On a notion of independence proposed by Teddy Seidenfeld
aa r X i v : . [ c s . A I] F e b On a notion of independence proposed by TeddySeidenfeld
Jasper De Bock and Gert de Cooman
Abstract
Teddy Seidenfeld has been arguing for quite a long time that binary pref-erence models are not powerful enough to deal with a number of crucial aspectsof imprecision and indeterminacy in uncertain inference and decision making. Itis at his insistence that we initiated our study of so-called sets of desirable optionsets, which we have argued elsewhere provides an elegant and powerful approachto dealing with general, binary as well as non-binary, decision-making under uncer-tainty. We use this approach here to explore an interesting notion of irrelevance (andindependence), first suggested by Seidenfeld in an example intended as a criticism ofa number of specific decision methodologies based on (convex) binary preferences.We show that the consequences of making such an irrelevance or independence as-sessment are very strong, and might be used to argue for the use of so-called mixingchoice functions, and E-admissibility as the resulting decision scheme.
In much of our earlier work on the foundations of imprecise—or indeterminate[21]—probabilities [1, 33] we availed ourselves of binary preference orders betweenuncertain rewards to model a subject’s decisions under uncertainty; see [3, 7, 14, 16,19, 23] for a few representative examples. In the field, the monikers ‘desirability’and ‘sets of desirable gambles’ are typically used to describe uncertainty modelsinvolving such (strict) binary preference orders [2, 14, 19, 22, 34]. In earlier work,Seidenfeld et al. [24] also introduced the term ‘favourability’ for this.
Jasper De BockFoundations Lab for Imprecise Probabilities, Ghent University, Technologiepark-Zwijnaarde 125,9052 Belgium e-mail: [email protected]
Gert de CoomanFoundations Lab for Imprecise Probabilities, Ghent University, Technologiepark-Zwijnaarde 125,9052 Belgium e-mail: [email protected]
Since the publication of that work, Teddy Seidenfeld has been developing argu-ments in favour of a more involved approach to uncertainty modelling and decisionmaking. If we really want to take indecision and imprecision seriously, he has in-sisted, we need to abandon binary preference models in favour of more generalchoice functions, as for instance described in [26]. For this reason, one of us (Gert)started to work with Arthur Van Camp—his then PhD student—on exploring theconnections between choice function theory and desirability. This led to a numberof joint papers [30, 31, 32], and, eventually, to Arthur’s PhD Thesis [29].Inspired by that work, the two of us decided to explore this connection further.A key insight we had, is that choice functions, when interpreted appropriately, canexpress statements such as “ at least one of these preferences is true”. Since a desirablegamble is by definition a preference assessment—it is an uncertain reward that isstrictly preferred to the status quo—this suggests that choice functions can dealwith ‘OR’-statements between assessments of desirability. In contrast, the languageof sets of desirable gambles typically only deals with ‘AND’-statements betweensuch assessments. This observation led us to the intriguing idea that general choicefunctions might be interpreted, axiomatised and represented using the languageof desirability, and that at the same time, they could enrich this language with‘OR’-statements. Investigating this idea and confirming our suspicions in all thenecessary detail has been part of an ongoing project, with a number of papersnearing completion. Early versions, which the present discussion is based on, havebeen published in conference proceedings [4, 8, 10, 12], and more detailed versionswith proofs are also available on ArXiv [5, 9, 11, 13]. We will summarise the relevantideas and results in Section 2 further on.Our results so far have led us to agree with Seidenfeld’s criticism, and to followhim in moving from binary preference models to choice functions. But they havenot led us to abandon desirability. On the contrary: on our account, desirabilityis also very well suited for describing and interpreting non-binary choice. Thisinterpretation—that (not) choosing an option from an option set with more than twoelements can be brought back to an ‘OR’ of desirability statements—inspires a setof axioms which allows us to cover much—if not all—of the literature on the subjectthat we have come across.Simply moving towards general choice functions doesn’t immunise us against allaspects of Seidenfelds’s criticism, however. For it is not merely the systematic useof binary decision schemes that he has been arguing against, it is also—and perhapsforemost—some of their features, which may also be shared by some types of non-binary decision schemes. In fact, he has a treasure trove of intricate little examplesthat he likes to pick apart other people’s pet theories about rational decision makingwith. We have no doubt that some of them may still be brought to bear on specificdecision schemes within our desirability-based theory of choice functions. In oneof his examples, which he typically mobilises to cast doubt on the indiscriminateassumption of convexity for a set of indeterminate probabilities, he introduced enpassant a requirement for ‘independence’ that hasn’t stopped fascinating us since the How to also deal with ‘NOT’ in this and related languages, was studied in quite some detail byone of us in an earlier collaboration [23].n a notion of independence proposed by Teddy Seidenfeld 3 fateful day he sent us a few hand-outs explaining the basic ideas behind it. Whereasthe hand-outs are not publicly available, the main idea expressed in them is, becausehe and his colleagues have also published a similar example in [26, Section 4]. Itinvolves the following very intuitive ‘rationality requirement’ about the value of‘independent information’, namely, that it ought to have none:
When two events, 𝐸 and 𝐹 , are ‘independent’ then it is not reasonable to spend resources in order to use the state of one, 𝐸 versus 𝐸 c , to decide between two gambles that dependsolely on the other event, 𝐹 versus 𝐹 c . Rather than reproduce his specific example here in its full detail, we will rephrasehis requirement in a more general abstract form, and without the symmetry that isimplicit in his formulation. We will refer to this asymmetric version as (an assessmentof)
S-irrelevance . The main goal of this paper will be to study its implications.We consider a possibility space Ω , and two events 𝐸, 𝐹 ⊆ Ω . The event 𝐹 could for instance refer to a(n unknown) medical condition of a patient in a Brusselshospital, and the event 𝐸 could refer to (unknown) specific weather conditions at theSouth Pole. Gambles are uncertain rewards expressed in units of some predeterminedlinear utility, modelled as bounded maps 𝑓 : Ω → R . We denote the set of all suchmaps by ℒ ( Ω ) , or more simply by ℒ when it is clear from the context what thedomain of the gambles is.A gamble on the occurrence of 𝐹 is a gamble of the type 𝜆 I 𝐹 + 𝜇 I 𝐹 c = ( 𝜆 if 𝐹 occurs 𝜇 if 𝐹 doesn’t occur , where 𝜆 and 𝜇 are real numbers, 𝐹 c ≔ Ω \ 𝐹 is the complement of 𝐹 , and I 𝐹 is theindicator of 𝐹 : the gamble that assumes the value 1 on 𝐹 and 0 elsewhere. We denoteby ℒ 𝐹 ≔ { 𝜆 I 𝐹 + 𝜇 I 𝐹 c : 𝜆, 𝜇 ∈ R } the set of all such gambles.To formalise his idea, Seidenfeld considers two gambles 𝑓 , 𝑔 ∈ ℒ 𝐹 on the occur-rence of 𝐹 . In our example, they could for instance represent the uncertain rewardsfor two possible courses of treatment for our patient in the Brussels hospital, whoseactual rewards are determined by her actual (but unknown) medical condition.We can use these two gambles to construct a composite gamble I 𝐸 𝑓 + I 𝐸 c 𝑔 , whoseoutcome also depends on the state of the event 𝐸 : it yields the uncertain reward 𝑓 if 𝐸 occurs, and 𝑔 if 𝐸 doesn’t occur. In our example, this would correspond to takingeither the first or the second treatment, depending on the weather condition at theSouth Pole, and I 𝐸 𝑓 + I 𝐸 c 𝑔 is then the reward function for this composite treatment.We now consider the problem of choosing between the gambles in the collection 𝐴 𝜖 ≔ { 𝑓 , 𝑔, I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 } for some real constant 𝜖 > We will use the language of events, rather than propositions, to express the things we are uncertainabout, but the difference is immaterial for what we have in mind. Our results can be developed using horse lotteries, but we opt here for a simplified version. De Bock and De Cooman
The third gamble consists in paying a price 𝜖 > 𝐸 which then determines whether we get the uncertain reward 𝑓 or 𝑔 on theoutcome of 𝐹 . Seidenfeld’s requirement states that when a subject believes 𝐸 and 𝐹 to be ‘independent’, then the third gamble must never be chosen, for any 𝜖 >
0. Oralternatively, in a language that stresses rejection rather than choice: the third optionmust be rejected from the set of options 𝐴 𝜖 for all 𝜖 > If, as we will explain in Section 2, we consider choice or rejection statements toprovide information about strict preferences between gambles, then this requirementstates that at least one option in the set { 𝑓 , 𝑔 } must be preferred over the rejectedoption I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 , which amounts to OR-ing these two preference assessments: 𝑓 is preferred over I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 OR 𝑔 is preferred over I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖. In this paper, we intend to explore the consequences of making such assessments,using our new approach to coherent choice functions, which is, as we have alreadymentioned, eminently suited for dealing with such ‘OR’-statements.Thus, in developing this independence idea, we can make two lines of researchcome together, both of which were inspired by Teddy Seidenfeld.We outline our approach to coherent choice in Section 2, and derive the far-reaching consequences of imposing the above-mentioned ‘independence’ require-ment in Section 3. We extend the discussion from events to variables in Section 4,and dwell on the implications of our findings in Section 5. A choice function 𝐶 is a set-valued operator on sets of options. In particular, for anyset of options 𝐴 , the corresponding value of 𝐶 is a subset 𝐶 ( 𝐴 ) of 𝐴 . We will beconsidering throughout the special case where these options are gambles in ℒ ( Ω ) :bounded real-valued maps on Ω , interpreted as uncertain rewards. The option sets 𝐴 are furthermore taken to be finite, and we denote the set of all such finite subsetsof ℒ ( Ω ) by 𝒬 ( Ω ) . Again, if it is clear from the context what the possibility space is,we will use the simpler notations ℒ and 𝒬 .Gambles can be ordered by the point-wise ordering, where 𝑓 ≥ 𝑔 means that 𝑓 ( 𝜔 ) ≥ 𝑔 ( 𝜔 ) for all 𝜔 ∈ Ω , and 𝑓 > 𝑔 means that 𝑓 ≥ 𝑔 but 𝑓 ≠ 𝑔 . We will alsouse the notation 𝑓 ≻ 𝑔 to mean that inf ( 𝑓 − 𝑔 ) > 𝐴 , for any 𝐴 ∈ 𝒬 . The terminology can be a bit misleading, though,because to say that 𝐶 ( 𝐴 ) = 𝐵 is not taken to mean that all options in 𝐵 are chosen.Rather, it means that, based on the available information, our subject is only disposed In both his hand-outs and the above-mentioned paper [26], Seidenfeld argues, similarly, for whathe calls the inadmissibility of the third option. For a more general approach to desirability-based choice functions, where options can take valuesin an abstract vector space, we refer the interested reader to one of our earlier papers [10, 11].n a notion of independence proposed by Teddy Seidenfeld 5 to rule out the options in 𝐴 \ 𝐵 , but remains undecided about the remaining optionsin 𝐵 . For this reason, it makes sense to focus on the options that are rejected—asin ‘not chosen’—and to consider the corresponding rejection function 𝑅 , definedby 𝑅 ( 𝐴 ) ≔ 𝐴 \ 𝐶 ( 𝐴 ) for all 𝐴 ∈ 𝒬 .Our interpretation for rejection—and hence also choice—functions now goesas follows. Consider a subject whose uncertainty is represented by a rejection func-tion 𝑅 , or equivalently, by a choice function 𝐶 . Then for a given option set 𝐴 ∈ 𝒬 , thestatement that an option 𝑓 ∈ 𝐴 is rejected from 𝐴 —that 𝑓 ∈ 𝑅 ( 𝐴 ) or 𝑓 ∉ 𝐶 ( 𝐴 ) —istaken to mean that there is at least one option 𝑔 in 𝐴 that our subject strictly prefersover 𝑓 .The connection with the language of desirability is now almost immediate, be-cause it is eminently suited for dealing with binary preference statements such as“the gamble 𝑔 is strictly preferred over the gamble 𝑓 ”. In terms of desirability, thissimply means that 𝑔 − 𝑓 is desirable [22, 34]. By applying this to our interpretationfor rejection in terms of preferences, we obtain an equivalent interpretation in termsof desirability: 𝑓 is rejected from 𝐴 if the option set 𝐴 ⊖ 𝑓 ≔ { 𝑔 − 𝑓 : 𝑔 ∈ 𝐴 \ { 𝑓 }} contains at least one desirable gamble. So we find that under our interpretation, thestudy of choice and rejection functions reduces to the study of sets of gambles thatcontain at least one desirable gamble.In order to formalise this, we have introduced the concept of a desirable optionset : a set 𝐴 ∈ 𝒬 that contains at least one desirable gamble. A subject’s uncertaintycan then be described by means of a set of such desirable option sets: a set 𝐾 ⊆ 𝒬 of option sets 𝐴 that she assesses to be desirable, in the sense that according to herbeliefs, every 𝐴 ∈ 𝐾 contains at least one desirable gamble. For any such set ofdesirable option sets 𝐾 , the corresponding rejection function and choice function arethen defined by 𝑅 ( 𝐴 ) ≔ { 𝑓 ∈ 𝐴 : 𝐴 ⊖ 𝑓 ∈ 𝐾 } and 𝐶 ( 𝐴 ) ≔ { 𝑓 ∈ 𝐴 : 𝐴 ⊖ 𝑓 ∉ 𝐾 } (1)In the remainder of this contribution, we will focus mainly on sets of desirableoptions sets 𝐾 , and will consider choice and rejection functions as derived objects,obtained by Equation (1). In particular, we will focus on sets of desirable option setsthat are coherent , in the sense that they satisfy the following rationality criteria forthe beliefs—or behavioural dispositions—expressed by 𝐾 . We will use ‘ ( 𝜆, 𝜇 ) > 𝜆 ≥ 𝜇 ≥ 𝜆 + 𝜇 > Definition 1 (Coherence for sets of desirable option sets)
A set of desirable optionsets 𝐾 ⊆ 𝒬 is called coherent if it satisfies the following axioms:K . if 𝐴 ∈ 𝐾 then also 𝐴 \ { } ∈ 𝐾 , for all 𝐴 ∈ 𝒬 ;K . { } ∉ 𝐾 ;K . { 𝑓 } ∈ 𝐾 , for all 𝑓 ∈ ℒ with inf 𝑓 > . if 𝐴 , 𝐴 ∈ 𝐾 and if, for all 𝑓 ∈ 𝐴 and 𝑔 ∈ 𝐴 , ( 𝜆 𝑓 ,𝑔 , 𝜇 𝑓 ,𝑔 ) >
0, then also
De Bock and De Cooman { 𝜆 𝑓 ,𝑔 𝑓 + 𝜇 𝑓 ,𝑔 𝑔 : 𝑓 ∈ 𝐴 , 𝑔 ∈ 𝐴 } ∈ 𝐾 ;K . if 𝐴 ∈ 𝐾 and 𝐴 ⊆ 𝐴 , then also 𝐴 ∈ 𝐾 , for all 𝐴 , 𝐴 ∈ 𝒬 .This axiomatisation is entirely based on—and motivated by—our interpretation andthe following three rationality principles for a notion of desirability:d . 0 is not desirable;d . if 𝑓 ≻
0, or in other words, inf 𝑓 >
0, then 𝑓 is desirable; d . if 𝑓 , 𝑔 are desirable and ( 𝜆, 𝜇 ) >
0, then 𝜆 𝑓 + 𝜇𝑔 is desirable.For a motivation and discussion of these principles, we refer to [22, 34]. For a detailedexplanation of why d –d indeed naturally lead to K –K , we refer to [10], which alsocontains a small example that illustrates the use of our workhorse axiom K . Moregenerally, that same reference [10] also provides much more information about—andmotivation for—the framework that we summarise here.For our present purposes, it will suffice to focus on a number of special casesthat play a central role in this paper: choice functions based on linear, and on lower,previsions. Perhaps the best-known method for choosing between uncertain rewards, is to choosethose options that have the highest expected utility with respect to some givenprobability measure. This measure is often taken to be countably additive, but wewill not impose this restriction here and work with finitely additive probabilitymeasures instead, defined on all events 𝐸 ⊆ Ω . The expectation operators thatcorrespond to such measures are linear previsions. Definition 2 (Linear prevision) A linear prevision 𝑃 on ℒ is a real-valued mapon ℒ that satisfiesP . 𝑃 ( 𝑓 ) ≥ inf 𝑓 for all 𝑓 ∈ ℒ ;P . 𝑃 ( 𝜆 𝑓 ) = 𝜆𝑃 ( 𝑓 ) for all 𝜆 ∈ R and 𝑓 ∈ ℒ ;P . 𝑃 ( 𝑓 + 𝑔 ) = 𝑃 ( 𝑓 ) + 𝑃 ( 𝑔 ) for all 𝑓 , 𝑔 ∈ ℒ .We denote the set of all linear previsions on ℒ by P .Conversely, every linear prevision 𝑃 has a corresponding finitely additive probabilitymeasure, also denoted by 𝑃 , and defined by 𝑃 ( 𝐸 ) ≔ 𝑃 ( I 𝐸 ) for all 𝐸 ⊆ Ω .For any given linear prevision—or, equivalently, any finitely additive probabilitymeasure— 𝑃 , we now let 𝐶 𝑃 be the choice function that corresponds to maximisingexpected utility. For all 𝐴 ∈ 𝒬 , it is defined by Our general approach [10, 11] allows for more general ‘background orderings’ ≻ to replace theordering based on ‘inf 𝑓 >
0’ considered here. It is also customary in much of the literature to furthermore remove from a choice set 𝐶 ( 𝐴 ) thoseoptions that are dominated by other options in 𝐴 for the point-wise ordering ≥ of options. We willleave this implicit here, as an operation that can always be performed afterwards .n a notion of independence proposed by Teddy Seidenfeld 7 𝐶 𝑃 ( 𝐴 ) ≔ { 𝑓 ∈ 𝐴 : (∀ 𝑔 ∈ 𝐴 ) 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑔 )} . (2)That this is a special case of our more general framework can be seen by definingthe set of desirable option sets 𝐾 𝑃 ≔ { 𝐴 ∈ 𝒬 : (∃ 𝑓 ∈ 𝐴 ) 𝑃 ( 𝑓 ) > } , (3)which is easily verified to be coherent. By applying Equation (1), we find that thecorresponding choice function is indeed given by 𝐶 𝑃 : for any 𝐴 ∈ 𝒬 and 𝑓 ∈ 𝐴 , wesee that 𝐴 ⊖ 𝑓 ∉ 𝐾 𝑃 ⇔ ¬(∃ 𝑔 ∈ 𝐴 \ { 𝑓 }) 𝑃 ( 𝑔 − 𝑓 ) > ⇔ (∀ 𝑔 ∈ 𝐴 \ { 𝑓 }) 𝑃 ( 𝑔 − 𝑓 ) ≤ ⇔ (∀ 𝑔 ∈ 𝐴 \ { 𝑓 }) 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑔 )⇔ (∀ 𝑔 ∈ 𝐴 ) 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑔 ) . It is clear that the choice models 𝐾 𝑃 and 𝐶 𝑃 are binary , in the sense that they arecompletely determined by a binary strict preference relation on gambles 𝑓 , 𝑔 ∈ ℒ ,in this case expressed by 𝑃 ( 𝑓 ) > 𝑃 ( 𝑔 ) .More generally, we can replace the linear prevision, or probability measure, 𝑃 by aset 𝒫 ⊆ P of such previsions. In that case, one possible approach to decision makingis to apply Levi’s E-admissibility criterion [20, 27], which amounts to consideringthe union of the choices of the individual 𝑃 ∈ 𝒫 , or equivalently, rejecting theoptions that are rejected under every 𝑃 ∈ 𝒫 , typically leading to a non-binarychoice model. We let 𝐶 𝒫 be the choice function that corresponds to this decisioncriterion, defined by 𝐶 𝒫 ( 𝐴 ) ≔ Ø { 𝐶 𝑃 ( 𝐴 ) : 𝑃 ∈ 𝒫 } = (cid:8) 𝑓 ∈ 𝐴 : (∃ 𝑃 ∈ 𝒫 ) (∀ 𝑔 ∈ 𝐴 ) 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑔 ) (cid:9) for all 𝐴 ∈ 𝒬 . (4)This too corresponds to a special case of our framework, as can be seen by applyingEquation (1) to the set of desirable option sets 𝐾 𝒫 ≔ Ù 𝑃 ∈ 𝒫 𝐾 𝑃 = { 𝐴 ∈ 𝒬 : (∀ 𝑃 ∈ 𝒫 ) (∃ 𝑓 ∈ 𝐴 ) 𝑃 ( 𝑓 ) > } . (5)That 𝐾 𝒫 is coherent can be seen by observing that coherence is preserved undertaking (non-empty) intersections of sets of desirable option sets. Strictly speaking, Levi only introduced, and argued for, this criterion in a context where he requiredthe set 𝒫 to be convex . We will still use the term ‘E-admissibility’ even when 𝒫 is not convex. Asmentioned before, we also leave the removal of ≥ -dominated options implicit, as something thatcan be done afterwards. De Bock and De Cooman While intuitive and straightforward, E-admissibility is not the only possible gener-alisation of expectation maximisation. Within the field of imprecise probabilities,(Walley–Sen) maximality [27, 33] is another extension that is often adopted; thistoo, as we will see, corresponds to a special case of our framework.The uncertainty models to which the decision criterion of maximality is typicallyapplied are not linear previsions, but rather a generalisation called coherent lowerprevisions . We will only give a very brief account of them here; many more detailsabout their interpretation and mathematical properties can be found in [28, 33].
Definition 3 (Coherent lower prevision) A coherent lower prevision 𝑃 on ℒ is areal-valued map on ℒ that satisfiesLP . 𝑃 ( 𝑓 ) ≥ inf 𝑓 for all 𝑓 ∈ ℒ ;LP . 𝑃 ( 𝜆 𝑓 ) = 𝜆𝑃 ( 𝑓 ) for all 𝜆 ∈ R > and 𝑓 ∈ ℒ ;LP . 𝑃 ( 𝑓 + 𝑔 ) ≥ 𝑃 ( 𝑓 ) + 𝑃 ( 𝑔 ) for all 𝑓 , 𝑔 ∈ ℒ .We denote the set of all coherent lower previsions on ℒ by P .For any event 𝐸 ⊆ Ω , we will also call 𝑃 ( 𝐸 ) ≔ 𝑃 ( I 𝐸 ) the lower probability of 𝐸 and 𝑃 ( 𝐸 ) ≔ 𝑃 ( I 𝐸 ) its upper probability .By comparing Definitions 2 and 3, we see that linear previsions indeed correspondto a special case of coherent lower previsions. In particular, they are coherent lowerprevisions 𝑃 that are precise , in the sense that they coincide with their conjugateupper prevision 𝑃 , defined by 𝑃 ( 𝑓 ) ≔ − 𝑃 (− 𝑓 ) for all 𝑓 ∈ ℒ . Proposition 1 ([33, Section 2.3.6])
Let 𝑃 be a coherent lower prevision on ℒ .Then 𝑃 is a linear prevision on ℒ if and only if 𝑃 ( 𝑓 ) = 𝑃 ( 𝑓 ) for all 𝑓 ∈ ℒ . Besides their defining properties LP –LP , coherent lower previsions also satisfyvarious other properties that are often conveniently used in proofs. We only mentiona small selection and refer to [33, Section 2.6.1] and [28] for more extensive lists,proofs and further discussion:LP . inf 𝑓 ≤ 𝑃 ( 𝑓 ) ≤ 𝑃 ( 𝑓 ) ≤ sup 𝑓 for all 𝑓 ∈ ℒ ;LP . if 𝑓 ≤ 𝑔 then 𝑃 ( 𝑓 ) ≤ 𝑃 ( 𝑔 ) and 𝑃 ( 𝑓 ) ≤ 𝑃 ( 𝑔 ) for all 𝑓 , 𝑔 ∈ ℒ ;LP . 𝑃 ( 𝑓 + 𝜇 ) = 𝑃 ( 𝑓 ) + 𝜇 for all 𝑓 ∈ ℒ and 𝜇 ∈ R ;LP . 𝑃 ( 𝑓 𝑛 ) → 𝑃 ( 𝑓 ) for all 𝑓 𝑛 , 𝑓 ∈ ℒ such that sup | 𝑓 𝑛 − 𝑓 | → . 𝑃 ( 𝑓 ) + 𝑃 ( 𝑔 ) ≤ 𝑃 ( 𝑓 + 𝑔 ) ≤ 𝑃 ( 𝑓 ) + 𝑃 ( 𝑔 ) ≤ 𝑃 ( 𝑓 + 𝑔 ) ≤ 𝑃 ( 𝑓 ) + 𝑃 ( 𝑔 ) for all 𝑓 , 𝑔 ∈ ℒ .Observe that we have identified the real number 𝜇 with the gamble that assumes thisconstant value.Given a coherent lower prevision 𝑃 on ℒ , we let 𝐶 𝑃 be the choice function thatis obtained by applying the criterion of maximality. For all 𝐴 ∈ 𝒬 , it is defined by n a notion of independence proposed by Teddy Seidenfeld 9 𝐶 𝑃 ( 𝐴 ) ≔ (cid:8) 𝑓 ∈ 𝐴 : (∀ 𝑔 ∈ 𝐴 ) 𝑃 ( 𝑔 − 𝑓 ) ≤ (cid:9) . (6)The idea here is that an option 𝑓 ∈ 𝐴 is rejected from an option set 𝐴 if there issome (other) option 𝑔 ∈ 𝐴 such that 𝑃 ( 𝑔 − 𝑓 ) > As for linear previsions, thisleads to a binary choice model, as it is completely determined by this binary strictpreference relation on gambles.Under the behavioural interpretation of lower previsions [33, Section 2.3.1], wherethe lower prevision of a gamble is interpreted as the supremum price for buying thatgamble, 𝑃 ( 𝑔 − 𝑓 ) > 𝑔 − 𝑓 , or equivalently, that she is willing to pay a strictly positiveprice to replace the uncertain reward 𝑓 by 𝑔 .Alternatively, the statement that 𝑃 ( 𝑔 − 𝑓 ) > 𝑃 : P ( 𝑃 ) ≔ { 𝑃 ∈ P : 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑓 ) for all 𝑓 ∈ ℒ } . This set is non-empty, convex and closed with respect to the topology of point-wiseconvergence of bounded linear real functionals, and it furthermore has 𝑃 and 𝑃 asits lower and upper envelopes [33, Section 3.3.3]: 𝑃 ( 𝑓 ) = min { 𝑃 ( 𝑓 ) : 𝑃 ∈ P ( 𝑃 )} and 𝑃 ( 𝑓 ) = max { 𝑃 ( 𝑓 ) : 𝑃 ∈ P ( 𝑃 )} . (7)The statement that 𝑃 ( 𝑔 − 𝑓 ) is strictly positive can therefore also be interpreted asstating that 𝑃 ( 𝑔 ) > 𝑃 ( 𝑓 ) for all 𝑃 ∈ P ( 𝑃 ) . This leads to the following alternativecharacterisation of 𝐶 𝑃 : 𝐶 𝑃 ( 𝐴 ) = (cid:8) 𝑓 ∈ 𝐴 : (∀ 𝑔 ∈ 𝐴 ) (∃ 𝑃 ∈ P ( 𝑃 )) 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑔 ) (cid:9) for all 𝐴 ∈ 𝒬 .By comparing this expression with Equation (4), we see that, generally speaking, 𝐶 𝑃 doesn’t coincide with 𝐶 P ( 𝑃 ) , which illustrates that maximality and E-admissibilityare distinct decision criteria. In the precise case, however, they do coincide andthen reduce to expectation maximisation. For E-admissibility, this is immediate; formaximality, this can be seen by comparing Equations (2) and (6) in the case that 𝑃 = 𝑃 is linear.That maximality is also a special case of our desirability-based theory of choicefunctions can be seen by defining the following set of desirable option sets 𝐾 𝑃 ≔ { 𝐴 ∈ 𝒬 : (∃ 𝑓 ∈ 𝐴 ) 𝑃 ( 𝑓 ) > } . (8)It is easily verified to be coherent [this follows from LP –LP ], and its correspondingchoice function is given by 𝐶 𝑃 .Since sets of desirable option sets of this kind are coherent, so are their intersec-tions. With any non-empty set 𝒫 ⊆ P of coherent lower previsions, we can thereforeassociate a coherent set of desirable option sets Again, we leave the removal of dominated options for the point-wise ordering ≥ implicit, assomething that can be done afterwards.0 De Bock and De Cooman 𝐾 𝒫 ≔ Ù { 𝐾 𝑃 : 𝑃 ∈ 𝒫 } = { 𝐴 ∈ 𝒬 : (∀ 𝑃 ∈ 𝒫 ) (∃ 𝑓 ∈ 𝐴 ) 𝑃 ( 𝑓 ) > } (9)and its corresponding choice function 𝐶 𝒫 , given by 𝐶 𝒫 ( 𝐴 ) ≔ Ø { 𝐶 𝑃 ( 𝐴 ) : 𝑃 ∈ 𝒫 } = (cid:8) 𝑓 ∈ 𝐴 : (∃ 𝑃 ∈ 𝒫 ) (∀ 𝑔 ∈ 𝐴 ) 𝑃 ( 𝑔 − 𝑓 ) ≤ (cid:9) for all 𝐴 ∈ 𝒬 . (10)If 𝒫 consists of linear previsions only, these respective expressions reduce to theEquations (5) and (4) for E-admissibility. If 𝒫 consists of a single lower prevision,we obtain maximality, and if this single lower prevision is furthermore linear, wearrive at expectation maximisation. So we see that this class of choice functionscontains all the special cases that we have seen so far. We will call all such choicemodels Archimedean . Definition 4
A set of desirable option sets 𝐾 is called Archimedean if there is somenon-empty set 𝒫 ⊆ P of coherent lower previsions such that 𝐾 = 𝐾 𝒫 . The largestsuch set 𝒫 is then P ( 𝐾 ) ≔ { 𝑃 ∈ P : 𝐾 ⊆ 𝐾 𝑃 } . In the remainder of this contribution, we will focus on Archimedean sets of desirableoption sets and their corresponding choice functions, either in their full generality orin particular cases. However, rather than impose this Archimedean property ad hoc ,we prefer to derive it from first principles by imposing additional axioms, besidescoherence, on the sets of desirable option sets that model a subject’s choices.In order to achieve this, we strengthen our interpretation for a set of desirableoption sets 𝐾 . That is, for any 𝐴 ∈ 𝒬 , 𝐴 ∈ 𝐾 is henceforth taken to mean that there isat least one gamble 𝑓 in 𝐴 that is strictly desirable [33, 34], in the sense that thereis some 𝜖 ∈ R > such that 𝑓 − 𝜖 is desirable, where—as before for 𝜇 —we identifythe real number 𝜖 with the constant gamble that takes the value 𝜖 .In an earlier paper [10], this interpretation in terms of strict desirability ledus to propose a notion of Archimedeanity for sets of desirable option sets. Withhindsight, we now prefer to call it strong Archimedeanity , and to reserve the termArchimedeanity for sets of desirable option sets 𝐾 𝒫 that correspond to a set of lowerprevisions 𝒫 ⊆ P . Definition 5 (Strongly Archimedean set of desirable option sets)
We call a set ofdesirable option sets 𝐾 strongly Archimedean if it is coherent and satisfies There may arise, due to Walley’s [33, 34] perhaps unfortunate introduction of this terminology,some confusion in the reader’s mind about the use of ‘strict’. In most accounts of preferencerelations, the term ‘strict preference’ refers to ‘(weak) preference without indifference’, and it isalso in this sense that we have used the term ‘strict preference’ in the Introduction. Walley usesthe moniker ‘strict’ for a stronger requirement, which is essentially based on some lower (or linear)prevision being strictly positive. We maintain this rather unhappy use of terminology here merelyfor historical reasons, but insist on warning the reader about the possible confusion this may entail.n a notion of independence proposed by Teddy Seidenfeld 11 K A . for all 𝐴 ∈ 𝐾 , there is some 𝜖 ∈ R > such that 𝐴 − 𝜖 ∈ 𝐾 .As we proved in earlier work [10], strongly Archimedean choice models are in aone-to-one correspondence with sets of coherent lower previsions that are closed inthe topology on bounded real functionals induced by point-wise convergence. Theorem 1 (Representation for strongly Archimedean choice models)
A set ofdesirable option sets 𝐾 is strongly Archimedean if and only if there is a non-emptyclosed set 𝒫 ⊆ P of coherent lower previsions such that 𝐾 = 𝐾 𝒫 . The largest suchset 𝒫 is then P ( 𝐾 ) . Proof
This result is a direct consequence of [10, Theorem 28 and Proposition 24]. (cid:3)
If we compare this result to Definition 4, we see that every strongly Archimedeanset of desirable options is—as the terminology also suggests—Archimedean. Theaxiom of strong Archimedeanity can therefore be employed as a justification forworking with an Archimedean choice model, or equivalently, with a set 𝒫 of co-herent lower previsions. However, strong Archimedeanity is a bit too strong forthat purpose—hence our change in terminology with respect to [10]—because itadditionally implies that 𝒫 is closed.In order to resolve this issue, one of us has developed alternative axioms thatweaken strong Archimedeanity in such a way that the closedness condition in The-orem 1 is no longer needed, which makes sure that these alternative axioms char-acterise Archimedeanity [4, 5, 6]. They are still based on an interpretation in termsof strict desirability, but employ this interpretation more subtly. Simply put, thesubtlety involves the fact that even if a subject assesses that there is some 𝑓 ∈ 𝐴 that is strictly desirable, meaning that there is some 𝜖 ∈ R > such that 𝑓 − 𝜖 isdesirable, she may not know for which specific 𝜖 this is the case. We are then nolonger justified in stating that 𝐴 − 𝜖 ∈ 𝐾 for some 𝜖 ∈ R > , as strong Archimedeanitydoes. Fortunately, however, we can still infer—more involved—conditions on 𝐾 fromsuch an assessment, which turn out to be equivalent to Archimedeanity. A detailedexposition of these ideas falls outside the scope of the present discussion though; formore information on Archimedean choice functions and how to axiomatise them, werefer to our most recent work on this topic [4, 5, 6, 12, 13]. For our present purposes,it will suffice to merely remember that Archimedeanity can be given an axiomaticbasis that motivates the use of general—not necessarily closed—sets of coherentlower previsions.In order to obtain a representation in terms of sets of linear rather than coherentlower previsions, it turns out that we need to impose one more axiom, which is due toSeidenfeld et al. [26]. It states that we can remove from a desirable option set thoseoptions that are positive linear combinations—mixtures—of a number of its otheroptions. It involves the following closure operator, which adds to a set of optionsall the positive linear combinations of any finite number of its elements: In this sense, it would perhaps be preferable to call it an ‘unmixing property’, as the term ‘mixing’is better suited for an approach that favours choice over rejection. Nevertheless, we have decided tostick to ‘mixing’, for reasons of consistency with the terminology introduced in [26].2 De Bock and De Cooman posi ( 𝐴 ) ≔ (cid:26) 𝑛 Õ 𝑘 = 𝜆 𝑘 𝑓 𝑘 : 𝑛 ∈ N , 𝜆 𝑘 > , 𝑓 𝑘 ∈ 𝐴 (cid:27) for all 𝐴 ⊆ ℒ . Definition 6 (Mixing property for sets of desirable option sets)
We call a set ofdesirable option sets 𝐾 mixing if it is coherent and satisfiesK M . if 𝐵 ∈ 𝐾 and 𝐴 ⊆ 𝐵 ⊆ posi ( 𝐴 ) , then also 𝐴 ∈ 𝐾 , for all 𝐴, 𝐵 ∈ 𝒬 . Proposition 2 ([4, Proposition 2])
Let 𝐾 be an Archimedean set of desirable optionsets that is mixing. Then every coherent lower prevision 𝑃 in P ( 𝐾 ) is linear. Proposition 3
Let 𝒫 ⊆ P be a non-empty set of linear previsions. Then 𝐾 𝒫 ismixing. Proof
Since mixingness is clearly preserved under taking (non-empty) intersections,it suffices to prove that for any 𝑃 ∈ P , 𝐾 𝑃 is mixing. To this end, consider any 𝑃 ∈ P and any 𝐴, 𝐵 ∈ 𝒬 such that 𝐵 ∈ 𝐾 𝑃 and 𝐴 ⊆ 𝐵 ⊆ posi ( 𝐴 ) . Since 𝐵 ∈ 𝐾 𝑃 , there issome 𝑔 ∈ 𝐵 such that 𝑃 ( 𝑔 ) >
0. Since 𝐵 ⊆ posi ( 𝐴 ) , there are 𝑛 ∈ N and, for each 𝑘 ∈ { , . . . , 𝑛 } , 𝜆 𝑘 > 𝑓 𝑘 ∈ 𝐴 such that 𝑔 = Í 𝑛𝑘 = 𝜆 𝑘 𝑓 𝑘 . Hence, it follows fromthe linearity of 𝑃 that0 < 𝑃 ( 𝑔 ) = 𝑃 (cid:18) 𝑛 Õ 𝑘 = 𝜆 𝑘 𝑓 𝑘 (cid:19) = 𝑛 Õ 𝑘 = 𝜆 𝑘 𝑃 ( 𝑓 𝑘 ) , which implies that there is at least one 𝑘 ∗ ∈ { , . . . , 𝑛 } such that 𝑃 ( 𝑓 𝑘 ∗ ) >
0. Since 𝑓 𝑘 ∗ ∈ 𝐴 , this implies that 𝐴 ∈ 𝐾 𝑃 . (cid:3) By combining these two results with Theorem 1 and Definition 4, we obtainrepresentation in terms of (closed) sets of linear previsions.
Theorem 2
A set of desirable option sets 𝐾 is Archimedean and mixing if and onlyif there is a non-empty set 𝒫 ⊆ P of coherent linear previsions such that 𝐾 = 𝐾 𝒫 .Similarly, 𝐾 is strongly Archimedean and mixing if and only if there is a non-emptyclosed set 𝒫 ⊆ P of linear previsions such that 𝐾 = 𝐾 𝒫 . In both cases, the largestsuch set 𝒫 is P ( 𝐾 ) ≔ { 𝑃 ∈ P : 𝐾 ⊆ 𝐾 𝑃 } , and P ( 𝐾 ) = P ( 𝐾 ) . Proof
First assume that 𝐾 is (strongly) Archimedean and mixing. Since 𝐾 is(strongly) Archimedean, we know from Definition 4 (Theorem 1) that there is anon-empty (closed) set 𝒫 ⊆ P of coherent lower previsions such that 𝐾 = 𝐾 𝒫 .Furthermore, since 𝐾 is mixing, it follows from Proposition 2 that 𝒫 ⊆ P .Conversely, assume that there is a non-empty (closed) set 𝒫 ⊆ P such that 𝐾 = 𝐾 𝒫 . Proposition 3 then implies that 𝐾 is mixing. Furthermore, since P ⊆ P , itfollows from Definition 4 (Theorem 1) that 𝐾 is (strongly) Archimedean. It remainsto show that any such 𝒫 is a subset of P ( 𝐾 ) , that P ( 𝐾 ) is non-empty (and closed),that P ( 𝐾 ) = P ( 𝐾 ) and that 𝐾 = 𝐾 P ( 𝐾 ) .That 𝒫 is a subset of P ( 𝐾 ) follows immediately from the definitions of 𝐾 𝒫 and P ( 𝐾 ) and from the fact that 𝐾 = 𝐾 𝒫 . On the one hand, since 𝒫 is non-empty, this establishes the non-emptiness of P ( K ) . On the other hand, this implies n a notion of independence proposed by Teddy Seidenfeld 13 that 𝐾 P ( 𝐾 ) ⊆ 𝐾 𝒫 = 𝐾 , and therefore, since 𝐾 is clearly a subset of 𝐾 P ( 𝐾 ) , that 𝐾 = 𝐾 P ( 𝐾 ) . That P ( 𝐾 ) = P ( 𝐾 ) follows from Proposition 2. That P ( 𝐾 ) is closed if 𝐾 is strongly Archimedean and mixing, finally, follows from the fact that P ( 𝐾 ) = P ( 𝐾 ) and Theorem 1. (cid:3) Let us now assume that our subject’s preferences are modelled by some coherentset of desirable option sets 𝐾 . We consider two events 𝐸 and 𝐹 , and investigate theimport of Teddy Seidenfeld’s operationalisation of the independence requirementdiscussed in the Introduction: for any 𝑓 and 𝑔 in ℒ 𝐹 and all real 𝜖 > I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 is rejected from the option set 𝐴 𝜖 ≔ { 𝑓 , 𝑔, I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 } . Taking into account that 𝑓 − ( I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 ) = ( 𝑓 − 𝑔 ) I 𝐸 c + 𝜖 and 𝑔 − ( I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 ) = ( 𝑔 − 𝑓 ) I 𝐸 + 𝜖, Equation (1) leads us to the following definition. It acknowledges that the aboverequirement is not necessarily symmetrical in 𝐸 and 𝐹 , and therefore leads to anotion of irrelevance; independence is then introduced as symmetrised irrelevance. Definition 7 (S-irrelevance and S-independence)
Consider two events
𝐸, 𝐹 ⊆ Ω .We say that 𝐸 is S-irrelevant to 𝐹 with respect to a coherent set of desirable optionsets 𝐾 if { I 𝐸 c 𝑓 + 𝜖, − I 𝐸 𝑓 + 𝜖 } ∈ 𝐾 for all 𝑓 ∈ ℒ 𝐹 and all 𝜖 > . (11)We say that 𝐸 and 𝐹 are S-independent with respect to 𝐾 if 𝐸 is S-irrelevant to 𝐹 and 𝐹 is S-irrelevant to 𝐸 .These irrelevance and independence notions are invariant under complementation,as follows easily from Definition 7. Proposition 4
Consider any two events
𝐸, 𝐹 ⊆ Ω , and any ˜ 𝐸 ∈ { 𝐸, 𝐸 c } and ˜ 𝐹 ∈{ 𝐹, 𝐹 c } . Then 𝐸 is S-irrelevant to 𝐹 with respect to a coherent set of desirable optionsets 𝐾 if and only if ˜ 𝐸 is S-irrelevant to ˜ 𝐹 with respect to 𝐾 . Similarly, 𝐸 and 𝐹 areS-independent with respect to 𝐾 if and only if ˜ 𝐸 and ˜ 𝐹 are. Proof
We concentrate on the statement concerning S-irrelevance, as the proof forthe statement about S-independence then follows immediately. Due to the symmetryof the statement, it clearly suffices to prove necessity.So assume that 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 . We start by observing that ℒ ˜ 𝐹 = ℒ 𝐹 . So if ˜ 𝐸 = 𝐸 , our assumption trivially implies that ˜ 𝐸 is S-irrelevant to ˜ 𝐹 with respect to 𝐾 . If ˜ 𝐸 = 𝐸 c , then for any 𝑓 ∈ ℒ ˜ 𝐹 and 𝜖 >
0, since − 𝑓 ∈ ℒ ˜ 𝐹 = ℒ 𝐹 ,it follows from the assumption that {− I ˜ 𝐸 𝑓 + 𝜖, I ˜ 𝐸 c 𝑓 + 𝜖 } = { I 𝐸 c (− 𝑓 ) + 𝜖, − I 𝐸 (− 𝑓 ) + 𝜖 } ∈ 𝐾. Hence, also in this case, ˜ 𝐸 is S-irrelevant to ˜ 𝐹 with respect to 𝐾 . (cid:3) If the coherent 𝐾 is moreover Archimedean, as we will henceforth typicallyassume, then S-irrelevance with respect to 𝐾 can be characterised more simply interms of S-irrelevance with respect to specific binary choice models of the types 𝐾 𝑃 and 𝐾 𝑃 . Similar statements will of course hold for S-independence. Proposition 5
Consider any events
𝐸, 𝐹 ⊆ Ω and a set of desirable option sets 𝐾 .If 𝐾 is Archimedean, then 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 if and onlyif 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 for all 𝑃 ∈ P ( 𝐾 ) . Similarly, if 𝐾 isArchimedean and mixing, then 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 if and onlyif 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 for all 𝑃 ∈ P ( 𝐾 ) . Proof
If we combine Definitions 4 and 7, we see that 𝐸 is S-irrelevant to 𝐹 withrespect to an Archimedean 𝐾 if and only if { I 𝐸 c 𝑓 + 𝜖, − I 𝐸 𝑓 + 𝜖 } ∈ Ù { 𝐾 𝑃 : 𝑃 ∈ P ( 𝐾 )} for all 𝑓 ∈ ℒ 𝐹 and all 𝜖 > , which proves the statement for Archimedeanity. If we also impose mixingness on 𝐾 ,the second statement follows at once from the first and the fact that then, accordingto Theorem 2, P ( 𝐾 ) = P ( 𝐾 ) . (cid:3) It therefore behoves us to study S-irrelevance with respect to such special binarymodels 𝐾 𝑃 and 𝐾 𝑃 . First, we consider any linear prevision 𝑃 on ℒ and study the implications of S-irrelevance with respect to the binary choice model 𝐾 𝑃 . For any events 𝐸 and 𝐹 ,Equation (3) then clearly implies that 𝐸 is S-irrelevant to 𝐹 if and only if 𝑃 ( I 𝐸 c 𝑓 + 𝜖 ) > 𝑃 (− I 𝐸 𝑓 + 𝜖 ) > 𝑓 ∈ ℒ 𝐹 \ { } and all 𝜖 > , or equivalently, since 𝑃 is constant additive [LP ], 𝑃 ( I 𝐸 c 𝑓 ) ≥ 𝑃 (− I 𝐸 𝑓 ) ≥ 𝑓 ∈ ℒ 𝐹 . (12)We now set out to investigate how this specific instance of S-irrelevance relates tothe usual independence condition. We will find that for this particular type of binarychoice model, S-irrelevance, S-independence and the usual independence notioncoincide. Definition 8 (Independent events with respect to a linear prevision)
We will callany two events 𝐸 and 𝐹 independent with respect to a linear prevision 𝑃 on ℒ if 𝑃 ( 𝐸 ∩ 𝐹 ) = 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) . n a notion of independence proposed by Teddy Seidenfeld 15 Proposition 6
Consider any two events
𝐸, 𝐹 ⊆ Ω , and consider any ˜ 𝐸 ∈ { 𝐸, 𝐸 c } and ˜ 𝐹 ∈ { 𝐹, 𝐹 c } . Then 𝐸 and 𝐹 are independent with respect to a linear prevision 𝑃 on ℒ if and only if ˜ 𝐸 and ˜ 𝐹 are. Proof
It clearly suffices to assume that 𝐸 and 𝐹 are independent, and to prove that 𝐸 c and 𝐹 are. So assume that 𝐸 and 𝐹 are independent. Then 𝑃 ( 𝐸 c ) 𝑃 ( 𝐹 ) = [ − 𝑃 ( 𝐸 )] 𝑃 ( 𝐹 ) = 𝑃 ( 𝐹 ) − 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) = 𝑃 ( 𝐹 ) − 𝑃 ( 𝐸 ∩ 𝐹 ) = 𝑃 ( 𝐹 \ ( 𝐸 ∩ 𝐹 )) = 𝑃 ( 𝐹 ∩ 𝐸 c ) , implying that 𝐸 c and 𝐹 are independent as well. (cid:3) Theorem 3
Consider any two events
𝐸, 𝐹 ⊆ Ω , and any linear prevision 𝑃 on ℒ .Then the following statements are equivalent: (i) 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 ; (ii) 𝐸 and 𝐹 are S-independent with respect to 𝐾 𝑃 ; (iii) 𝐸 and 𝐹 are independent with respect to 𝑃 . Proof
Since 𝐸 is independent of 𝐹 if and only if 𝐹 is independent of 𝐸 , it clearlysuffices to prove the equivalence of S-irrelevance and independence.We begin with the ‘only if’ part. Assume ex absurdo that 𝐸 and 𝐹 are notindependent with respect to 𝑃 , so 𝑃 ( 𝐸 ∩ 𝐹 ) ≠ 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) . We will assume that 𝑃 ( 𝐸 ∩ 𝐹 ) > 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) , but the arguments are completely analogous for the casethat 𝑃 ( 𝐸 ∩ 𝐹 ) < 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) : simply reverse the roles of 𝐸 and 𝐸 c .Let 𝛿 ≔ 𝑃 ( 𝐸 ∩ 𝐹 ) − 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) >
0. Then 𝑃 ( 𝐸 c ∩ 𝐹 ) − 𝑃 ( 𝐸 c ) 𝑃 ( 𝐹 ) = 𝑃 ( 𝐹 ) − 𝑃 ( 𝐸 ∩ 𝐹 ) − 𝑃 ( 𝐸 c ) 𝑃 ( 𝐹 ) = 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) − 𝑃 ( 𝐸 ∩ 𝐹 ) = − 𝛿 < , so 𝑃 ( 𝐸 c ∩ 𝐹 ) < 𝑃 ( 𝐸 c ) 𝑃 ( 𝐹 ) . Now let 𝑓 ≔ I 𝐹 − 𝑃 ( 𝐹 ) ∈ ℒ 𝐹 , then 𝑃 ( I 𝐸 c 𝑓 ) = 𝑃 ( I 𝐸 c I 𝐹 ) − 𝑃 ( 𝐹 ) 𝑃 ( I 𝐸 c ) = 𝑃 ( 𝐸 c ∩ 𝐹 ) − 𝑃 ( 𝐸 c ) 𝑃 ( 𝐹 ) = − 𝛿 < 𝑃 (− I 𝐸 𝑓 ) = − 𝑃 ( I 𝐸 I 𝐹 ) + 𝑃 ( 𝐹 ) 𝑃 ( I 𝐸 ) = 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) − 𝑃 ( 𝐸 ∩ 𝐹 ) = − 𝛿 < . It therefore follows from the S-irrelevance criterion (12) that 𝐸 is not S-irrelevantto 𝐹 with respect to 𝐾 𝑃 , a contradiction. We conclude that 𝐸 and 𝐹 are independentwith respect to 𝑃 .It remains to prove the ‘if’ part, so let us assume that 𝐸 and 𝐹 are independentwith respect to 𝑃 . We will use Definition 7 to prove that 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 . Fix any 𝑓 ∈ ℒ 𝐹 , then we need to show that the S-irrelevancecriterion (12) is satisfied. Since 𝐸 and 𝐹 are independent with respect to 𝑃 —andalso using Proposition 6—this criterion simplifies to 𝑃 ( 𝐸 c ) 𝑃 ( 𝑓 ) ≥ − 𝑃 ( 𝐸 ) 𝑃 ( 𝑓 ) ≥ , which is clearly always satisfied, because the first inequality holds when 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑓 ) ≤ (cid:3) Since we have seen that sets of desirable option sets 𝐾 that are Archimedean andmixing are completely determined by the linear prevision models 𝐾 𝑃 that includethem, it ought not to surprise us, in view of Theorem 3, that for such 𝐾 , we canreduce S-irrelevance (and S-independence) to independence with respect to theirrepresenting linear previsions 𝑃 . Theorem 4
Let 𝐾 be an Archimedean and mixing set of desirable option sets, andconsider any events 𝐸, 𝐹 ⊆ Ω . Then the following statements are equivalent: (i) 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 ; (ii) 𝐸 and 𝐹 are S-independent with respect to 𝐾 ; (iii) 𝐸 and 𝐹 are independent with respect to 𝑃 , for all 𝑃 ∈ P ( 𝐾 ) . Proof
This is an immediate consequence of Proposition 5 and Theorem 3. (cid:3)
Next, we consider a coherent lower prevision 𝑃 on ℒ and study the implicationsof S-irrelevance with respect to the binary choice model 𝐾 𝑃 . For any two events 𝐸 and 𝐹 , Equation (8) then clearly implies that 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 if and only if 𝑃 ( I 𝐸 c 𝑓 + 𝜖 ) > 𝑃 (− I 𝐸 𝑓 + 𝜖 ) > 𝑓 ∈ ℒ 𝐹 and all 𝜖 > , or equivalently, since 𝑃 is constant additive [LP ], 𝑃 ( I 𝐸 c 𝑓 ) ≥ 𝑃 (− I 𝐸 𝑓 ) ≥ 𝑓 ∈ ℒ 𝐹 . (13)Here too, we set out to investigate how this particular instance of S-irrelevancerelates to the usual independence condition. A first important result is that in thecontext of lower previsions, S-irrelevance implies independence in the usual sensefor every dominating linear prevision. Proposition 7
Consider any two events
𝐸, 𝐹 ⊆ Ω and any coherent lower previ-sion 𝑃 on ℒ , and let 𝐸 be S-irrelevant to 𝐹 with respect to 𝐾 𝑃 . Then 𝐸 and 𝐹 areindependent with respect to all 𝑃 ∈ P ( 𝑃 ) . Proof
Assume ex absurdo that there is some 𝑃 ∈ P ( 𝑃 ) with respect to which 𝐸 and 𝐹 are not independent. It then follows from Theorem 3 that, with respect to thecorresponding 𝐾 𝑃 , 𝐸 is not S-irrelevant to 𝐹 , implying that there is some 𝑓 ∈ ℒ 𝐹 such that 𝑃 ( I 𝐸 c 𝑓 ) < 𝑃 (− I 𝐸 𝑓 ) <
0. Since 𝑃 dominates 𝑃 , this immediatelyimplies that also 𝑃 ( I 𝐸 c 𝑓 ) < 𝑃 (− I 𝐸 𝑓 ) <
0. Hence, 𝐸 is not S-irrelevant to 𝐹 with respect to 𝐾 𝑃 , a contradiction. (cid:3) n a notion of independence proposed by Teddy Seidenfeld 17 Given this result, and drawing inspiration from Theorem 3, one might be led tothink that independence of the dominating linear previsions is not only necessary,but also sufficient for S-irrelevance. This is not the case, though; the non-linearityof 𝑃 makes for a slightly more involved picture. To arrive at a condition that is bothnecessary and sufficient, we start by introducing a notion of triviality for events: wesay that an event 𝐸 is trivial with respect to 𝑃 if (either) 𝑃 ( 𝐸 ) = 𝑃 ( 𝐸 c ) = Proposition 8
If the events 𝐸 or 𝐹 are trivial with respect to the coherent lowerprevision 𝑃 on ℒ , then 𝐸 and 𝐹 are S-independent with respect to 𝐾 𝑃 . Our proof for this result, as well as that of Theorem 7 further on, makes use of thefollowing simple technical lemma, which basically states that the lower and upperprevision of a gamble don’t depend on the values of that gamble on any event withzero upper probability.
Lemma 1
Consider any event 𝐺 ⊆ Ω such that 𝑃 ( 𝐺 ) = . Then 𝑃 ( 𝑔 I 𝐺 + ℎ ) = 𝑃 ( ℎ ) and 𝑃 ( 𝑔 I 𝐺 + ℎ ) = 𝑃 ( ℎ ) for all gambles 𝑔 and ℎ on Ω . Proof
It follows from conjugacy that we need only prove the equality for lowerprevisions. To see that it holds for lower previsions, observe that the inequality I 𝐺 inf 𝑔 ≤ 𝑔 I 𝐺 ≤ I 𝐺 sup 𝑔 and the coherence of 𝑃 [use LP and LP ] indeed lead to 𝑃 ( ℎ ) = 𝑃 ( I 𝐺 inf 𝑔 ) + 𝑃 ( ℎ ) ≤ 𝑃 ( 𝑔 I 𝐺 + ℎ ) ≤ 𝑃 ( I 𝐺 sup 𝑔 ) + 𝑃 ( ℎ ) = 𝑃 ( ℎ ) , where the equalities hold because for all real 𝜆 , also by coherence [use LP ], 𝑃 ( 𝜆 I 𝐺 ) = 𝑃 ( 𝜆 I 𝐺 ) = 𝑃 ( 𝐺 ) = 𝑃 ( 𝐺 ) = 𝑃 ( 𝐺 ) = 𝑃 ( 𝐺 ) = ]. (cid:3) Proof of Proposition 8
Assume that 𝐸 or 𝐹 are trivial with respect to 𝑃 , then itsuffices to prove that 𝐸 is S-irrelevant to 𝐹 . So consider any 𝑓 ≔ 𝜆 I 𝐹 + 𝜇 I 𝐹 c , with ( 𝜆, 𝜇 ) ∈ R . Then we have to show that 𝑃 ( I 𝐸 𝑓 ) ≥ 𝑃 (− I 𝐸 c 𝑓 ) ≥
0. Due to thesymmetry, it suffices to consider the following two possible cases.The first case is that 𝑃 ( 𝐸 ) =
0. Lemma 1 then guarantees that 𝑃 ( I 𝐸 𝑓 ) = 𝑃 ( ) = ].The second case is that 𝑃 ( 𝐹 ) =
0. Considering that I 𝐸 𝑓 = 𝜆 I 𝐸 I 𝐹 + 𝜇 I 𝐸 I 𝐹 c and − I 𝐸 c 𝑓 = − 𝜆 I 𝐸 c I 𝐹 − 𝜇 I 𝐸 c I 𝐹 c , we now infer from Lemma 1 that 𝑃 ( I 𝐸 𝑓 ) = 𝑃 ( 𝜇 I 𝐸 I 𝐹 c ) and 𝑃 (− I 𝐸 c 𝑓 ) = 𝑃 (− 𝜇 I 𝐸 c I 𝐹 c ) . There are now two possibilities. If 𝜇 ≥
0, then we infer [use LP and LP ] from thefirst equality that 𝑃 ( I 𝐸 𝑓 ) = 𝑃 ( 𝜇 I 𝐸 I 𝐹 c ) = 𝜇𝑃 ( I 𝐸 I 𝐹 c ) ≥
0. If 𝜇 ≤
0, then we infer[again use LP and LP ] from the second equality that 𝑃 (− I 𝐸 c 𝑓 ) = 𝑃 (− 𝜇 I 𝐸 c I 𝐹 c ) = (− 𝜇 ) 𝑃 ( I 𝐸 c I 𝐹 c ) ≥ (cid:3) So triviality is a sufficient condition for S-irrelevance and S-independence. Thefollowing crucial proposition shows, on the other hand, that whenever 𝐸 is not trivial,S-irrelevance of 𝐸 to 𝐹 implies precision for gambles on 𝐹 . Proposition 9
Consider any two events
𝐸, 𝐹 ⊆ Ω and any coherent lower previ-sion 𝑃 on ℒ . If 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 , then 𝐸 is trivial or 𝑃 ( 𝐹 ) = 𝑃 ( 𝐹 ) and hence also 𝑃 ( 𝑓 ) = 𝑃 ( 𝑓 ) for all 𝑓 ∈ ℒ 𝐹 . Our proof for this result makes use of two technical lemmas. The first of these twois fairly straightforward; it establishes that if the probability of an event is precise—meaning that its lower and upper probabilities coincide—then every gamble on theoccurrence of this event has a precise prevision—meaning that its lower and upperprevision coincide.
Lemma 2
Consider any event 𝐹 ⊆ Ω and any coherent lower prevision 𝑃 on ℒ .If 𝑃 ( 𝐹 ) = 𝑃 ( 𝐹 ) , then also 𝑃 ( 𝑓 ) = 𝑃 ( 𝑓 ) for all 𝑓 ∈ ℒ 𝐹 . Proof
Since 𝑓 ∈ ℒ 𝐹 , we know that there are 𝜆, 𝜇 ∈ R such that 𝑓 = 𝜆 I 𝐹 + 𝜇 I 𝐹 c . Let 𝛼 ≔ 𝑃 ( 𝐹 ) = 𝑃 ( 𝐹 ) . For all 𝑃 ∈ P ( 𝑃 ) , we then have that 𝑃 ( 𝑓 ) = 𝑃 ( 𝜆 I 𝐹 + 𝜇 I 𝐹 c ) = 𝜆𝑃 ( I 𝐹 ) + 𝜇𝑃 ( I 𝐹 c ) = 𝜆𝑃 ( 𝐹 ) + 𝜇𝑃 ( 𝐹 c ) = 𝜆𝛼 + 𝜇 ( − 𝛼 ) . The result therefore follows from Equation (7). (cid:3)
The second lemma on which our proof for Proposition 9 depends, is more involvedand less intuitive. It states that if 𝐸 is S-irrelevant to 𝐹 for some lower prevision,then any two linear previsions that dominate this lower prevision but disagree on theprobability of 𝐹 must either both assign zero probability to 𝐸 , or both assign zeroprobability to 𝐸 c . Lemma 3
Consider any two events
𝐸, 𝐹 ⊆ Ω and any coherent lower prevision 𝑃 on ℒ . Let 𝐸 be S-irrelevant to 𝐹 with respect to 𝐾 𝑃 . Then for any 𝑃 , 𝑃 ∈ P ( 𝑃 ) such that 𝑃 ( 𝐹 ) ≠ 𝑃 ( 𝐹 ) , there is some ˜ 𝐸 ∈ { 𝐸, 𝐸 c } such that 𝑃 ( ˜ 𝐸 ) = 𝑃 ( ˜ 𝐸 ) = . Proof
Recall from Proposition 7 that the assumptions imply in particular that 𝐸 and 𝐹 are independent with respect to both 𝑃 and 𝑃 .We will assume without loss of generality that 𝑃 ( 𝐹 ) < 𝑃 ( 𝐹 ) , and therefore that 𝛿 ≔ ( 𝑃 ( 𝐹 ) − 𝑃 ( 𝐹 ))/ >
0. In the other case, simply reverse the roles of 𝑃 and 𝑃 and repeat the argument.If we let 𝜅 ≔ ( 𝑃 ( 𝐹 ) + 𝑃 ( 𝐹 ))/
2, then 𝑃 ( 𝐹 ) < 𝜅 < 𝑃 ( 𝐹 ) . So if we now let 𝑓 ≔ I 𝐹 − 𝜅 ∈ ℒ 𝐹 , then 𝑃 ( 𝑓 ) = − 𝛿 < < 𝛿 = 𝑃 ( 𝑓 ) .First, assume ex absurdo that 𝑃 ( 𝐸 c ) > 𝑃 ( 𝐸 ) >
0. Then 𝑃 ( I 𝐸 c 𝑓 ) = 𝑃 ( 𝐸 c ∩ 𝐹 ) − 𝜅𝑃 ( 𝐸 c ) = 𝑃 ( 𝐸 c ) 𝑃 ( 𝐹 ) − 𝜅𝑃 ( 𝐸 c ) = − 𝛿𝑃 ( 𝐸 c ) < n a notion of independence proposed by Teddy Seidenfeld 19 𝑃 (− I 𝐸 𝑓 ) = − 𝑃 ( 𝐸 ∩ 𝐹 ) + 𝜅𝑃 ( 𝐸 ) = − 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) + 𝜅𝑃 ( 𝐸 ) = − 𝛿𝑃 ( 𝐸 ) < , where we have used the independence of 𝐸 and 𝐹 with respect to both 𝑃 and 𝑃 ,together with Proposition 6 for the second case. This implies that also 𝑃 ( I 𝐸 c 𝑓 ) < 𝑃 (− I 𝐸 𝑓 ) < , contradicting the assumption that 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 ; seecriterion (13). We therefore conclude that 𝑃 ( 𝐸 c ) = 𝑃 ( 𝐸 ) = ex absurdo that 𝑃 ( 𝐸 ) > 𝑃 ( 𝐸 c ) >
0. Then 𝑃 ( I 𝐸 𝑓 ) = 𝑃 ( 𝐸 ∩ 𝐹 ) − 𝜅𝑃 ( 𝐸 ) = 𝑃 ( 𝐸 ) 𝑃 ( 𝐹 ) − 𝜅𝑃 ( 𝐸 ) = − 𝛿𝑃 ( 𝐸 ) < 𝑃 (− I 𝐸 c 𝑓 ) = − 𝑃 ( 𝐸 c ∩ 𝐹 ) + 𝜅𝑃 ( 𝐸 c ) = − 𝑃 ( 𝐸 c ) 𝑃 ( 𝐹 ) + 𝜅𝑃 ( 𝐸 c ) = − 𝛿𝑃 ( 𝐸 c ) < , where we have again used the independence of 𝐸 and 𝐹 with respect to both 𝑃 and 𝑃 , together with Proposition 6 for the second case. It follows that 𝑃 ( I 𝐸 𝑓 ) < 𝑃 (− I 𝐸 c 𝑓 ) < , contradicting the assumption that 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 ; seecriterion (13) for the gamble − 𝑓 this time. We therefore conclude that 𝑃 ( 𝐸 ) = 𝑃 ( 𝐸 c ) = ( 𝑃 ( 𝐸 c ) = 𝑃 ( 𝐸 ) = ) and ( 𝑃 ( 𝐸 ) = 𝑃 ( 𝐸 c ) = ) . After applying the distributivity of ‘and’ over ‘or’, and removing two contradictions,we see that this is equivalent to ( 𝑃 ( 𝐸 c ) = 𝑃 ( 𝐸 c ) = ) or ( 𝑃 ( 𝐸 ) = 𝑃 ( 𝐸 ) = ) , as claimed in the statement of the lemma. (cid:3) Proof of Proposition 9
Assume that 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 andthat 𝑃 ( 𝐹 ) < 𝑃 ( 𝐹 ) . We prove that this implies that either 𝑃 ( 𝐸 ) = 𝑃 ( 𝐸 c ) = 𝑃 ( 𝐹 ) < 𝑃 ( 𝐹 ) , there are 𝑃 , 𝑃 ∈ P ( 𝑃 ) such that 𝑃 ( 𝐹 ) ≠ 𝑃 ( 𝐹 ) . Since 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 , we know from Lemma 3 that there is some˜ 𝐸 ∈ { 𝐸, 𝐸 c } such that 𝑃 ( ˜ 𝐸 ) = 𝑃 ( ˜ 𝐸 ) = 𝑃 ∈ P ( 𝑃 ) . Since 𝑃 ( 𝐹 ) ≠ 𝑃 ( 𝐹 ) , there is at least one 𝑘 ∈{ , } such that 𝑃 𝑘 ( 𝐹 ) ≠ 𝑃 ( 𝐹 ) . Since 𝑃, 𝑃 𝑘 ∈ P ( 𝑃 ) and since 𝐸 is S-irrelevantto 𝐹 with respect to 𝐾 𝑃 , Lemma 3 tells us that there is some ˜ 𝐸 𝑘 ∈ { 𝐸, 𝐸 c } suchthat 𝑃 ( ˜ 𝐸 𝑘 ) = 𝑃 𝑘 ( ˜ 𝐸 𝑘 ) =
0. Assume ex absurdo that ˜ 𝐸 𝑘 ≠ ˜ 𝐸 . Since ˜ 𝐸 𝑘 and ˜ 𝐸 belong to { 𝐸, 𝐸 c } and both 𝑃 𝑘 ( ˜ 𝐸 ) = 𝑃 𝑘 ( ˜ 𝐸 𝑘 ) =
0, this would imply that 𝑃 𝑘 ( 𝐸 ) = 𝑃 𝑘 ( 𝐸 c ) =
0, a contradiction because these probabilities must sum to one.
So we find that ˜ 𝐸 𝑘 = ˜ 𝐸 , and therefore also that 𝑃 ( ˜ 𝐸 ) = 𝑃 ( ˜ 𝐸 𝑘 ) =
0. Since this istrue for any 𝑃 ∈ P ( 𝑃 ) , we conclude that 𝑃 ( ˜ 𝐸 ) = (cid:3) We can now combine Propositions 7, 8 and 9 into the following result, whichis the counterpart of Theorem 3 for lower prevision models. It yields a necessaryand sufficient condition for S-irrelevance and S-independence, expressed in termsof triviality, precision and an interval version of the factorisation property. To statethis factorisation condition, we adopt 𝑃 ( 𝑔 ) as a shorthand notation for the interval [ 𝑃 ( 𝑔 ) , 𝑃 ( 𝑔 )] and employ the so-called interval product [ 𝑎, 𝑏 ] ⊙ 𝑐 of a real interval [ 𝑎, 𝑏 ] with a real number 𝑐 , defined as [ 𝑎, 𝑏 ] ⊙ 𝑐 ≔ [ min { 𝑎𝑐, 𝑏𝑐 } , max { 𝑎𝑐, 𝑏𝑐 }] . We also adopt the convention that a singleton is identified with its unique element,which allows us to write 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) as a shorthand for 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) = : 𝑃 ( 𝑔 ) . Theorem 5
Consider any two events
𝐸, 𝐹 ⊆ Ω and any coherent lower prevision 𝑃 on ℒ . Then 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 if and only if 𝐸 is trivial or 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) and 𝑃 ( 𝑓 𝑔 ) = 𝑃 ( 𝑓 ) ⊙ 𝑃 ( 𝑔 ) for all 𝑓 ∈ ℒ 𝐸 and 𝑔 ∈ ℒ 𝐹 . (14) Similarly, 𝐸 and 𝐹 are S-independent if and only if 𝐸 or 𝐹 are trivial or 𝑃 ( 𝑓 ) = 𝑃 ( 𝑓 ) , 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) and 𝑃 ( 𝑓 𝑔 ) = 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) for all 𝑓 ∈ ℒ 𝐸 and 𝑔 ∈ ℒ 𝐹 . (15) Proof
We begin with the first statement. For necessity, assume that 𝐸 is non-trivialand S-irrelevant to 𝐹 . Consider any 𝑓 ∈ ℒ 𝐸 and 𝑔 ∈ ℒ 𝐹 . Proposition 9 thenguarantees that 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) ≕ 𝑃 ( 𝑔 ) . Consider now any 𝑄 ∈ P ( 𝑃 ) , then on theone hand 𝑄 ( 𝑔 ) = 𝑃 ( 𝑔 ) and on the other hand 𝑄 ( 𝑓 𝑔 ) = 𝑄 ( 𝑓 ) 𝑄 ( 𝑔 ) = 𝑄 ( 𝑓 ) 𝑃 ( 𝑔 ) by Proposition 7. Hence by taking infima and suprema over all 𝑄 ∈ P ( 𝑃 ) on bothsides, we get that 𝑃 ( 𝑓 𝑔 ) = ( 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) if 𝑃 ( 𝑔 ) ≥ 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) if 𝑃 ( 𝑔 ) ≤ 𝑃 ( 𝑓 𝑔 ) = ( 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) if 𝑃 ( 𝑔 ) ≥ 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) if 𝑃 ( 𝑔 ) ≤ , which can indeed be summarised as 𝑃 ( 𝑓 𝑔 ) = 𝑃 ( 𝑓 ) ⊙ 𝑃 ( 𝑔 ) .Next, we address sufficiency. Since we know from Proposition 8 that the trivi-ality of 𝐸 implies its S-irrelevance to 𝐹 , we can assume without loss of general-ity that Equation (14) holds, and prove that 𝐸 is S-irrelevant to 𝐹 . Consider any 𝑓 ∈ ℒ 𝐹 . Equation (14) then tells us that 𝑃 ( 𝑓 ) = 𝑃 ( 𝑓 ) and 𝑃 (− 𝑓 ) = 𝑃 (− 𝑓 ) . Fur-thermore, since 𝑃 ( 𝑓 ) = − 𝑃 (− 𝑓 ) , we find that 𝑃 ( 𝑓 ) = − 𝑃 (− 𝑓 ) . We now considertwo cases: 𝑃 ( 𝑓 ) ≥ 𝑃 ( 𝑓 ) ≤
0. If 𝑃 ( 𝑓 ) ≥
0, we infer from Equation (14) that 𝑃 ( I 𝐸 c 𝑓 ) = 𝑃 ( 𝐸 c ) 𝑃 ( 𝑓 ) ≥
0. If 𝑃 ( 𝑓 ) ≤
0, then 𝑃 (− 𝑓 ) = − 𝑃 ( 𝑓 ) ≥
0, so it followsfrom Equation (14) that 𝑃 (− I 𝐸 𝑓 ) = 𝑃 ( I 𝐸 (− 𝑓 )) = 𝑃 ( 𝐸 ) 𝑃 (− 𝑓 ) ≥
0. We conclude n a notion of independence proposed by Teddy Seidenfeld 21 that in all cases, 𝑃 ( I 𝐸 c 𝑓 ) ≥ 𝑃 (− I 𝐸 𝑓 ) ≥
0. Since this is true for every 𝑓 ∈ ℒ 𝐹 ,it follows that, indeed, 𝐸 is S-irrelevant to 𝐹 with respect to 𝐾 𝑃 .Next, we turn to the second statement. For necessity, assume that 𝐸 and 𝐹 are non-trivial and S-independent. Proposition 9 then guarantees that 𝑃 ( 𝑓 ) = 𝑃 ( 𝑓 ) ≕ 𝑃 ( 𝑓 ) and 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) ≕ 𝑃 ( 𝑔 ) for all 𝑓 ∈ ℒ 𝐸 and 𝑔 ∈ ℒ 𝐹 . For all 𝑄 ∈ P ( 𝑃 ) , Propos-ition 7 then guarantees that 𝑄 ( 𝑓 𝑔 ) = 𝑄 ( 𝑓 ) 𝑄 ( 𝑔 ) = 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) , and therefore, bytaking infima and suprema over all 𝑄 ∈ P ( 𝑃 ) on both sides, we get that, indeed, 𝑃 ( 𝑓 𝑔 ) = 𝑃 ( 𝑓 𝑔 ) = 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) .We now turn to sufficiency. Since we know from Proposition 8 that the trivialityof 𝐸 or 𝐹 implies their S-independence, we can assume without loss of generality thatEquation (15) holds, and prove that 𝐸 and 𝐹 are S-independent. Since Equation (15)implies Equation (14), the S-irrelevance of 𝐸 to 𝐹 follows from the first part ofthis theorem. Since Equation (15) is symmetric in 𝐸 and 𝐹 , the S-irrelevance of 𝐹 to 𝐸 follows in exactly the same way. Hence, we find that, indeed, 𝐸 and 𝐹 areS-independent with respect to 𝐾 𝑃 . (cid:3) In combination with Proposition 5, this Theorem 5 also yields characterisationsfor S-irrelevance and S-independence for Archimedean sets of desirable optionsets 𝐾 , in terms of their representing lower previsions. In particular, we see thatin the absence of triviality, S-independence implies precision and factorisation for(products of) gambles on 𝐸 and 𝐹 , without the need for imposing mixingness . In theremainder of this section, we seek to exclude the trivial cases by considering a newpotential property for events with respect to a coherent set of desirable option sets 𝐾 .We start by introducing the notion of credibility: we say that 𝐸 is credible withrespect to a coherent set of desirable option sets 𝐾 whenever (∃ 𝜖 > ){ I 𝐸 − 𝜖 } ∈ 𝐾, meaning that our subject is willing to bet on 𝐸 at some positive—but possibly verysmall—betting rate 𝜖 . For an Archimedean 𝐾 , this is equivalent to the lower prob-ability of 𝐸 being strictly bounded below by 𝜖 for all representing lower previsions. Proposition 10
If the set of desirable option sets 𝐾 is Archimedean, an event 𝐸 ⊆ Ω is credible with respect to 𝐾 if and only if there is some real 𝜖 > such that 𝑃 ( 𝐸 ) > 𝜖 for all 𝑃 ∈ P ( 𝐾 ) . Proof
Consider any 𝜖 >
0. Then for any 𝑃 ∈ P ( 𝐾 ) , { I 𝐸 − 𝜖 } ∈ 𝐾 𝑃 if and onlyif 𝑃 ( I 𝐸 − 𝜖 ) >
0, or equivalently—since lower previsions are constant additive[LP ]— 𝑃 ( I 𝐸 ) > 𝜖 . The result is therefore an immediate consequence of the fact that 𝐾 = Ñ { 𝐾 𝑃 : 𝑃 ∈ P ( 𝐾 )} . (cid:3) We now say that an event 𝐸 is credibly indeterminate with respect to a coherent setof desirable option sets 𝐾 if 𝐸 and 𝐸 c are both credible with respect to 𝐾 , meaning thatour subject is willing to bet both on and against 𝐸 at some positive betting rate. Thiscondition of credible indeterminacy, when combined with S-independence, allowsus to infer both precision and factorisation for every representing lower prevision ofan Archimedean set of desirable option sets 𝐾 , even without our having to imposemixingness! Theorem 6
Consider any two events
𝐸, 𝐹 ⊆ Ω and an Archimedean set of desirableoption sets 𝐾 . If 𝐸 is credibly indeterminate and S-irrelevant to 𝐹 with respect to 𝐾 ,then for all 𝑃 ∈ P ( 𝐾 ) : 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) and 𝑃 ( 𝑓 𝑔 ) = 𝑃 ( 𝑓 ) ⊙ 𝑃 ( 𝑔 ) for all 𝑓 ∈ ℒ 𝐸 and 𝑔 ∈ ℒ 𝐹 .Similarly, if 𝐸 and 𝐹 are credibly indeterminate and S-independent with respectto 𝐾 , then for all 𝑃 ∈ P ( 𝐾 ) : 𝑃 ( 𝑓 ) = 𝑃 ( 𝑓 ) , 𝑃 ( 𝑔 ) = 𝑃 ( 𝑔 ) and 𝑃 ( 𝑓 𝑔 ) = 𝑃 ( 𝑓 ) 𝑃 ( 𝑔 ) for all 𝑓 ∈ ℒ 𝐸 and 𝑔 ∈ ℒ 𝐹 . Proof
Due to Proposition 5 and Theorem 5, it clearly suffices to show that thecredible indeterminacy of an event 𝐸 implies that 𝐸 is non-trivial with respect toevery 𝑃 ∈ P ( 𝐾 ) . So assume that 𝐸 is credibly indeterminate with respect to 𝐾 ,meaning that ˜ 𝐸 is credible for each ˜ 𝐸 ∈ { 𝐸, 𝐸 c } . For any 𝑃 ∈ P ( 𝐾 ) , it then followsfrom Proposition 10 that there is some 𝜖 > 𝑃 ( ˜ 𝐸 ) > 𝜖 . Hence, since 𝑃 ( ˜ 𝐸 ) ≥ 𝑃 ( ˜ 𝐸 ) > 𝜖 >
0, we see that 𝐸 is indeed non-trivial with respect to 𝑃 . (cid:3) Let us now extend the discussion from events to variables. We still assume that oursubject’s preferences are modelled by a set of desirable option sets 𝐾 , where thepossible options are the gambles on the possibility space Ω .We will follow the often used device of representing a variable 𝑍 as a map that isdefined on the possibility space Ω : 𝑍 : Ω → 𝒵 : 𝜔 ↦→ 𝑍 ( 𝜔 ) , where we denote by 𝒵 the set of possible values of the variable 𝑍 . The idea behindthis device is that since our subject is uncertain about the value that 𝜔 assumes in Ω ,she will typically also be uncertain about the value assumed by 𝑍 ( 𝜔 ) in 𝒵 .If we want to talk about decisions involving the value of the variable 𝑍 , we needto consider uncertain rewards whose value depends only on the value of 𝑍 , or morespecifically, gambles on Ω of the type ℎ ( 𝑍 ) ≔ ℎ ◦ 𝑍 : Ω → R : 𝜔 ↦→ ℎ ( 𝑍 ( 𝜔 )) , where ℎ is any gamble on 𝒵 .In particular, with any 𝐸 ⊆ 𝒵 , we can associate the indicator (gamble) I 𝐸 on 𝒵 ,which corresponds to a gamble I 𝐸 ( 𝑍 ) on Ω . Since I 𝐸 ( 𝑍 ) = I 𝐸 ◦ 𝑍 = I 𝑍 − ( 𝐸 ) , we see Although it is related to what we call a variable in spirit, we will refrain from using the term‘random variable’, as that is typically associated with precise and countable additive probabilitymodels, and typically comes with a measurability requirement.n a notion of independence proposed by Teddy Seidenfeld 23 that I 𝐸 ( 𝑍 ) is an indicator on Ω . The event 𝑍 − ( 𝐸 ) ⊆ Ω that it indicates, correspondsto the proposition ‘ 𝑍 ∈ 𝐸 ’. To see what an assessment of S-irrelevance might mean for variables, we considertwo variables 𝑋 and 𝑌 , which we model as maps on the possibility space Ω : 𝑋 : Ω → 𝒳 and 𝑌 : Ω → 𝒴 , where the respective non-empty sets 𝒳 and 𝒴 are the sets of possible values for 𝑋 and 𝑌 . Seidenfeld’s ‘independence’ requirement can then be extended straightfor-wardly from events to variables as follows: When two variables, 𝑋 and 𝑌 , are ‘independent’ then it is not reasonable to spend resources in order to use the observed value of one of them, say 𝑋 , to choose between options thatdepend solely on the value of the other variable, 𝑌 . As before for events, we recognise the essentially asymmetrical nature of this require-ment, and will try to formulate a requirement of S-irrelevance of 𝑋 to 𝑌 . Because wecare about the operational meaning of our criterion, we will allow the variables 𝑋 and 𝑌 to assume infinitely many values, but want to keep our observations of thevalues of 𝑋 and the choices between gambles on 𝑌 finitary. Consequently, observingthe value of 𝑋 will be modelled by choosing a finite partition P of the set 𝒳 , andfinding out which event 𝐸 ∈ P in that partition obtains, meaning that 𝑋 ∈ 𝐸 . Foreach such possible observation 𝐸 , we consider a gamble 𝑠 𝐸 : 𝒴 → R on the valueof 𝑌 , which we will assume to be finite-valued —simple—in accordance with ourfinitary approach. Let us denote by 𝒮 𝒴 the set of all simple gambles on 𝒴 .In summary, our subject needs to choose between the gambles 𝑠 𝐸 , 𝐸 ∈ P onthe value of 𝑌 , or in other words, between the gambles 𝑠 𝐸 ( 𝑌 ) ≔ 𝑠 𝐸 ◦ 𝑌 on thepossibility space Ω . Observing the value of 𝑋 to choose between these gamblesthen corresponds to the composite gamble Í 𝐺 ∈P I 𝐺 ( 𝑋 ) 𝑠 𝐺 ( 𝑌 ) on the possibilityspace Ω . The purport of Seidenfeld’s requirement is that when our subject judges 𝑋 to be irrelevant to 𝑌 , she should reject the composite option Í 𝐺 ∈P I 𝐺 ( 𝑋 ) 𝑠 𝐺 ( 𝑌 ) − 𝜖 from the set of options { 𝑠 𝐸 ( 𝑌 ) : 𝐸 ∈ P} ∪ (cid:26) Õ 𝐺 ∈P I 𝐺 ( 𝑋 ) 𝑠 𝐺 ( 𝑌 ) − 𝜖 (cid:27) , for all real 𝜖 >
0. Following the discussion in Sections 2 and 3, and Equation (1) inparticular, this means that the option set An engaged reader will be able to verify further on that, despite our insistence on a finitaryapproach, this doesn’t really matter from a mathematical point of view. In particular, none of ourproofs—even the ones that establish sufficient conditions for S-irrelevance or S-independence forvariables—will actually require the restriction that the gambles 𝑠 𝐸 —or 𝑠 𝐺 —should be simple.4 De Bock and De Cooman (cid:26) 𝑠 𝐸 ( 𝑌 ) − Õ 𝐺 ∈P I 𝐺 ( 𝑋 ) 𝑠 𝐺 ( 𝑌 ) + 𝜖 : 𝐸 ∈ P (cid:27) = (cid:26) Õ 𝐺 ∈P I 𝐺 ( 𝑋 ) 𝑠 𝐸 ( 𝑌 ) − Õ 𝐺 ∈P I 𝐺 ( 𝑋 ) 𝑠 𝐺 ( 𝑌 ) + 𝜖 : 𝐸 ∈ P (cid:27) = (cid:26) Õ 𝐺 ∈P\{ 𝐸 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] + 𝜖 : 𝐸 ∈ P (cid:27) must be desirable for our subject. This leads to the following definitions. Definition 9 (S-irrelevance and S-independence for variables)
Consider two vari-ables 𝑋 and 𝑌 and a coherent set of desirable option sets 𝐾 . We say that 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 if (cid:26) Õ 𝐺 ∈P\{ 𝐸 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] + 𝜖 : 𝐸 ∈ P (cid:27) ∈ 𝐾 for all finite partitions P of 𝒳 , all 𝑠 𝐸 ∈ 𝒮 𝒴 and all 𝜖 > . (16)We say that 𝑋 and 𝑌 are S-independent with respect to 𝐾 if 𝑋 is S-irrelevant to 𝑌 and 𝑌 is S-irrelevant to 𝑋 .Interestingly, the assessments for variables imply similar assessments for events. Proposition 11
Consider two variables 𝑋 and 𝑌 and a coherent set of desirableoption sets 𝐾 . If 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 , then for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , the event 𝑋 − ( 𝐸 ) is S-irrelevant to the event 𝑌 − ( 𝐹 ) with respect to 𝐾 .Similarly, if 𝑋 and 𝑌 are S-independent with respect to 𝐾 , then for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , the events 𝑋 − ( 𝐸 ) and 𝑌 − ( 𝐹 ) are S-independent with respect to 𝐾 . Proof
It clearly suffices to give the proof for S-irrelevance. So consider any 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 and any gamble 𝑓 ∈ ℒ 𝑌 − ( 𝐹 ) ( Ω ) . Then 𝑓 is completely characterisedby the real values 𝑓 𝐹 and 𝑓 𝐹 c it assumes on 𝑌 − ( 𝐹 ) and ( 𝑌 − ( 𝐹 )) c = 𝑌 − ( 𝐹 c ) ,respectively: 𝑓 = 𝑓 𝐹 I 𝑌 − ( 𝐹 ) + 𝑓 𝐹 c I 𝑌 − ( 𝐹 c ) . If we define the simple gamble 𝑔 ≔ 𝑓 𝐹 I 𝐹 + 𝑓 𝐹 c I 𝐹 c ∈ 𝒮 𝒴 , then clearly 𝑓 = 𝑔 ( 𝑌 ) . We now consider the partition P ≔ { 𝐸, 𝐸 c } of 𝒳 and the corresponding simple gambles 𝑠 𝐸 ≔ 𝑔 and 𝑠 𝐸 c ≔ 𝒴 . Then theS-irrelevance of 𝑋 to 𝑌 implies in particular that for all 𝜖 > 𝐾 ∋ (cid:8) I 𝐸 c ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐸 c ( 𝑌 )] + 𝜖, I 𝐸 ( 𝑋 ) [ 𝑠 𝐸 c ( 𝑌 ) − 𝑠 𝐸 ( 𝑌 )] + 𝜖 (cid:9) = (cid:8) I 𝐸 c ( 𝑋 ) 𝑔 ( 𝑌 ) + 𝜖, − I 𝐸 ( 𝑋 ) 𝑔 ( 𝑌 ) + 𝜖 (cid:9) = (cid:8) I 𝑋 − ( 𝐸 c ) 𝑓 + 𝜖, − I 𝑋 − ( 𝐸 ) 𝑓 + 𝜖 (cid:9) . Since 𝑋 − ( 𝐸 c ) = ( 𝑋 − ( 𝐸 )) c , this indeed tells us that 𝑋 − ( 𝐸 ) is S-irrelevantto 𝑌 − ( 𝐹 ) with respect to 𝐾 . (cid:3) A straightforward generalisation, mutatis mutandis , of Proposition 12 indicatesthat S-irrelevance and S-independence of variables with respect to Archimedean n a notion of independence proposed by Teddy Seidenfeld 25 (and mixing) models are completely determined by the corresponding notions fortheir dominating binary models.
Proposition 12
Consider two variables 𝑋 and 𝑌 and a set of desirable option sets 𝐾 .If 𝐾 is Archimedean, then 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 if and only if 𝑋 isS-irrelevant to 𝑌 with respect to 𝐾 𝑃 for all 𝑃 ∈ P ( 𝐾 ) . Similarly, if 𝐾 is Archimedeanand mixing, then 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 if and only if 𝑋 is S-irrelevantto 𝑌 with respect to 𝐾 𝑃 for all 𝑃 ∈ P ( 𝐾 ) . Proof
If we combine Definitions 4 and 9, we see that 𝑋 is S-irrelevant to 𝑌 withrespect to an Archimedean 𝐾 if and only if (cid:26) Õ 𝐺 ∈P\{ 𝐸 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] + 𝜖 : 𝐸 ∈ P (cid:27) ∈ Ù { 𝐾 𝑃 : 𝑃 ∈ P ( 𝐾 )} for all finite partitions P of 𝒳 , all 𝑠 𝐸 ∈ 𝒮 𝒴 and all 𝜖 > . which proves the statement for Archimedeanity. If we also impose mixingness on 𝐾 ,the second statement follows at once from the first and the fact that then, accordingto Theorem 2, P ( 𝐾 ) = P ( 𝐾 ) . (cid:3) Because of Proposition 12, it will be useful in this section to consider and studyS-irrelevance with respect to the binary model 𝐾 𝑃 associated with a coherent lowerprevision 𝑃 on ℒ . Clearly, 𝑋 will be S-irrelevant to 𝑌 with respect to 𝐾 𝑃 if and onlyif max (cid:26) 𝑃 (cid:18) Õ 𝐺 ∈P\{ 𝐸 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] (cid:19) : 𝐸 ∈ P (cid:27) ≥ P of 𝒳 and all 𝑠 𝐸 ∈ 𝒮 𝒴 . (17)As a first step, we recall from Proposition 11 that S-irrelevance for variablesimplies S-irrelevance for events. This allows us to apply some of the results inSection 3 and generalise them to the present context of variables. To get our feetwet, we look at Proposition 7, whose generalisation requires a notion of independentvariables for linear previsions. Definition 10 (Independent variables with respect to a linear prevision)
We calltwo variables 𝑋 and 𝑌 independent with respect to a linear prevision 𝑃 on ℒ if forall 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , the events 𝑋 − ( 𝐸 ) and 𝑌 − ( 𝐹 ) are independent with respectto 𝑃 .Its characterisation in terms of gambles on 𝑋 and 𝑌 follows a fairly standard argumentbased on the uniform density of the simple gambles for the set of all gambles. Weinclude the simple proof for the sake of completeness. Proposition 13
For any linear prevision 𝑃 , two variables 𝑋 and 𝑌 are independentwith respect to 𝑃 if and only if 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 ( 𝑓 ( 𝑋 )) 𝑃 ( 𝑔 ( 𝑌 )) for all gambles 𝑓 on 𝒳 and all gambles 𝑔 on 𝒴 . Proof
The sufficiency part of the proof is immediate: it suffices to let 𝑓 ≔ I 𝐸 and 𝑔 ≔ I 𝐹 . For the necessity part, we assume that 𝑋 and 𝑌 are independent withrespect to 𝑃 and consider any gamble 𝑓 on 𝒳 and 𝑔 on 𝒴 . We need to provethat 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 ( 𝑓 ( 𝑋 )) 𝑃 ( 𝑔 ( 𝑌 )) . It suffices to give the proof for simple—finite valued—gambles 𝑓 and 𝑔 , because any gamble is a uniform limit of simplegambles, and because coherent lower previsions, and hence also linear previsions,are guaranteed to be uniformly continuous [see LP ]. We may therefore assumethat 𝑓 = Í 𝐸 ∈P 𝒳 𝑓 𝐸 I 𝐸 for some partition P 𝒳 of 𝒳 and some choice of the realnumbers 𝑓 𝐸 . Similarly, we may assume that 𝑔 = Í 𝐹 ∈P 𝒴 𝑔 𝐹 I 𝐹 for some partition P 𝒳 of 𝒴 and some choice of the real numbers 𝑔 𝐹 . But then 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 (cid:18) Õ 𝐸 ∈P 𝒳 Õ 𝐹 ∈P 𝒴 𝑓 𝐸 𝑔 𝐹 I 𝐸 ( 𝑋 ) I 𝐹 ( 𝑌 ) (cid:19) = Õ 𝐸 ∈P 𝒳 Õ 𝐹 ∈P 𝒴 𝑓 𝐸 𝑔 𝐹 𝑃 ( I 𝐸 ( 𝑋 ) I 𝐹 ( 𝑌 )) = Õ 𝐸 ∈P 𝒳 Õ 𝐹 ∈P 𝒴 𝑓 𝐸 𝑔 𝐹 𝑃 ( I 𝐸 ( 𝑋 )) 𝑃 ( I 𝐹 ( 𝑌 )) = Õ 𝐸 ∈P 𝒳 𝑓 𝐸 𝑃 ( I 𝐸 ( 𝑋 )) Õ 𝐹 ∈P 𝒴 𝑔 𝐹 𝑃 ( I 𝐹 ( 𝑌 )) = 𝑃 (cid:18) Õ 𝐸 ∈P 𝒳 𝑓 𝐸 I 𝐸 ( 𝑋 ) (cid:19) 𝑃 (cid:18) Õ 𝐹 ∈P 𝒴 𝑔 𝐹 I 𝐹 ( 𝑌 ) (cid:19) = 𝑃 ( 𝑓 ( 𝑋 )) 𝑃 ( 𝑔 ( 𝑌 )) . where the second and fifth equality follow from coherence [use P and P ], and thecrucial third equality follows from the assumption that 𝑋 and 𝑌 are independent withrespect to 𝑃 . (cid:3) With independence out of the way, we can now address, as announced, thegeneralisation of Proposition 7 from events to variables.
Proposition 14
Consider any coherent lower prevision 𝑃 on ℒ , and two variables 𝑋 and 𝑌 . If for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) with respectto 𝐾 𝑃 , then 𝑋 and 𝑌 are independent with respect to all 𝑃 ∈ P ( 𝑃 ) . Proof
Consider any 𝑃 ∈ P ( 𝑃 ) and any 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 . Then since 𝑋 − ( 𝐸 ) isS-irrelevant to 𝑌 − ( 𝐹 ) with respect to 𝐾 𝑃 , it follows from Proposition 7 that 𝑋 − ( 𝐸 ) and 𝑌 − ( 𝐹 ) are independent with respect to 𝑃 . Now use Proposition 13. (cid:3) To extend Proposition 9 to the context of variables, we first need to introduceconcepts of triviality and precision for variables. n a notion of independence proposed by Teddy Seidenfeld 27
We begin with precision. It makes perfect sense to call the lower prevision 𝑃 𝑍 onthe set ℒ ( 𝒵 ) of all gambles on 𝒵 , defined by 𝑃 𝑍 ( ℎ ) ≔ 𝑃 ( ℎ ◦ 𝑍 ) for all gambles ℎ on 𝒵 the (lower) distribution of the variable 𝑍 with respect to the coherent lowerprevision 𝑃 . We will say that a variable 𝑍 : Ω → 𝒵 has a precise distribu-tion with respect to 𝑃 if the gamble ℎ ( 𝑍 ) = ℎ ◦ 𝑍 on Ω has a precise previ-sion 𝑃 ( ℎ ◦ 𝑍 ) = 𝑃 ( ℎ ◦ 𝑍 ) ≕ 𝑃 𝑍 ( ℎ ) for all gambles ℎ on 𝒵 , or in other words if thedistribution 𝑃 𝑍 of 𝑍 with respect to 𝑃 is a linear prevision, then denoted by 𝑃 𝑍 .This notion of precision can also be expressed in terms of events. Proposition 15
A variable 𝑍 : Ω → 𝒵 has a precise distribution with respect to acoherent lower prevision 𝑃 if and only if the events 𝑍 − ( 𝐸 ) have a precise probabilitywith respect to 𝑃 for all 𝐸 ⊆ 𝒵 . Proof
Necessity is immediate, so we concentrate on sufficiency. We assume that theevents 𝑍 − ( 𝐸 ) have a precise probability for all 𝐸 ⊆ 𝒵 , and prove that the gambles ℎ ( 𝑍 ) = ℎ ◦ 𝑍 have a precise prevision for all gambles ℎ on 𝒵 . Since a coherent lowerprevision is uniformly continuous [see LP ], and since all gambles are uniform limitsof simple gambles, it suffices to give the proof for simple gambles 𝑠 = Í 𝑛𝑘 = 𝑠 𝑘 I 𝐸 𝑘 ,where the 𝐸 𝑘 constitute a partition of 𝒵 and the 𝑠 𝑘 ∈ R . We may assume withoutloss of generality that 𝑠 is non-negative, so all 𝑠 𝑘 ≥
0, due to the constant additivity[LP ] of a coherent lower prevision. Hence, indeed, 𝑃 ( 𝑠 ◦ 𝑍 ) = 𝑃 (cid:18) 𝑛 Õ 𝑘 = 𝑠 𝑘 I 𝑍 − ( 𝐸 𝑘 ) (cid:19) ≥ 𝑛 Õ 𝑘 = 𝑠 𝑘 𝑃 ( 𝑍 − ( 𝐸 𝑘 )) = 𝑛 Õ 𝑘 = 𝑠 𝑘 𝑃 ( 𝑍 − ( 𝐸 𝑘 )) ≥ 𝑃 (cid:18) 𝑛 Õ 𝑘 = 𝑠 𝑘 I 𝑍 − ( 𝐸 𝑘 ) (cid:19) = 𝑃 ( 𝑠 ◦ 𝑍 ) , where the first inequality follows from the super-linearity of the coherent lowerprevision 𝑃 [combine LP and LP ], the second equality from the assumption,and the second inequality from the sub-linearity of the coherent upper prevision 𝑃 [combine LP and LP ]. (cid:3) Let us call a variable 𝑍 : Ω → 𝒵 trivial with respect to a coherent lower pre-vision 𝑃 if 𝑍 − ( 𝐺 ) is trivial with respect to 𝑃 for all subsets 𝐺 ⊆ 𝒵 , meaningthat 𝑃 𝑍 ( 𝐺 ) = 𝑃 𝑍 ( 𝐺 c ) = 𝐺 ⊆ 𝒵 . The distribution 𝑃 𝑍 is then clearlyprecise on all events, and therefore also a linear prevision 𝑃 𝑍 on all gambles [useProposition 15]. It is the—degenerate—linear prevision given by 𝑃 𝑍 ( ℎ ) = sup 𝐸 ∈ 𝒰 𝑃,𝑍 inf 𝑧 ∈ 𝐸 ℎ ( 𝑧 ) = inf 𝐸 ∈ 𝒰 𝑃 ,𝑍 sup 𝑧 ∈ 𝐸 ℎ ( 𝑧 ) for all gambles ℎ on 𝒵 , where the collection of practically certain events 𝒰 𝑃,𝑍 ≔ { 𝐺 ⊆ 𝒵 : 𝑃 ( 𝑍 − ( 𝐺 )) = } = { 𝐺 ⊆ 𝒵 : 𝑃 𝑍 ( 𝐺 ) = } is an ultrafilter of events on 𝒵 ; see for instance [28, Section 5.5] and [33, Sec-tions 2.9.8 and 3.2.6]. If the ultrafilter 𝒰 𝑃,𝑍 is fixed , meaning that Ñ 𝒰 𝑃,𝑍 = { 𝑧 𝑜 } for some 𝑧 𝑜 ∈ 𝒵 , then 𝑃 𝑍 ( ℎ ) = ℎ ( 𝑧 𝑜 ) , so all probability mass of the precise dis-tribution 𝑃 𝑍 is concentrated in 𝑧 𝑜 . The only other possibility is that the ultrafilter 𝒰 𝑃,𝑍 is free , meaning that Ñ 𝒰 𝑃,𝑍 = ∅ , and then typically all probability mass willlie infinitesimally close to some 𝑧 𝑜 in 𝒵 , or to some ‘point on the boundary’ of 𝒵 .In both cases, this represents a model for our subject’s certainty that 𝑍 assumes afixed value; see also the extensive discussion in Section 5.5.5 of [28].Proposition 9 now generalises fairly easily from events to variables. Proposition 16
Consider a coherent lower prevision 𝑃 and two variables 𝑋 and 𝑌 .If for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) with respect to 𝐾 𝑃 ,then 𝑋 is trivial with respect to 𝑃 or 𝑌 has a precise distribution with respect to 𝑃 . Proof
Assume that 𝑋 is not trivial with respect to 𝑃 . We then need to show that 𝑌 has a precise distribution with respect to 𝑃 . Due to Proposition 15, it suffices toconsider any 𝐹 ⊆ 𝒴 and prove that 𝑃 ( 𝑌 − ( 𝐹 )) = 𝑃 ( 𝑌 − ( 𝐹 )) .Since 𝑋 is not trivial with respect to 𝑃 , there is some 𝐸 ⊆ 𝒳 such that the event 𝑋 − ( 𝐸 ) is not trivial with respect to 𝑃 . Since 𝑋 is S-irrelevant to 𝑌 with respectto 𝐾 𝑃 , we also know that 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) with respect to 𝐾 𝑃 , byProposition 11. It therefore follows from Proposition 9that 𝑃 ( 𝑌 − ( 𝐹 )) = 𝑃 ( 𝑌 − ( 𝐹 )) ,as required. (cid:3) We showed in Proposition 11 that S-irrelevance for variables implies S-irrelevancefor the corresponding families of events. It turns out that for sets of desirable optionsets that are Archimedean, these notions are equivalent; see Theorem 8 further on.We start out by establishing this result for binary sets of desirable option sets of theform 𝐾 𝑃 , using the results in Propositions 11, 13, 14 and 16. Theorem 7
Consider a coherent lower prevision 𝑃 on ℒ , and two variables 𝑋 and 𝑌 . Then 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 𝑃 if and only if, for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) with respect to 𝐾 𝑃 . Besides on the mentioned propositions, our proof for this theorem also dependson two lemmas: Lemma 1 from before, and the following simple consequence oftriviality for variables.
Lemma 4
Consider any variable 𝑍 : Ω → 𝒵 and any finite partition P of 𝒵 .Assume that 𝑍 is trivial with respect to a coherent lower prevision 𝑃 . Then there issome 𝐸 𝑜 ∈ P such that 𝑃 ( 𝑍 − ( 𝐸 )) = for all 𝐸 ∈ P \ { 𝐸 𝑜 } . Proof If 𝑃 ( 𝑍 − ( 𝐸 )) = 𝐸 ∈ P , then we are done. Without loss of generality,we may therefore assume that there is at least one 𝐸 𝑜 ∈ P such that 𝑃 ( 𝑍 − ( 𝐸 𝑜 )) >
0. Hence, since the triviality of 𝑍 implies the triviality of 𝐸 𝑜 , it must be that 𝑃 ( 𝑍 − ( 𝐸 c 𝑜 )) =
0. Consider now any 𝐸 ∈ P \ { 𝐸 𝑜 } . Since 𝐸 and 𝐸 𝑜 are disjoint,we have that 𝐸 ⊆ 𝐸 c 𝑜 . It therefore follows from coherence [use LP and LP ] that0 ≤ 𝑃 ( 𝑍 − ( 𝐸 )) ≤ 𝑃 ( 𝑍 − ( 𝐸 c 𝑜 )) =
0, so 𝑃 ( 𝑍 − ( 𝐸 )) = (cid:3) n a notion of independence proposed by Teddy Seidenfeld 29 Proof of Theorem 7
Necessity is immediate from Proposition 11, so it remains toprove sufficiency. So let us assume that 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) with respectto 𝐾 𝑃 , for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 . We need to prove that 𝑋 is S-irrelevant to 𝑌 withrespect to 𝐾 𝑃 . We will do this using the criterion (17), and consider, to this end, anypartition P of 𝒳 and any choice of simple gambles 𝑠 𝐸 ∈ 𝒮 𝒴 for 𝐸 ∈ P . There arenow two possible cases.The first case we consider is that 𝑋 is trivial with respect to 𝑃 . We then infer fromLemma 4 that there is some 𝐸 𝑜 ∈ P such that 𝑃 ( 𝑋 − ( 𝐸 )) = 𝐸 ∈ P \ { 𝐸 𝑜 } .Repeated invocation of Lemma 1 then guarantees that 𝑃 (cid:18) Õ 𝐺 ∈P\{ 𝐸 𝑜 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 𝑜 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] (cid:19) = , so this case is dealt with.Next, we consider the case where 𝑋 is not trivial with respect to 𝑃 . In that case,it follows from the assumption and Proposition 16 that 𝑌 has a precise distribution 𝑃 𝑌 with respect to 𝑃 . For any 𝐸 ∈ P and any 𝑄 ∈ P ( 𝑃 ) , we then get that 𝑄 (cid:18) Õ 𝐺 ∈P\{ 𝐸 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] (cid:19) = 𝑄 (cid:18) 𝑠 𝐸 ( 𝑌 ) − Õ 𝐺 ∈P I 𝐺 ( 𝑋 ) 𝑠 𝐺 ( 𝑌 ) (cid:19) = 𝑄 ( 𝑠 𝐸 ( 𝑌 )) − Õ 𝐺 ∈P 𝑄 (cid:0) I 𝐺 ( 𝑋 ) 𝑠 𝐺 ( 𝑌 ) (cid:1) = 𝑄 ( 𝑠 𝐸 ( 𝑌 )) − Õ 𝐺 ∈P 𝑄 (cid:0) I 𝐺 ( 𝑋 ) (cid:1) 𝑄 (cid:0) 𝑠 𝐺 ( 𝑌 ) (cid:1) = 𝑃 𝑌 ( 𝑠 𝐸 ) − Õ 𝐺 ∈P 𝑄 ( I 𝐺 ( 𝑋 )) 𝑃 𝑌 ( 𝑠 𝐺 ) , where the crucial third equality follows from Propositions 14 and 13. Hence, for any 𝐸 ∈ P : 𝑃 (cid:18) Õ 𝐺 ∈P\{ 𝐸 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] (cid:19) = min 𝑄 ∈ P ( 𝑃 ) (cid:18) 𝑃 𝑌 ( 𝑠 𝐸 ) − Õ 𝐺 ∈P 𝑄 ( I 𝐺 ( 𝑋 )) 𝑃 𝑌 ( 𝑠 𝐺 ) (cid:19) = 𝑃 𝑌 ( 𝑠 𝐸 ) − max 𝑄 ∈ P ( 𝑃 ) Õ 𝐺 ∈P 𝑄 ( I 𝐺 ( 𝑋 )) 𝑃 𝑌 ( 𝑠 𝐺 )≥ 𝑃 𝑌 ( 𝑠 𝐸 ) − max 𝐺 ∈P 𝑃 𝑌 ( 𝑠 𝐺 ) , where the inequality holds because Í 𝐺 ∈P 𝑄 ( I 𝐺 ( 𝑋 )) 𝑃 𝑌 ( 𝑠 𝐺 ) is a convex combinationof the terms 𝑃 𝑌 ( 𝑠 𝐺 ) , 𝐺 ∈ P , and is therefore dominated by their maximum. Thistells us thatmax 𝐸 ∈P 𝑃 (cid:18) Õ 𝐺 ∈P\{ 𝐸 } I 𝐺 ( 𝑋 ) [ 𝑠 𝐸 ( 𝑌 ) − 𝑠 𝐺 ( 𝑌 )] (cid:19) ≥ max 𝐸 ∈P 𝑃 𝑌 ( 𝑠 𝐸 ) − max 𝐺 ∈P 𝑃 𝑌 ( 𝑠 𝐺 ) = , as required. (cid:3) Theorem 7 generalises easily to general Archimedean sets of desirable option setsbecause for those, S-irrelevance can be expressed in terms of the representing lowerprevisions; see Propositions 5 and 12. This yields the following simple characterisa-tion of S-irrelevance for variables in terms of S-irrelevance for events. It provides an ex post justification for our having focused on the latter first, and for having paid somuch attention to it in Section 3.
Theorem 8
Let 𝐾 be an Archimedean set of desirable option sets and consider twovariables 𝑋 and 𝑌 . Then 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 if and only if, forall 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) with respect to 𝐾 . Proof
Immediate from Theorem 7 and Propositions 5 and 12. (cid:3)
The characterisation of S-irrelevance for variables in terms of S-irrelevance forevents in Theorem 7 also leads to a fairly easily proven generalisation of Proposition 8,describing the implications of triviality.
Proposition 17
If the variables 𝑋 or 𝑌 are trivial with respect to a coherent lowerprevision 𝑃 , then 𝑋 and 𝑌 are S-independent with respect to 𝐾 𝑃 . Proof
Assume that 𝑋 or 𝑌 is trivial with respect to 𝑃 . Consider any 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 . If the variable 𝑋 is trivial with respect to 𝑃 , then the event 𝑋 − ( 𝐸 ) is trivialwith respect to 𝑃 . Similarly, if 𝑌 is trivial with respect to 𝑃 , then 𝑌 − ( 𝐹 ) is as well. Inboth cases, it follows from Proposition 8 that 𝑋 − ( 𝐸 ) and 𝑌 − ( 𝐹 ) are S-independentwith respect to 𝐾 𝑃 . Since this is true for every 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , it follows fromTheorem 7 that 𝑋 and 𝑌 are S-independent with respect to 𝐾 𝑃 . (cid:3) All this preparatory work is about to bear fruit in the final two theorems of thissection. The following characterisation provides better insight into what—and howsurprisingly strong—the implications of an S-irrelevance assessment really are.
Theorem 9
Consider a coherent lower prevision 𝑃 on ℒ and two variables 𝑋 and 𝑌 .Then 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 𝑃 if and only if 𝑋 is trivial with respectto 𝑃 , or if 𝑌 has a precise distribution 𝑃 𝑌 with respect to 𝑃 and 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 ( 𝑓 ( 𝑋 )) ⊙ 𝑃 𝑌 ( 𝑔 ) for all gambles 𝑓 on 𝒳 and 𝑔 on 𝒴 . (18) Similarly, 𝑋 and 𝑌 are S-independent with respect to 𝐾 𝑃 if and only if 𝑋 or 𝑌 aretrivial with respect to 𝑃 , or if they both have precise distributions 𝑃 𝑋 and 𝑃 𝑌 withrespect to 𝑃 and 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 𝑋 ( 𝑓 ) 𝑃 𝑌 ( 𝑔 ) for all gambles 𝑓 on 𝒳 and 𝑔 on 𝒴 . (19) Since ‘factorisation’ of this kind for multiple variables leads to a version of the law of largenumbers [15, 18], it doesn’t seem too farfetched to envision extensions of S-irrelevance and S-independence from two to multiple variables that allow us to prove similar laws of large numbers.n a notion of independence proposed by Teddy Seidenfeld 31
Proof
We begin with the first statement. For necessity, assume that 𝑋 is non-trivialwith respect to 𝑃 and S-irrelevant to 𝑌 with respect to 𝐾 𝑃 . Theorem 7 and Proposi-tion 16 then guarantee that 𝑌 has a precise distribution 𝑃 𝑌 with respect to 𝑃 . Considernow any 𝑓 ∈ ℒ ( 𝒳 ) , 𝑔 ∈ ℒ ( 𝒴 ) and 𝑄 ∈ P ( 𝑃 ) , then on the one hand 𝑄 ( 𝑔 ( 𝑌 )) = 𝑃 𝑌 ( 𝑔 ) and on the other hand 𝑄 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑄 ( 𝑓 ( 𝑋 )) 𝑄 ( 𝑔 ( 𝑌 )) = 𝑄 ( 𝑓 ( 𝑋 )) 𝑃 𝑌 ( 𝑔 ) by Theorem 7 and Propositions 14 and 13. Hence by taking minima and maximaover all 𝑄 ∈ P ( 𝑃 ) on both sides, we get that 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = ( 𝑃 ( 𝑓 ( 𝑋 )) 𝑃 𝑌 ( 𝑔 ) if 𝑃 𝑌 ( 𝑔 ) ≥ 𝑃 ( 𝑓 ( 𝑋 )) 𝑃 𝑌 ( 𝑔 ) if 𝑃 𝑌 ( 𝑔 ) ≤ 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = ( 𝑃 ( 𝑓 ( 𝑋 )) 𝑃 𝑌 ( 𝑔 ) if 𝑃 𝑌 ( 𝑔 ) ≥ 𝑃 ( 𝑓 ( 𝑋 )) 𝑃 𝑌 ( 𝑔 ) if 𝑃 𝑌 ( 𝑔 ) ≤ 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 ( 𝑓 ( 𝑋 )) ⊙ 𝑃 𝑌 ( 𝑔 ) .We now turn to sufficiency. If 𝑋 is trivial with respect to 𝑃 , it follows immediatelyfrom Proposition 17 that 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 𝑃 . We can thereforeassume, without loss of generality, that 𝑌 has a precise distribution 𝑃 𝑌 with respectto 𝑃 and that Equation (18) holds. Consider now any 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 . Then for any 𝑓 ( 𝑋 ) ∈ ℒ 𝑋 − ( 𝐸 ) and 𝑔 ( 𝑌 ) ∈ ℒ 𝑌 − ( 𝐹 ) , we have that 𝑃 ( 𝑔 ( 𝑌 )) = 𝑃 𝑌 ( 𝑔 ) because 𝑌 hasa precise distribution with respect to 𝑃 , and that 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 ( 𝑓 ( 𝑋 )) ⊙ 𝑃 𝑌 ( 𝑔 ) because of Equation (18). It therefore follows from Theorem 5 that 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) . Since 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 were arbitrary, it follows fromTheorem 7 that 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 𝑃 .Next, we turn to the second statement. For necessity, assume that 𝑋 and 𝑌 arenon-trivial with respect to 𝑃 and S-independent with respect to 𝐾 𝑃 . Theorem 7 andProposition 16 then guarantee that 𝑋 and 𝑌 respectively have precise distributions 𝑃 𝑋 and 𝑃 𝑌 with respect to 𝑃 . Consider now any 𝑓 ∈ ℒ ( 𝒳 ) , 𝑔 ∈ ℒ ( 𝒴 ) and 𝑄 ∈ P ( 𝑃 ) , then on the one hand 𝑄 ( 𝑓 ( 𝑋 )) = 𝑃 𝑋 ( 𝑓 ) and 𝑄 ( 𝑔 ( 𝑌 )) = 𝑃 𝑌 ( 𝑔 ) and onthe other hand 𝑄 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑄 ( 𝑓 ( 𝑋 )) 𝑄 ( 𝑔 ( 𝑌 )) = 𝑃 𝑋 ( 𝑓 ) 𝑃 𝑌 ( 𝑔 ) by Theorem 7and Propositions 14 and 13. Hence by taking minima and maxima over all 𝑄 ∈ P ( 𝑃 ) on both sides, we get that, indeed, 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 𝑋 ( 𝑓 ) 𝑃 𝑌 ( 𝑔 ) .We now turn to sufficiency. If 𝑋 or 𝑌 are trivial with respect to 𝑃 , then it followsimmediately from Proposition 17 that 𝑋 and 𝑌 are S-independent with respect to 𝐾 𝑃 .We can therefore assume, without loss of generality, that 𝑋 and 𝑌 respectively haveprecise distributions 𝑃 𝑋 and 𝑃 𝑌 with respect to 𝑃 and that Equation (19) holds, andprove that 𝑋 and 𝑌 are S-independent with respect to 𝐾 𝑃 . Since Equation (19) impliesEquation (18), the S-irrelevance of 𝑋 to 𝑌 with respect to 𝐾 𝑃 follows from the firstpart of this theorem. Since Equation (19) is symmetric in 𝑋 and 𝑌 , the S-irrelevanceof 𝑌 to 𝑋 with respect to 𝐾 𝑃 follows in exactly the same way. Hence, we find that,indeed, 𝑋 and 𝑌 are S-independent with respect to 𝐾 𝑃 . (cid:3) By combining this result with Proposition 12, we immediately obtain charac-terisations for S-irrelevance and S-independence for Archimedean sets of desirableoption sets 𝐾 , in terms of their representing lower previsions. If we ignore the trivial cases, we see that each of these lower previsions features both precision and fac-torisation. As we did in Section 3.2, we now seek to exclude the trivial cases byimposing credible indeterminacy, this time for variables instead of events. We saythat the variable 𝑍 is credibly indeterminate with respect to a coherent set of desir-able option sets 𝐾 if there is at least one event 𝐺 ⊆ 𝒵 such that 𝑍 − ( 𝐺 ) is crediblyindeterminate with respect to 𝐾 . If 𝐾 is Archimedean, then due to Proposition 10,this means that there is some 𝜖 > 𝑃 ∈ P ( 𝐾 ) , both 𝑃 𝑍 ( 𝐺 ) > 𝜖 and 𝑃 𝑍 ( 𝐺 c ) > 𝜖 .Similarly to what we found for events, the condition of credible indetermin-acy, when combined with S-independence, allows us to infer both precision—for 𝑋 and 𝑌 —and factorisation for every representing lower prevision of an Archimedeanset of desirable option sets 𝐾 , without having to impose mixingness . This is a surpris-ingly strong implication, we think, and especially so since credible indeterminacyfor a variable 𝑍 is such a weak requirement, as it only requires one single event aboutthis variable 𝑍 to be credibly indeterminate. Theorem 10
Let 𝐾 be an Archimedean set of desirable option sets and consider twovariables 𝑋 and 𝑌 . If 𝑋 is credibly indeterminate and S-irrelevant to 𝑌 with respectto 𝐾 , then for all 𝑃 ∈ P ( 𝐾 ) , 𝑌 has a precise distribution 𝑃 𝑌 with respect to 𝑃 and 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 ( 𝑓 ( 𝑋 )) ⊙ 𝑃 𝑌 ( 𝑔 ) for all gambles 𝑓 on 𝒳 and 𝑔 on 𝒴 .Similarly, if 𝑋 and 𝑌 are credibly indeterminate and S-independent with respectto 𝐾 , then for all 𝑃 ∈ P ( 𝐾 ) , 𝑋 and 𝑌 have a precise distribution 𝑃 𝑋 and 𝑃 𝑌 withrespect to 𝑃 , respectively, and 𝑃 ( 𝑓 ( 𝑋 ) 𝑔 ( 𝑌 )) = 𝑃 𝑋 ( 𝑓 ) 𝑃 𝑌 ( 𝑔 ) for all gambles 𝑓 on 𝒳 and 𝑔 on 𝒴 . Proof
Due to Proposition 12 and Theorem 9, it suffices to show that the credibleindeterminacy of 𝑋 implies that 𝑋 is non-trivial with respect to every 𝑃 ∈ P ( 𝐾 ) .So assume that 𝑋 is credibly indeterminate with respect to 𝐾 . Then there is some 𝐸 ⊆ 𝒳 that is credibly indeterminate with respect to 𝐾 , meaning that ˜ 𝐸 is crediblefor each ˜ 𝐸 ∈ { 𝐸, 𝐸 c } . For any 𝑃 ∈ P ( 𝐾 ) , it then follows from Proposition 10 thatthere is some 𝜖 > 𝑃 ( ˜ 𝐸 ) > 𝜖 . Hence, since 𝑃 ( ˜ 𝐸 ) ≥ 𝑃 ( ˜ 𝐸 ) > 𝜖 >
0, wesee that 𝐸 is non-trivial with respect to 𝑃 , implying that 𝑋 is non-trivial with respectto 𝑃 as well. (cid:3) We now want to reward those readers who are fans of decision-making with lin-ear previsions—or precise probability models—and who have nevertheless had thecourage and determination to follow our arguments all the way to this point. Dueto the heavy lifting already done for the more general cases of lower previsions andArchimedean models in the previous section, we are now able, without further ado, n a notion of independence proposed by Teddy Seidenfeld 33 to present our results for the special case of linear previsions, in Theorem 11, andfor the more involved, non-binary case of mixing models, in Theorem 12 below.Observe, first of all, that Theorem 7 also applies in particular in the linear pre-visions context of the present section. It allows us to apply arguments for events—Theorem 3 in particular—in order to obtain the following results about variables ina fairly straightforward manner.
Theorem 11
Consider two variables 𝑋 and 𝑌 and a linear prevision 𝑃 on ℒ . Thenthe following statements are equivalent: (i) 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 𝑃 ; (ii) 𝑋 and 𝑌 are S-independent with respect to 𝐾 𝑃 ; (iii) 𝑋 and 𝑌 are independent with respect to 𝑃 . Proof
Since 𝑃 is a linear prevision and hence definitely a coherent lower prevision,it follows from Theorem 7 that condition (i) holds if and only if for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , 𝑋 − ( 𝐸 ) is S-irrelevant to 𝑌 − ( 𝐹 ) with respect to 𝐾 𝑃 .Similarly, condition (ii) holds if and only if for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , 𝑋 − ( 𝐸 ) isS-independent to 𝑌 − ( 𝐹 ) with respect to 𝐾 𝑃 .Furthermore, because of Definition 10, condition (iii) holds if and only if for all 𝐸 ⊆ 𝒳 and 𝐹 ⊆ 𝒴 , the events 𝑋 − ( 𝐸 ) and 𝑌 − ( 𝐹 ) are independent with respectto 𝑃 .Given these observations, the equivalence of (i), (ii) and (iii) follows immediatelyfrom Theorem 3. (cid:3) Since we know from Theorem 2 that Archimedean and mixing models corres-pond to sets of linear previsions, the result above can be extended to Archimedeanand mixing models too. Observe that in this case, due to the mixingness property,credible indeterminacy is not required for factorisation to appear—here in the formof independence; see Proposition 13. Note also that, as a direct result of Theorem 8,the conditions (i) and (ii) can be equivalently expressed in terms of events as well.
Theorem 12
Consider two variables 𝑋 and 𝑌 and an Archimedean and mixing setof desirable option sets 𝐾 . Then the following statements are equivalent: (i) 𝑋 is S-irrelevant to 𝑌 with respect to 𝐾 ; (ii) 𝑋 and 𝑌 are S-independent with respect to 𝐾 ; (iii) 𝑋 and 𝑌 are independent with respect to 𝑃 , for all 𝑃 ∈ P ( 𝐾 ) . Proof
This result follows directly from Theorem 11 and Proposition 12. (cid:3)
After the detailed mathematical analysis of the previous sections, let us now takea moment to consider what these mathematical results imply, and how far-reaching we believe these implications to be. In doing so, we will also lay the foundationsfor talking about inferences and decisions involving variables and non-binary choicemodels.If our subject has a choice model 𝐾 for choosing between gambles on Ω , we canderive from that her choice model 𝐾 𝑍 for choosing between gambles on the value ofa variable 𝑍 : Ω → 𝒵 . We will use the following (notational) device: for any optionset 𝐶 ∈ 𝒬 ( 𝒵 ) of gambles on the possibility space 𝒵 , we let 𝐶 ( 𝑍 ) ≔ { ℎ ( 𝑍 ) : ℎ ∈ 𝐶 } ∈ 𝒬 ( Ω ) be the corresponding option set of gambles ℎ ( 𝑍 ) ≔ ℎ ◦ 𝑍 on the variable 𝑍 , whichare, of course, gambles whose domain is the possibility space Ω . Then clearly, 𝐾 𝑍 ≔ { 𝐶 ∈ 𝒬 ( 𝒵 ) : 𝐶 ( 𝑍 ) ∈ 𝐾 } is the set of desirable option sets on 𝒵 that represents the choices between gamblesthat depend on the variable 𝑍 , implicit in 𝐾 . It is completely in the spirit of theexisting terminology in standard probability theory to call this choice model 𝐾 𝑍 the distribution of the variable 𝑍 , as it is a full decision-theoretic model for the subject’suncertainty about the value that 𝑍 assumes in 𝒵 .It is also a matter of simple and direct verification that this operation preservescoherence, mixingness and Archimedeanity. Moreover, if 𝑃 is a coherent lowerprevision on ℒ ( Ω ) , then this operation turns the Archimedean 𝐾 = 𝐾 𝑃 into theArchimedean 𝐾 𝑍 = 𝐾 𝑃 𝑍 , where the coherent lower prevision 𝑃 𝑍 on ℒ ( 𝒵 ) is thelower distribution of 𝑍 with respect to 𝑃 , introduced in Section 4.2. The same goesfor a linear prevision 𝑃 and the corresponding precise distribution 𝑃 𝑍 . We provesome of these claims involving Archimedean (and mixing) models explicitly in thefollowing proposition. Proposition 18
Consider an Archimedean set of desirable option sets 𝐾 and a vari-able 𝑍 . Then 𝐾 𝑍 is Archimedean too, and has { 𝑃 𝑍 : 𝑃 ∈ P ( 𝐾 )} as a set of represent-ing coherent lower previsions. If 𝐾 is furthermore mixing, then 𝐾 𝑍 is Archimedeanand mixing, and has { 𝑃 𝑍 : 𝑃 ∈ P ( 𝐾 )} as a set of representing linear previsions. Proof
First assume that 𝐾 is Archimedean. Definition 4 then implies that we canconsider the following chain of equivalences for any 𝐶 in 𝒬 ( 𝒵 ) : 𝐶 ∈ 𝐾 𝑍 ⇔ 𝐶 ( 𝑍 ) ∈ 𝐾 ⇔ (∀ 𝑃 ∈ P ( 𝐾 )) 𝐶 ( 𝑍 ) ∈ 𝐾 𝑃 ⇔ (∀ 𝑃 ∈ P ( 𝐾 )) (∃ ℎ ∈ 𝐶 ) 𝑃 ( ℎ ( 𝑍 )) > ⇔ (∀ 𝑃 ∈ P ( 𝐾 )) (∃ ℎ ∈ 𝐶 ) 𝑃 𝑍 ( ℎ ) > ⇔ (∀ 𝑃 ∈ P ( 𝐾 )) 𝐶 ∈ 𝐾 𝑃 𝑍 . So 𝐾 𝑍 is indeed Archimedean, and has (cid:8) 𝑃 𝑍 : 𝑃 ∈ P ( 𝐾 ) (cid:9) as a set of representingcoherent lower previsions. If 𝐾 is furthermore mixing, we know from Theorem 2that P ( 𝐾 ) = P ( 𝐾 ) , which implies that (cid:8) 𝑃 𝑍 : 𝑃 ∈ P ( 𝐾 ) (cid:9) = (cid:8) 𝑃 𝑍 : 𝑃 ∈ P ( 𝐾 ) (cid:9) . Asecond application of Theorem 2 therefore implies that 𝐾 𝑍 is indeed mixing. (cid:3) n a notion of independence proposed by Teddy Seidenfeld 35 In order to explore the implications of what we have discovered in the previoussections, let us now focus on a decision problem involving gambles that dependon two variables 𝑋 and 𝑌 . These are gambles of the type ℎ ( 𝑋, 𝑌 ) ≔ ℎ ◦ ( 𝑋, 𝑌 ) ,where ℎ is some gamble on 𝒳 × 𝒴 . Of course ( 𝑋, 𝑌 ) can be seen as a new variable ( 𝑋, 𝑌 ) : Ω → 𝒳 × 𝒴 : 𝜔 ↦→ ( 𝑋 ( 𝜔 ) , 𝑌 ( 𝜔 )) , and all we have said above about thedistribution of a variable can also be brought to bear on this variable ( 𝑋, 𝑌 ) . Inparticular, if our subject has a coherent set of desirable option sets 𝐾 , then theso-called joint distribution 𝐾 ( 𝑋,𝑌 ) of the variable ( 𝑋, 𝑌 ) is given by 𝐾 ( 𝑋,𝑌 ) ≔ { 𝐶 ∈ 𝒬 ( 𝒳 × 𝒴 ) : 𝐶 ( 𝑋, 𝑌 ) ∈ 𝐾 } , and, of course, choices between gambles on the value of, say, 𝑌 separately aremodelled by the so-called marginal distribution 𝐾 𝑌 of 𝑌 , given by 𝐾 𝑌 ≔ { 𝐴 ∈ 𝒬 ( 𝒴 ) : 𝐴 ( 𝑌 ) ∈ 𝐾 } = 𝐾 ( 𝑋,𝑌 ) ∩ 𝒬 ( 𝒴 ) , where the rightmost equality also indicates that the marginal distribution 𝐾 𝑌 can bederived from the joint distribution 𝐾 ( 𝑋,𝑌 ) by a marginalisation operation , similarlyto what is done for sets of desirable gambles [3, 7, 17].Assume now that our subject’s choice model 𝐾 is Archimedean—but not necessar-ily mixing—and that she has furthermore made the assessment that 𝑋 is S-irrelevantto 𝑌 , and that this assessment is reflected in her Archimedean model 𝐾 . We willalso assume that 𝐾 reflects her beliefs that 𝑋 is credibly indeterminate, which wehave argued is a rather weak requirement to impose. The strong consequences ofthese assumptions have been derived in Theorem 10. In particular, it guaranteesfactorisation properties for the binary distributions of 𝑋 and 𝑌 in the representationof the Archimedean model 𝐾 : the distribution 𝐾 ( 𝑋,𝑌 ) is represented by a set of factor-ising lower previsions with precise (linear) marginals for 𝑌 . And if we furthermoresymmetrise the assessment of our subject—that is, if 𝑋 and 𝑌 are (both) crediblyindeterminate and S-independent—then the same is true for the marginals of 𝑋 .What is perhaps the most striking about Theorem 10, however, are its implicationsfor the choice model 𝐾 𝑌 . Since 𝐾 is Archimedean, we know from Proposition 18that 𝐾 𝑌 is Archimedean as well. What is very surprising, though, is that our subject’sassessments—that 𝑋 is credibly indeterminate and S-irrelevant to 𝑌 —imply that itmust be mixing as well. Corollary 1
Suppose that a variable 𝑋 is credibly indeterminate and S-irrelevant toa variable 𝑌 with respect to an Archimedean set of desirable option sets 𝐾 . Then thedistribution 𝐾 𝑌 of 𝑌 is an Archimedean and mixing set of desirable option sets. Proof
For any 𝐵 ∈ 𝒬 ( 𝒴 ) , we have that 𝐵 ∈ 𝐾 𝑌 ⇔ (∀ 𝑃 ∈ P ( 𝐾 )) 𝐵 ∈ 𝐾 𝑃 𝑌 ⇔ (∀ 𝑃 ∈ P ( 𝐾 )) 𝐵 ∈ 𝐾 𝑃 𝑌 , This notation uses the implicit convention that gambles with domain 𝒴 are considered as specialinstances of gambles with domain 𝒳 × 𝒴 .6 De Bock and De Cooman where the first equivalence follows from Proposition 18, and the second from The-orem 10. Since we know from Theorem 1 that P ( 𝐾 ) is non-empty, it therefore followsfrom Theorem 2 that 𝐾 𝑌 is Archimedean and mixing. (cid:3) If the Archimedean set of desirable option sets 𝐾 is mixing, then the mixingnessof 𝐾 𝑌 follows easily from the fact that mixingness is preserved under marginalisation;see for example Proposition 18. The striking thing about Corollary 1 is that itdoesn’t require 𝐾 to be mixing: we obtain the mixingness of 𝐾 𝑌 using only credibleindeterminacy and S-irrelevance.Since mixing sets of desirable options sets correspond to choice functions gov-erned by E-admissibility, the implications of this result are far-reaching: we find thatchoices between gambles that depend only on 𝑌 will be governed by E-admissibilitywith respect to a set of linear previsions, for example P ( 𝐾 𝑌 ) . This is a very curiousand amazingly strong result. As soon as a subject assumes that there is some cred-ibly indeterminate variable 𝑋 that is S-irrelevant to a variable 𝑌 , which seems a veryweak assumption to make, she is forced by coherence—and Archimedeanity—to usea mixing model for 𝑌 , and to use E-admissibility as her decision scheme for choosingbetween gambles on 𝑌 . To give a simple example, we believe that our flipping acoin here in Ghent today will not affect in any way whether Teddy will have pickledherring for breakfast tomorrow morning. As soon as we translate this belief into anassessment that the outcome of our coin flip today is credibly indeterminate—whichseems uncontroversial for a coin flip—and S-irrelevant to Teddy’s choice of break-fast tomorrow, we are forced by coherence—and Archimedeanity—to use a mixingmodel for our uncertainty about Teddy’s breakfast choice.It would seem, then, that our mathematical derivations in this paper lead to anargument in favour of using mixing models and decision schemes based on E-admissibility. It is indeed very easy to imagine that there are experiments whoseoutcomes—variables 𝑋 —are indeterminate and have nothing whatsoever to do withthe outcome—variable 𝑌 —of the experiment that we are currently considering. Assoon as we translate this ‘being indeterminate and having nothing whatsoever to dowith’ by an assessment of credible indeterminacy and S-irrelevance, we are led tousing mixing models and E-admissibility only.We don’t want to take this discussion too far, but still feel inclined to suggestthat, perhaps, it is the translation that constitutes the Achilles’ heel of this argument.Going back to binary variables, or events, in the interest of simplicity, isn’t requiringthat the composite gamble I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 must always be rejected from the set { 𝑓 , 𝑔, I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 } for all 𝑓 , 𝑔 ∈ ℒ 𝐹 too strong if we want to express that‘whether the event 𝐸 obtains has nothing whatsoever to do with whether 𝐹 obtains’?At least one of us isn’t entirely convinced of the validity of this requirement. Even ifour subject believes that the event 𝐸 has no effect on the event 𝐹 , why should she thenreject the gamble I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 from the set { 𝑓 , 𝑔, I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 } ? Or equivalently,why should she then necessarily find I 𝐸 c ( 𝑓 − 𝑔 ) + 𝜖 or I 𝐸 ( 𝑔 − 𝑓 ) + 𝜖 desirable? Forexample, if 𝑓 and 𝑔 are deemed incomparable by our subject, meaning that 𝑓 − 𝑔 nor 𝑔 − 𝑓 are desirable, what would then compel her to find I 𝐸 𝑓 + I 𝐸 c 𝑔 − 𝜖 comparableto—even dominated by— 𝑓 or 𝑔 . Or to rephrase it one more time: if 𝑓 − 𝑔 nor 𝑔 − 𝑓 n a notion of independence proposed by Teddy Seidenfeld 37 are deemed desirable, why then should I 𝐸 c ( 𝑓 − 𝑔 ) + 𝜖 or I 𝐸 ( 𝑔 − 𝑓 ) + 𝜖 be desirable?We definitely think that these and related questions merit further attention. Acknowledgements
We would like to thank Teddy Seidenfeld for the many discussions, throughoutthe years, on so many issues related to imprecise probabilities and the foundations of decision-making. This paper, and our related earlier work on choice functions, would not have existedwithout his constructive and destructive criticism of our earlier work on binary choice.We would also like to thank the editors of this Festschrift for giving us the opportunity tocontribute to it, and two anonymous reviewers for their valuable and constructive feedback.Jasper De Bock’s work was partially supported by his BOF Starting Grant “Rational decisionmaking under uncertainty: a new paradigm based on choice functions”, number 01N04819.As with most of our joint work, there is no telling, after a while, which of us two had what idea,or did what, exactly. An irrelevant coin flip may have determined the actual order we are listed in.
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