Coherent and Archimedean choice in general Banach spaces

02/13/2020
by   Gert de Cooman, et al.
Ghent University
0

I introduce and study a new notion of Archimedeanity for binary and non-binary choice between options that live in an abstract Banach space, through a very general class of choice models, called sets of desirable option sets. In order to be able to bring horse lottery options into the fold, I pay special attention to the case where these linear spaces do not include all `constant' options. I consider the frameworks of conservative inference associated with Archimedean (and coherent) choice models, and also pay quite a lot of attention to representation of general (non-binary) choice models in terms of the simpler, binary ones. The representation theorems proved here provide an axiomatic characterisation of, amongst other choice methods, Levi's E-admissibility and Walley–Sen maximality.

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1. Introduction

This paper is about rational decision making under uncertainty using choice functions, along the lines established by Teddy Seidenfeld and colleagues [seidenfeld2010]. What are the underlying ideas? A subject is to choose between options , which are typically uncertain rewards, and which live in a so-called option space . Her choices are typically represented using a rejection function or a choice function . For any finite set of options, contains those options that our subject rejects from the option set , and the remaining options in are the ones that are then still being considered. It is important to note that is not necessarily a singleton, so this approach allows for indecision. Also, the binary choices are the ones where has only two elements, and I will not be assuming that these binary choices completely determine the behaviour of  or  on option sets with more than two elements: I will be considering choice behaviour that is not necessarily binary in nature.

My aim here is to present a theory of coherent and Archimedean choice (functions), complete with a notion of conservative inference and representation results, for very general option spaces: general Banach spaces that need not have constants.

For the basic theory of coherent choice (functions) on general linear option spaces but without Archimedeanity, I will rely fairly heavily on earlier work by Jasper De Bock and myself [debock2018:choice:arxiv, debock2018:choice:smps, debock2019:choice:arxiv, debock2019:choice:isipta]. The present paper expands that work to include a discussion of a novel notion of Archimedeanity. Since this approach needs a notion of closeness, I will need to focus on option spaces that are Banach, but I still want to keep the treatment general enough so as to avoid the need for including constant options.

The reasons for working with option spaces that are general linear spaces are manifold, and were discussed at length in [debock2019:choice:arxiv, debock2019:choice:isipta]. In summary, doing so allows us to deal with options that are gambles [walley1991, troffaes2013:lp], i.e. bounded real-valued maps on some set of possible states , that are considered as uncertain rewards. These maps constitute a linear space

, closed under point-wise addition and point-wise multiplication with real numbers. But it also brings in, in one fell swoop, vector-valued gambles

[zaffalon2017:incomplete:preferences, 2017vancamp:phdthesis], polynomial gambles to deal with exchangeability [cooman2009c, vancamp2018:exchangeable:choice], equivalence classes of gambles to deal with indifference [vancamp2015:indifference], and abstract gambles defined without an underlying possibility space [williams2007]. In all these cases, the space of options essentially includes all real constants—or constant gambles. But when we want our approach to also be able to deal generically with options that are horse lotteries, we need to consider option spaces that do not include all real constants.

Indeed, in that case we consider a finite set of rewards, a set of possible states , and the set of state-dependent probability mass functions on rewards, also called horse lotteries, where

Horse lotteries are often considered as options between which preferences can be expressed [anscombe1963, seidenfeld1995, seidenfeld2010, zaffalon2017:incomplete:preferences], but they are rather cumbersome to work with, as the set of all horse lotteries with state space  and reward set  does not constitute a linear space, but is only closed under convex mixtures. It turns out we can remedy this—without adding or losing ‘information’—by embedding into the smallest linear subspace of that includes it, namely

But since , is the only real constant that this linear space of real maps contains. This indicates why, in order to build a theory general enough to incorporate the horse lottery approach without too many restrictions, we must also pay attention to linear option spaces that don’t necessarily include (all) real constants.

In order to keep the length of this paper manageable, I have decided to focus on the mathematical developments only, and to keep the discussion fairly abstract. For a detailed exposition of the motivation for and the interpretation of the choice models discussed below, I refer to earlier joint papers by Jasper De Bock and myself [debock2018:choice:arxiv, debock2018:choice:smps, debock2019:choice:arxiv, debock2019:choice:isipta]. I also recommend Jasper De Bock’s most recent paper on Archimedeanity (add reference), as it contains a persuasive motivation for the new Archimedeanity condition, in the more restrictive and concrete context where options are gambles.

How do I plan to proceed? I briefly introduce binary choice models on abstract option spaces in Section 2, and extend the discussion to general—not necessarily binary—choice in Section 3. I rely on results in earlier papers to provide an axiomatisation and a conservative inference framework for these choice models, and recall that there are general theorems that allow for representation of general models in terms of binary ones. After these introductory sections, I focus on adding Archimedeanity to the picture. The basic representation tools that will turn out to be useful in this more restricted context, namely linear and superlinear bounded real functionals, are discussed in Section 4. The classical approach to Archimedeanity [walley1991, seidenfeld1995] for binary choice—which I will call essential Archimedeanity—is given an abstract treatment in Section 5. I discuss the trick that allows us to deal elegantly with option spaces without constants—namely proclaiming some option to be constant as far as representation is concerned—in Section 6. Sections 7 and 8 then deal with the new notion of Archimedeanity in the binary and general case, and discuss conservative inference and the representation of general Archimedean choice models in terms of binary (essentially) Archimedean ones. I conclude in Section 9 by stressing the relevance of my findings: they show that the axioms presented here allow for a complete characterisation of several decision methods in the literature, amongst which Levi’s E-admissibility [levi1980a] and Walley–Sen maximality [walley1991].

2. Coherent sets of desirable options

We begin by considering a linear space , whose elements are called options, and which represent the objects that a subject can choose between. This option space has some so-called background ordering , which is ‘natural’ in that we will assume that our subject’s choices will always at least respect this ordering, even before she has started reflecting on her preferences. This background ordering  is taken to be a strict vector ordering on , so an irreflexive and transitive binary relation that is compatible with the addition and scalar multiplication of options.

We will assume that our subject’s binary choices between options can be modelled by so-called set of desirable options , where an option is called desirable when the subject strictly prefers it to the zero option. We will denote the set of all possible set of desirable option sets—all subsets of —by . Of course, a set of desirable options strictly speaking only covers the strict preferences between options and : . For other strict preferences, it is assumed that they are compatible with the vector addition of options: .

We impose the following rationality requirements on a subject’s strict preferences. A set of desirable options is called coherent [walley2000, cooman2010, couso2011] if it satisfies the following axioms:

  1. [label=.,ref=,leftmargin=*]

  2. ;

  3. if and , then ;

  4. .

We will use the notation to mean that are non-negative real numbers such that . We denote the set of all coherent sets of desirable options by .

It is easy to see that is an intersection structure: for any non-empty family of sets of desirable options , , its intersection also belongs to . This also implies that we can introduce a coherent closure operator by letting

be the smallest—if any—coherent set of desirable options that includes . We call an assessment consistent if , or equivalently, if is included in some coherent set of desirable options. The closure operator implements conservative inference with respect to the coherence axioms, in that it extends a consistent assessment to the most conservative—smallest possible—coherent set of desirable options .

A coherent set of desirable option is called maximal if none of its supersets is coherent: . This turns out to be equivalent to the following so-called totality condition on [cooman2009c, couso2011]:

  1. [label=.,ref=,leftmargin=*]

  2. for all , either or .

The set of all maximal sets of desirable options is denoted by . These maximal elements can be used to represent all coherent sets of desirable options via intersection.

Theorem 1 (Closure [cooman2009c]).

For any , . Hence, a consistent is coherent if and only if .

Corollary 2 (Representation).

A set of desirable options is coherent if and only if there is some non-empty such that . In that case, the largest such set is .

For more details on these issues, and more ‘constructive’ expressions for , see [cooman2009c, debock2019:choice:isipta, debock2019:choice:arxiv].

I also want to mention another, additional, rationality property, central in Teddy Seidenfeld’s work [seidenfeld1995, seidenfeld2010], but introduced there in a form more appropriate for strict preferences between horse lotteries. We can get to the appropriate counterpart here when we introduce the operator, which, for any subset  of , returns the set of all positive linear combinations of its elements:

We call a set of desirable options mixing if it is coherent and satisfies the following mixingness axiom:

  1. [label=.,ref=,leftmargin=*]

  2. for all , if , then also .

We denote the set of all mixing sets of desirable options by . They can be characterised as follows.

Proposition 3 ([2017vancamp:phdthesis, 2018vancamp:lexicographic]).

Consider any set of desirable options and let . Then is mixing if and only if , or equivalently, .

They are therefore identical to the so-called lexicographic sets of desirable option sets introduced by Van Camp et al. [2017vancamp:phdthesis, 2018vancamp:lexicographic]. For more details, see also [2017vancamp:phdthesis, 2018vancamp:lexicographic, debock2019:choice:isipta, debock2019:choice:arxiv].

3. Coherent sets of desirable option sets

We now turn from strict binary preferences—of one option  over another option —to more general ones. The simplest way to introduce those more general choice models in the present context goes as follows. We call any finite subset of an option set, and we collect all such option sets into the set . We call an option set desirable to a subject if she assesses that at least one option in is desirable, meaning that it is strictly preferred to . We collect a subject’s desirable option sets into her set of desirable option sets . We denote the set of all such possible sets of desirable option sets—all subsets of —by .

The rationality requirements we will impose on such sets of desirable option sets turn out to be fairly natural generalisations of those for sets of desirable options. A set of desirable option sets  is called coherent [debock2019:choice:isipta, debock2019:choice:arxiv] if it satisfies the following axioms:

  1. [label=.,ref=,leftmargin=*,start=0]

  2. if then also , for all ;

  3. ;

  4. if and if, for all and , , then also

  5. if and , then also , for all ;

  6. , for all .

We denote the set of all coherent sets of desirable option sets by .

A coherent set of desirable option sets  contains singletons, doubletons, … Moreover, it also contains all supersets of its elements, by Axiom 4. The singletons in represent the binary choices, or in other words, the pure desirability aspects. We let

(1)

be the set of desirable gambles that represents the binary choices present in the model . Its elements are the options that—according to —are definitely desirable. But there may be elements of of higher cardinality that are minimal in the sense that has none of its strict subsets. This means that our subject holds that at least one option in is desirable, but her model holds no more specific information about which of these options actually are desirable. This indicates that the choice model has non-binary aspects. If such is not the case, or in other words, if every element of goes back to some singleton in , meaning that

then we call the choice model binary. With any , our interpretation inspires us to associate a set of desirable option sets , defined by

(2)

It turns out that a set of desirable option sets  is binary if and only if it has the form , and the unique representing is then given by .

Proposition 4 ([debock2019:choice:arxiv, debock2019:choice:isipta]).

A set of desirable option sets  is binary if and only if there is some  such that . This is then necessarily unique, and equal to .

The coherence of a binary set of desirable option sets is completely determined by the coherence of its corresponding set of desirable options.

Proposition 5 ([debock2019:choice:arxiv, debock2019:choice:isipta]).

Consider any binary set of desirable option sets  and let be its corresponding set of desirable options. Then is coherent if and only if  is. Conversely, consider any set of desirable options and let be its corresponding binary set of desirable option sets, then is coherent if and only if is.

So the binary coherent sets of desirable option sets are given by , allowing us to call any coherent set of desirable option sets in non-binary. If we replace such a non-binary coherent set of desirable option sets  by its corresponding set of desirable options , we lose information, because then necessarily . Sets of desirable option sets are therefore more expressive than sets of desirable options. But our coherence axioms lead to a representation result that allows us to still use sets of desirable options, or rather, sets of them, to completely characterise any coherent choice model.

Theorem 6 (Representation [debock2019:choice:arxiv, debock2019:choice:isipta]).

A set of desirable option sets  is coherent if and only if there is some non-empty set of coherent sets of desirable options such that . The largest such set is then .

It is also easy to see that is an intersection structure: if we consider any non-empty family of coherent sets of desirable options , , then their intersection is still coherent. This implies that we can introduce a coherent closure operator by letting

be the smallest—if any—coherent set of desirable option sets that includes . We call an assessment consistent if , or equivalently, if is included in some coherent set of desirable option sets. The closure operator implements conservative inference with respect to the coherence axioms, in that it extends a consistent assessment to the most conservative—smallest possible—coherent set of desirable option sets . In combination with Theorem 6, this leads to the following important result.

Theorem 7 (Closure [debock2018:choice:arxiv, debock2018:choice:smps, debock2019:choice:arxiv, debock2019:choice:isipta]).

Consider , then . Hence, a consistent is coherent if and only if .

We can also lift the mixingness property from binary to general choice models, as Seidenfeld et al. have done [seidenfeld2010]. Converted to our language [2017vancamp:phdthesis, 2018vancamp:lexicographic], this condition becomes:

  1. [label=.,ref=,leftmargin=*]

  2. if and , then also , for all .

We call a set of desirable option sets mixing if it is coherent and satisfies 1. The set of all mixing sets of desirable option sets is denoted by , and it also constitutes an intersection structure. It therefore comes with its own mixing closure operator and associated conservative inference system.

The binary elements of are precisely the ones based on a mixing set of desirable options.

Proposition 8 (Binary embedding [debock2019:choice:arxiv, debock2019:choice:isipta]).

For any set of desirable options , is mixing if and only if  is, so .

For general mixing sets of desirable option sets that are not necessarily binary, we still have a representation theorem analogous to Theorem 6.

Theorem 9 (Representation [debock2019:choice:arxiv, debock2019:choice:isipta]).

A set of desirable option sets is mixing if and only if there is some non-empty set of mixing sets of desirable options such that . The largest such set is then .

How can we connect this choice of model, sets of desirable option sets, to the rejection and choice functions that I mentioned in the Introduction, and which are much more prevalent in the literature? Their interpretation provides the clue. Consider any option set , and any option . Then, with ,

In these equivalences, the first one follows from compatibility of the rejection function with vector addition, the second one follows from a particular interpretation we give to the rejection function, and the third one follows from the definition of the set of desirable option sets . This tells us that, given this particular interpretation, choice and rejection functions are in a one-to-one relation with sets of desirable option sets.

4. Linear and superlinear functionals

Because the notions of essential Archimedeanity and Archimedeanity that I intend to introduce further on rely on an idea of openness—and therefore closeness—I will assume from now on that the option space  constitutes a Banach space with a norm  and a corresponding topological closure operator  and interior operator . In this section, I have gathered a few useful definitions and basic results for linear and superlinear bounded real functionals on the space . These functionals are intended to generalise to our more general context the linear and coherent lower previsions defined by Peter Walley [walley1991] on spaces of gambles.

A real functional on  is called bounded if its operator norm , where we let

We will denote by the linear space of such bounded real functionals on .

This space can be topologised by the operator norm , which leads to the so-called initial topology on . If we associate with any the so-called evaluation functional , defined by

then is clearly a real linear functional on the normed linear space , whose operator norm

is finite, which implies that is a continuous real linear functional on [schechter1997, Section 23.1].

We will also retopologise with the topology of pointwise convergence on , which is the weakest topology that makes all evaluation functionals , continuous. It is therefore weaker than the (so-called) initial topology induced by the norm .

An interesting subspace of  is the linear space  of all linear bounded—and therefore continuous [schechter1997, Section 23.1]—real functionals on . We will also consider the set of all superlinear bounded real functionals  on , meaning that they are elements of  that are furthermore superadditive and non-negatively homogeneous:

  1. [label=.,ref=,leftmargin=*]

  2. for all ; [superadditivity]

  3. for all and all real . [non-negative homogeneity]

Obviously, is a convex cone, and .

With any we can associate its conjugate (functional) defined by

It is also bounded, because obviously

Clearly, a bounded real functional is linear if and only if it is superlinear and self-conjugate, i.e. equal to its conjugate. Let us list a few useful properties of superlinear bounded real functionals and their conjugates.

Proposition 10.

Consider any and its conjugate , then

  1. [label=(),leftmargin=*]

  2. for all ;

  3. for all ;

  4. for all .

Proof.

For 1, observe that and therefore

where the first equality follows from Axiom 2, and the inequality from Axiom 1.

For 2, the first and last inequalities follow from Axiom 1 (and conjugacy). We prove the second inequality; the third then follows again by considering the consequences of conjugacy. Since , we infer from Axiom 1 and conjugacy that, indeed,

For 3, it suffices to prove that , because replacing with and with then completes the proof. Observe that it follows from Axiom 1 and conjugacy that, since , , and therefore

(3)

There are now two possibilities. The first is that , and then Equation (3) guarantees that

The second case is that , and then simply exchanging the roles of and in the argument above leads to the (same) inequality:

It is now but a small step to proving the continuity of all superlinear bounded real functionals.

Proposition 11.

Any , as well as its conjugate , are (uniformly) continuous.111It is easy to extend the argument to prove that for superlinear real functionals boundedness and (uniform) continuity are equivalent, as they also are for linear real functionals.

Proof.

We prove the uniform continuity of . The argument for is essentially the same. Consider any , then we have to prove that there is some such that

(4)

It follows from the boundedness of that and that

This allows us to infer from Proposition 103 that

which indeed guarantees that statement (4) holds, if we let . ∎

If we consider, for any its set of dominating continuous linear functionals

then a well-known version of the Hahn–Banach Theorem [schechter1997, Section 28.4, HB17] leads to the following representation result. An important condition for using this version is that should be both superlinear and continuous.

Theorem 12 (Lower envelope theorem).

For all and there is some such that , or in other words, .

5. Essential Archimedeanity for sets of desirable options

The background ordering on introduced in Section 2 allows us to define convex cones of positive real functionals:

(5)
(6)

Observe that . We will implicitly assume from now on that the strict vector ordering is such that , which then of course also implies that . We will also require for the remainder of this paper that : the background cone of positive options has a non-empty interior.

With any , we can associate a set of desirable gambles as follows:

(7)

Also, given a set of desirable options , we let

(8)
(9)

where we used Equation (7) for the second equalities. Clearly, . These sets are convex subcones of the convex cone , and is also a convex cone in the dual linear space  of continuous real linear functionals on .

Inspired by Walley’s [walley1991] discussion of ‘strict desirability’, we will call a set of desirable gambles essentially Archimedean if it is coherent and open.

It turns out that there is a close connection between essentially Archimedean sets of desirable options and superlinear bounded real functionals. Before we can lay it bare in Propositions 1315, we need to find a way to associate a superlinear bounded real functional with a set of desirable options . There are a number of different ways to achieve this, but I have found the following approach to be especially productive. Since we assumed from the outset that , we can fix any . We use this special option  to associate with the set of desirable options a specific (possibly extended) real functional by letting

(10)

and for its conjugate functional

Proposition 13.

If the set of desirable options is coherent, then . Moreover, for all and for all .

Proof.

We begin by proving that . Indeed, assume ex absurdo that there is some for which , so there are real such that and . We then infer from coherence [Axiom 2] that , which is impossible because and [indeed, Axiom 2 would then imply that , contradicting Axiom 1].

Next, we show that is positively homogeneous. Consider any and any real . By the coherence of [Axiom 2], if and only if . This guarantees that, indeed, . It follows immediately that is positively homogeneous too.

Next, we show that —and therefore also —is bounded, and therefore real-valued. Assume ex absurdo that it isn’t, so , implying that for any , there is some such that . Due to the positive homogeneity of we have just proved, we may assume without loss of generality that , so we find that or . The latter alternative is equivalent to , and therefore implies that also , because we have just proved that . We may therefore assume without loss of generality that , and therefore that , or equivalently—by coherence [Axiom 2]—that for all . But , so we are led to the conclusion that . This contradicts that [use Axiom 3].

To show that is non-negatively homogeneous, it now simply suffices to prove that . Since for and at the same time for , by coherence [use Axioms 2 and 1], this is immediate from Equation (10).

For the superadditivity of , consider any . Consider any such that both and , so and . But then by the coherence of [Axiom 2], implying that . Hence, indeed, .

Finally, for any , we have that for all , by coherence [Axioms 2 and 1], and therefore indeed . And for any , we get that for all , by coherence [Axiom 2], and therefore indeed . ∎

Proposition 14.

If the set of desirable options is coherent, then .

Proof.

The coherence of and Proposition 13 guarantee that . It therefore suffices to show that for any , .

First, assume that , then since , there is some sequence such that , and therefore also , by the continuity of [use and Proposition 11]. Since by Proposition 13, we see that, indeed, .

Next, assume that , so for some real . Let for all real , then , so if we pick any , we find that and therefore , whence

where the inequality follows from Proposition 13, and the last equality from Equation (10). Hence, indeed, . ∎

Proposition 15 (Representation).

A set of desirable options is essentially Archimedean if and only if there is some such that . In that case, we always have that , and therefore .

Proof.

For sufficiency, we we need to consider any and prove that is essentially Archimedean. That is open, follows at once from the continuity of , so we only need to focus on coherence. Since 1 implies that , and since is real-valued, we know that , so Equation (7) implies that . Hence, satisfies 1. For 3, it suffices to observe that for any , because , so by Equation (7). Let us now prove 2. Consider any and any . It then follows from Equation (7) that and , and from 2 and 1 that . Since , this implies that , which in turn implies that by Equation (7). So satisfies 2.

To prove necessity, we consider any essentially Archimedean , so is both coherent and open. Proposition 14 then guarantees that . Also recall that, by coherence [Axiom 3], , and therefore also for all . Hence also , which completes the proof of the ‘if and only if’ statement. For the rest of the proof, assume that there is some such that . But then is open, because is continuous, and therefore . Since Proposition 14 guarantees that also , we are done. ∎

For sets of desirable options that are essentially Archimedean and mixing, we have similar results in terms of linear rather than superlinear bounded real functionals.

Proposition 16.

If the set of desirable options is mixing, then , and we will then denote this linear bounded real functional by . Moreover, for all and for all .

Proof.

We only need to prove, by Proposition 15, that