conditionally countable order dense subset Z n (cid:118) O • n . By means of [14, Lemma 2.33], Z := (cid:116) Z n is a conditionally countable set. By means of Lemma A.2, straightforward inspection shows that Z is (cid:60) -conditionally order dense. Therefore, once again by means of Theorem 4.6, (cid:60) admits a conditionalnumerical representation. Theorem 7.3 guarantees that such a conditional numerical representation canbe chosen σ ( X ∗ , X ) -upper semi-continuous. (cid:3) A.4. Proof of the Automatic Continuity Result.
Proposition A.4.
Let X be a conditional Fréchet lattice and Z (cid:118) X be conditionally monotone andconvex. If f − ( Z ) is conditionally closed in [0 , for every given pair x, y ∈ X , where f : [0 , → X , α (cid:55)→ αx + (1 − α ) y , then Z is conditionally closed in X .Proof. Denote by d the conditional Fréchet distance on X , and let ( x n ) a conditional sequence of ele-ments in Z conditionally converging to x ∈ X . Up to a rapid conditional subsequence, we may supposethat d ( x n , x ) ≤ − n /n , n ∈ N . It follows that (cid:80) k (cid:62) k ( x k − x ) + is conditionally converging. Indeed,since the conditional Fréchet distance respects the conditional absolute value, it follows for n < md (cid:88) (cid:54) k (cid:54) n k ( x k − x ) + , (cid:88) (cid:54) k (cid:54) m k ( x k − x ) + (cid:54) d , (cid:88) n Fix an x ∈ X and denote by Z := U ( x ) . Then Z is conditionally convexand monotone since (cid:60) is so. We show that I := f − ( Z ) is conditionally closed in [0 , where f :[0 , → X , α (cid:55)→ αx + (1 − α ) y for any given x, y ∈ X . Up to conditioning, we may assume that I liveson Ω – in particular I is not conditionally empty . Since Z is conditionally convex and f conditionallyaffine, it follows that I is conditionally convex. Therefore, I is an interval ( s, t ) (cid:118) [0 , where s = inf I and t = sup I . If s = t , then I is a singleton and therefore is conditionally closed. Otherwise, let s < t without of loss of generality. Suppose now, for the sake of contradiction, s ∈ I (cid:64) . That is to say, αx + (1 − α ) x (cid:60) y (cid:31) sx + (1 − s ) x for every α ∈ ] s, t [ . The one-sided Archimedean axiom yields a β ∈ ]0 , such that y (cid:31) β ( αx + (1 − α ) z ) + (1 − β )( sx + (1 − s ) z ) = γx + (1 − γ ) z where γ = βα + (1 − β ) s . Since β > and s < α , it follows that γ > s contradicting the definition of s .Hence, sx + (1 − s ) z ∈ Z , and thus s ∈ ( s, t ) . 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