Introduction to Imprecise Probabilities

By Gert de Cooman and Matthias C. M. Troffaes

In recent years, the theory has become widely accepted and has been further developed, but a detailed introduction is needed in order to make the material available and accessible to a wide audience. This will be the first book providing such an introduction, covering core theory and recent developments which can be applied to many application areas. All authors of individual chapters are leading researchers on the specific topics, assuring high quality and up-to-date contents.

An Introduction to Imprecise Probabilities provides a comprehensive introduction to imprecise probabilities, including theory and applications reflecting the current state if the art. Each chapter is written by experts on the respective topics, including: Sets of desirable gambles; Coherent lower (conditional) previsions; Special cases and links to literature; Decision making; Graphical models; Classification; Reliability and risk assessment; Statistical inference; Structural judgments; Aspects of implementation (including elicitation and computation); Models in finance; Game-theoretic probability; Stochastic processes (including Markov chains); Engineering applications.

Essential reading for researchers in academia, research institutes and other organizations, as well as practitioners engaged in areas such as risk analysis and engineering.

• Language: English
• Publisher: Wiley
• Release date: Apr 11, 2014
• ISBN: 9781118763148

Book preview

Chapter 1

Desirability

Erik Quaeghebeur

SYSTeMS Research Group, Ghent University, Belgium

1.1 Introduction

There are many ways to model uncertainty. The most widely used type of model in the literature is a function that maps something we are uncertain about to a value that expresses what we know or believe to know about it. Examples are probabilities, which may specify a degree of belief that an event will occur, and previsions, which specify acceptable prices for gambles (cf. Section 1.6 and Chapter 2).

In this chapter, we show that other types of models that are conceptually and intuitively attractive can be built and used as well. The focus lies on the notion of desirability and the theory of sets of desirable gambles. Next to introducing its concepts and structure, we also use it as a nexus for clarifying the relationships between many of the equivalent or almost equivalent models for uncertainty appearing in the imprecise-probability literature: partial preference orders, credal sets, and lower previsions.

We formulate desirability in the context of an abstract betting framework:

We—short for ‘an intentional system’—are uncertain about the outcome of an experiment. A possibility space for the experiment is a finite or infinite set of elementary events— i.e., mutually exclusive outcomes—that is exhaustive in the sense that other outcomes are deemed practically or pragmatically impossible. A bounded real-valued function on a possibility space is called a gamble and interpreted as an uncertain payoff. The set of all gambles combined—as we do—with pointwise addition of gambles, pointwise multiplication with real numbers, and the supremum norm (topology), forms a normed real vector space, the bounded function (Banach) space to be precise [256].

Consider a possibility space c01-math-0001 , the associated set of all gambles c01-math-0002 , and one of its elements, c01-math-0003 . After the experiment's outcome c01-math-0004 is determined, the gamble's owner receives the (possibly negative) payoff c01-math-0005 .

We will supplement the formal exposition by intuition-building toy examples or illustrations, as in Figure 1.1.

c01f001

Figure 1.1 A depiction of the set of gambles on the possibility space c01-math-0006 : The plane depicts the two-dimensional vector space c01-math-0007 ; a gamble c01-math-0008 and its component payoffs are shown.

A special class of gambles are indicators of events c01-math-0009 , denoted c01-math-0010 : they are one on c01-math-0011 and zero elsewhere. Indicators of elementary events c01-math-0012 are written c01-math-0013 .

Payoffs have a certain utility. We assume that the gambles' payoffs are expressed in units of utility that are linear for us. So if we, e.g., receive double the payoff, we consider this to be twice as good, or bad.

A gamble is desirable for us if we accept ownership of it when offered to us. We model our uncertainty about the experiment's outcome using a set of desirable gambles.

1.2 Reasoning about and with sets of desirable gambles

We first need to establish what constitutes a reasonable set of desirable gambles. This forms the foundation of the theory and provides us with the basic rules that allow us to reason with sets of desirable gambles: make decisions and draw conclusions. This is the topic of this section.

1.2.1 Rationality criteria

So we use a set of desirable gambles c01-math-0014 to model our uncertainty about the experiment's outcome. Do we need to specify for each gamble individually whether we consider it desirable? Or can we argue that if some gambles are desirable, then others should be as well, i.e., can we automatically extend a partial specification? And must we consider some specific set of gambles to be desirable or not desirable? If we adopt some rationality criteria, the answer to the second and third questions is yes, and thus no to the first.

The first two, constructive rationality criteria express that we are working with a linear utility scale and say that a gamble's desirability should be independent of the stake and that the combination of two desirable gambles should also be desirable:

1.1

equation

1.2

equation

where c01-math-0017 and c01-math-0018 define the elementwise operations used. These two criteria enable the positive answer to question two.

So, given a partial desirability specification, or assessment c01-math-0019 , these criteria extend desirability to all gambles in its so-called positive hull:

1.3

equation

The constructive rationality criteria define a convex cone and therefore the positive hull operator generates cones of gambles. Its effect is illustrated in Figure 1.2; note the exclusion of the zero gamble in the extension. The positive hull operator moreover allows us to express the constructive rationality criteria succinctly: c01-math-0021 .

c01f002

Figure 1.2 An illustration of the positive hull operation (left) and the positive and negative orthants (right).

The third question effectively asks whether there are compelling reasons to insist that some gambles should be desirable or not desirable, independent of the information available to us about the experiment's outcome. And indeed there are: any gamble that might give a positive payoff without ever giving a negative one is certainly desirable, and any gamble that might give a negative payoff without ever giving a positive one is not. This results in a second pair of rationality criteria:

1.4

equation

1.5

equation

where we have used gamble (vector) inequalities and the zero gamble; i.e., c01-math-0024 if and only if c01-math-0025 and c01-math-0026 if moreover c01-math-0027 , or c01-math-0028 .

We see that the so-called positive orthant c01-math-0029 and negative orthant c01-math-0030 must, respectively, be included in and excluded from the set of desirable gambles. Their definition is implicit in Criteria (1.4) and (1.5). A toy illustration of them is shown in Figure 1.2. By replacing the positive and negative orthant in Criteria (1.4) and (1.5) by their interior, we obtain a weaker pair of criteria, used only in Section 1.6.4 on simplified variants of desirability:

1.6

equation

1.7

equation

1.2.2 Assessments avoiding partial or sure loss

We say that an assessment c01-math-0033 incurs partial loss if any desirable gamble, including those implied by the constructive rationality criteria, incurs a partial loss, or formally, if c01-math-0034 . Conversely, an assessment c01-math-0035 avoids partial loss if

1.8 equation

Again, we talk about a sure loss when c01-math-0037 is replaced by its interior c01-math-0038 . In the illustration on the left in Figure 1.3, the assessment c01-math-0039 incurs a sure loss because the combination of two of its elements, c01-math-0040 and c01-math-0041 , always has a strictly negative outcome.

c01f003

Figure 1.3 An illustration of an assessment c01-math-0042 incurring sure loss (left) and an illustration of a coherent set of desirable gambles c01-math-0043 (right).

1.2.3 Coherent sets of desirable gambles

A set of desirable gambles c01-math-0044 is coherent if it satisfies the four rationality criteria (1.1), (1.2), (1.4), and (1.5). So a coherent set of desirable gambles is a convex cone excluding c01-math-0045 and including c01-math-0046 , which is the smallest coherent set of desirable gambles. One is drawn as an illustration on the right in Figure 1.3. The set of all coherent sets of desirable gambles on the possibility space c01-math-0047 is denoted c01-math-0048 .

No coherent set of desirable gambles includes an assessment that incurs partial loss. However, an assessment that avoids partial loss can, in general, be included in an infinity of coherent sets of desirable gambles. This is illustrated for an assessment c01-math-0049 in Figure 1.4: six of its coherent extensions are shown. (Ignore the dashed lines for now.) The set c01-math-0050 contains all the coherent desirable gambles that include an assessment c01-math-0051 .

c01f004

Figure 1.4 Illustration of the multiplicity of the coherent extensions of an assessment c01-math-0052 .

The coherent sets of desirable gambles can be ordered according to set inclusion c01-math-0053 . A set of desirable gambles that includes another one is called more committal than the other, because it models a state in which we are committed to accepting more gambles. We know that this partial order has a least element, c01-math-0054 .

The set of coherent extensions of an assessment that avoids partial loss inherit this order. The Hasse diagram for the selection of extensions given in Figure 1.4 is given in Figure 1.5. Based on the sets shown there, our intuition tells us that c01-math-0055 is the least committal coherent extension of the assessment c01-math-0056 shown, i.e., the intersection c01-math-0057 of all coherent extensions. The intuition that there is such an extension is correct and we will show that this is a general feature in the next section. Later on in Section 1.5, we will discuss the class of maximally committal coherent extensions.

c01f005

Figure 1.5 Illustration of the partial order of coherent sets of desirable gambles induced by the inclusion relation.

1.2.4 Natural extension

Given an assessment c01-math-0058 , the fact that all gambles in c01-math-0059 must be desirable, and the constructive rationality criteria, there is a natural extension, defined as

1.9

equation

The rightmost expression follows from Equation (1.3) and the fact that c01-math-0061 is already a convex cone.

An important result links the natural and the least committal coherent extensions:

Theorem 1.1

The natural extension c01-math-0062 of c01-math-0063 coincides with its least committal coherent extension c01-math-0064 if and only if c01-math-0065 avoids partial loss.

Proof.

By construction, the natural extension c01-math-0066 must be included in any coherent extension, if they exist, as they must satisfy Criteria (1.1), (1.2), and (1.4): it is therefore the least committal one if it is coherent itself. This is the case if and only if it also satisfies (1.5). From Equation (1.9) we see that c01-math-0067 's pointwise smallest gambles lie in c01-math-0068 or c01-math-0069 , which proves the necessary equivalence of c01-math-0070 avoiding partial loss, i.e., c01-math-0071 , and c01-math-0072 .

Natural extension is the prime tool for deductive inference in desirability: given an initial assessment, it allows us to straightforwardly deduce which gambles must also be desirable in order to satisfy coherence, but makes no further commitments.

1.2.5 Desirability relative to subspaces with arbitrary vector orderings

In this section, we have considered desirability relative to the linear space of all gambles c01-math-0073 with the ordinary vector ordering: c01-math-0074 and c01-math-0075 for all gambles c01-math-0076 and c01-math-0077 on c01-math-0078 . The theory can be developed, mutatis mutandis, for linear subspaces c01-math-0079 with a vector ordering determined by an arbitrary cone c01-math-0080 instead of c01-math-0081 [213]: for all gambles c01-math-0082 and c01-math-0083 in c01-math-0084 , c01-math-0085 and c01-math-0086 . (We let the cone be a strict subset of the subspace to prevent triviality of the vector ordering.)

Restricting attention to linear subspaces can be especially useful for infinite possibility spaces. An example is the set of polynomials on the unit simplex [213]. Considering vector orderings different from the ordinary can be useful when looking at transformations between different vector spaces that do not map the positive orthant to the positive orthant, but to some other cone.

We do not pursue a treatment of desirability in this generality here, not to overly burden the notation. Nevertheless, it is good to keep this possibility in mind, especially when dealing with transformations, as in the upcoming section; it allows us to deal straightforwardly with a much wider class of them.

1.3 Deriving and combining sets of desirable gambles

Sometimes, we want to be able to look at the experiment from a different viewpoint, focus on some specific aspect, or combine different aspects. We will then also want to carry over the information we have formalized as a set of desirable gambles on one possibility space to another. In this section we are first going to get acquainted with gamble space transformations, the basic mathematical tool to deal with this kind of problem, then look at how one coherent set of desirable gambles can be derived from another and investigate two special instances, conditioning and marginalization, and finally explain how coherent sets of desirable gambles can be combined.

1.3.1 Gamble space transformations

Given two possibility spaces c01-math-0087 and c01-math-0088 and the corresponding sets of gambles, relationships between them can be expressed using transformations c01-math-0089 from c01-math-0090 to c01-math-0091 that—to preserve topological structure—are continuous. To work with such transformations, we first introduce some concepts:

1.10

equation

1.11 equation

The direct image of a set of gambles c01-math-0094 is c01-math-0095 ; in particular, the vector (sub)space c01-math-0096 is the range. The inverse image of a set of gambles c01-math-0097 is c01-math-0098 ; in particular, c01-math-0099 is the kernel. Whenever c01-math-0100 is injective, its inverse c01-math-0101 is well-defined on c01-math-0102 by c01-math-0103 . The inverse of a linear increasing injective transformation is also linear, but need not be increasing ([213, Example 9]). (This may lead one to use non-ordinary vector orderings.)

While other transformations may also be interesting, we here restrict attention to a class of transformations that are guaranteed to leave the partial-loss criterion invariant and commute with natural extension. For brevity's sake, we call linear increasing injective transformations with an increasing inverse liiwii transformations.

Lemma 1.1

Given a liiwii transformation c01-math-0104 from c01-math-0105 to c01-math-0106 , then (i) c01-math-0107 and c01-math-0108 commute, i.e., c01-math-0109 , and (ii) c01-math-0110 and c01-math-0111 .

Proof.

Claim (i) is a consequence of linearity. Claim (ii) follows from both c01-math-0112 and c01-math-0113 being increasing.

Proposition 1.1

Given a liiwii transformation c01-math-0114 from c01-math-0115 to c01-math-0116 , then: (i) for any c01-math-0117 , c01-math-0118 ; (ii) c01-math-0119 avoids partial loss if and only if c01-math-0120 does.

Proof.

To prove Claim (i), we transform the right-hand side of Equation (1.9). Given Lemma 1.1, we need to prove that

c01-math-0121

. Its left-hand side can be rewritten as c01-math-0122 because of linearity; the right-hand side can be put in this form by first applying commutativity of c01-math-0123 and c01-math-0124 and then realizing the first intersection factor can only contain elements outside of c01-math-0125 due to elements in c01-math-0126 outside of c01-math-0127 .

For Claim (ii), we show that Equation (1.8) remains invariant:

c01-math-0128

where the first step is immediate and the second follows from Lemma 1.1; similarly

c01-math-0129

Liiwii transformations appear when combining coherent sets of desirable gambles; their inverse is used when deriving one coherent set of desirable gambles from another.

1.3.2 Derived coherent sets of desirable gambles

Now, given a set of desirable gambles c01-math-0130 , we can use a liiwii transformation c01-math-0131 from c01-math-0132 to c01-math-0133 to derive a set of desirable gambles c01-math-0134 on c01-math-0135 .

Proposition 1.2

Given a liiwii transformation c01-math-0136 from c01-math-0137 to c01-math-0138 , then c01-math-0139 is coherent if c01-math-0140 is coherent.

Proof.

The transformation c01-math-0141 from c01-math-0142 to c01-math-0143 is also liiwii. Let c01-math-0144 , which avoids partial loss, then Proposition 1.1 tells us c01-math-0145 avoids partial loss. Lemma 1.1 allows us to conclude c01-math-0146 is a convex cone because c01-math-0147 is. We know c01-math-0148 ; applying c01-math-0149 allows us to conclude, using Lemma 1.1, that c01-math-0150 . So c01-math-0151 satisfies Criteria (1.1), (1.2), (1.4), and (1.5) and is therefore coherent.

In the illustration on the left in Figure 1.6, c01-math-0152 maps from c01-math-0153 to c01-math-0154 and is defined for any gamble c01-math-0155 on c01-math-0156 : c01-math-0157 and c01-math-0158 .

c01f006

Figure 1.6 Illustrations of derived coherent sets of desirable gambles.

Linear transformations map linear vector spaces to linear vector spaces. So the range c01-math-0159 will either coincide with c01-math-0160 or be a strict subspace of it, that, because c01-math-0161 is increasing, includes part of the positive and negative orthants. In the latter case, one can conceptually visualize an important part of the definition of c01-math-0162 as taking a slice of c01-math-0163 by intersecting it with the linear subspace c01-math-0164 .

Let us give a more involved illustration on the right in Figure 1.6 that clarifies this. First, look at the left figure on the right. To understand it, we first note that the intersection of many a convex cone with an appropriate hyperplane results in a convex polytope. In light grey, we have depicted the intersection of a coherent set of desirable gambles c01-math-0165 and the hyperplane consisting of those gambles with components that sum to one. The triangle that is the convex hull of the indicators of elementary events indicates the intersection with the positive orthant. Dotted lines indicate open parts of the border and undecorated ones indicate closed parts. Darkly-filled dots are closure points of interest belonging to c01-math-0166 ; white-filled ones do not belong to c01-math-0167 . For each, we indicate which gamble c01-math-0168 it depicts, explicitly or as a vector c01-math-0169 .

Rightmost, we have indicated two transformations that slice our cone c01-math-0170 , with resulting derived sets of coherent desirable gambles depicted: c01-math-0171 from c01-math-0172 to c01-math-0173 is defined by c01-math-0174 and c01-math-0175 ; c01-math-0176 from c01-math-0177 to c01-math-0178 is defined by c01-math-0179 , c01-math-0180 and c01-math-0181 .

1.3.3 Conditional sets of desirable gambles

We may be interested in obtaining an uncertainty model for the situation in which the experiment's outcome belongs to a conditioning event c01-math-0182 . In that case, we can focus on gambles that are contingent on c01-math-0183 occurring: these are gambles such that if c01-math-0184 does not occur, no payoff is received or due—status quo is maintained.

This idea can be formalized by introducing the liiwii transformation c01-math-0185 that maps gambles on any conditioning event c01-math-0186 to contingent gambles on c01-math-0187 , defined for every gamble c01-math-0188 on c01-math-0189 by

1.12 equation

where c01-math-0191 is c01-math-0192 's complement. Given a set of desirable gambles c01-math-0193 , we call

equation

its set of desirable gambles conditional on c01-math-0195 . Often, this name is also given to c01-math-0196 and

equation

which are equivalent to c01-math-0198 and each other as uncertainty models.

As an illustration, we depict, in Figure 1.7, the three conditional sets of desirable gambles that can be associated to the set of desirable gambles c01-math-0199 from the illustration on the right in Figure 1.6. These are obtained by slicing along subspaces corresponding to the span of subsets of axes. We have explicitly included the gambles that correspond to the contingent border gambles of the full uncertainty model.

c01f007

Figure 1.7 An illustration of conditional sets of desirable gambles.

A conditional set of desirable gambles expresses current commitments contingent on the occurrence of the conditioning event. Although it is not required by the rationality criteria we have adopted, conditional models are also often assumed to express current or future commitments after (nothing but) the conditioning event is learned to have occurred: they can be used as updated sets of desirable gambles.

1.3.4 Marginal sets of desirable gambles

When the experiment's possibility space has a Cartesian product structure, so if c01-math-0200 , we may wish to focus on one of the component possibility spaces, meaning that we can ignore one aspect of the experiment's outcome. So, assume we focus on c01-math-0201 —or ignore c01-math-0202 . Then the interesting gambles are those whose payoffs do not depend on the c01-math-0203 -component.

This idea can be formalized by introducing the liiwii transformation c01-math-0204 that performs the so-called cylindrical extension of a gamble on any possibility space c01-math-0205 to its Cartesian product with c01-math-0206 ; it is defined for any gamble c01-math-0207 on c01-math-0208 and outcome c01-math-0209 in c01-math-0210 by

1.13 equation

Given a set of desirable gambles c01-math-0212 , we call

equation

its c01-math-0214 -marginal.

To get a feel for what the effect is of cylindrical extension, visualization limitations force us to look, in Figure 1.8, at the trivial possibility space c01-math-0215 . Gambles on c01-math-0216 are constants, and c01-math-0217 maps them to the corresponding slice, the main diagonal of c01-math-0218 .

c01f008

Figure 1.8 Illustration of marginal sets of desirable gambles.

Cylindrical extension c01-math-0219 works between gamble spaces, but it can be expressed using a surjective coordinate projection c01-math-0220 that maps elements of the Cartesian product possibility space c01-math-0221 to c01-math-0222 ; it is defined for any outcome pair c01-math-0223 by c01-math-0224 . Namely, c01-math-0225 , where c01-math-0226 denotes function composition. This projection generates the partition

c01-math-0227

, so the c01-math-0228 -marginal can therefore also be seen as an uncertainty model with this partition as the possibility space.

Based on these observations, we can generalize the concept of marginalization to general surjective maps c01-math-0229 from c01-math-0230 to c01-math-0231 , where c01-math-0232 does not need to have a Cartesian product space structure. The associated transformation c01-math-0233 defined for any gamble c01-math-0234 on c01-math-0235 is obtained by the so-called lifting procedure:

1.14 equation

Now, given a set of desirable gambles c01-math-0237 , the derived set of desirable gambles

c01-math-0238

can be called its c01-math-0239 -marginal. And c01-math-0240 and the partition c01-math-0241 can be interchangeably used as its possibility space.

Given that surjective maps and partitions encode the same type of information, we can again translate the above concepts when a partition c01-math-0242 of the possibility space c01-math-0243 is given: c01-math-0244 denotes the associated transformation and c01-math-0245 the derived c01-math-0246 -marginal. Return to the involved illustration on the right in Figure 1.6 for a moment: c01-math-0247 actually coincides with c01-math-0248 under the identification c01-math-0249 and c01-math-0250 .

1.3.5 Combining sets of desirable gambles

Now that we have gained some insight into how a set of desirable gambles and its derived sets of desirable gambles are related, combining sets of desirable gambles is straightforward: we view them as being derived from the unknown, sought for joint set of desirable gambles. This means that we must first make explicit the transformations between the gamble spaces of the sets of desirable gambles to be combined and the joint one, i.e., we must provide an interpretation of their relationships. The union of the transformed sets of desirable gambles is then taken as an assessment. If this assessment avoids partial loss in the joint possibility space, the sets of desirable gambles are compatible and its natural extension is the joint.

Let us clarify this with two examples.

Example 1.1

For the first, consider three sets of desirable gambles, c01-math-0251 , c01-math-0252 , and c01-math-0253 , interpreted as conditional sets of desirable gambles of a joint on c01-math-0254 . Respective resulting contingent desirable gambles are c01-math-0255 , c01-math-0256 , and c01-math-0257 ; their sum, desirable by additivity, is c01-math-0258 , so the coherent sets of desirable gambles are incompatible in the sense that their joint incurs sure loss.

Example 1.2

For our second example, illustrated in Figure 1.9, we return to the illustration on the right in Figure 1.6 and the associated illustration in Figure 1.7. There, we started from a coherent set of desirable gambles c01-math-0259 , and obtained the derived sets of desirable gambles c01-math-0260 and c01-math-0261 , and conditional ones c01-math-0262 , c01-math-0263 , and c01-math-0264 . Here, we work back from those as the sets of desirable gambles to be combined: let

equation

be the assessment, which avoids partial loss, given that we are only putting slices back in place. The resulting joint c01-math-0266 is depicted in Figure 1.9 we see that it is strictly less committal than the original c01-math-0267 because the particular sets of desirable gambles we derived do not contain all the information present in c01-math-0268 .

c01f009

Figure 1.9 Illustration of combining sets of desirable gambles.

Let us make our remark about ‘putting slices back in place’ precise with the following immediate result:

Proposition 1.3

If c01-math-0269 is coherent and c01-math-0270 , then c01-math-0271 is coherent.

Our first example showed that individual coherence—i.e., coherence of the individual sets of desirable gambles to be combined—does not imply coherence of the joint. However, there is an often-occurring specific situation where this does hold. Individual conditional sets of desirable gambles are separately specified if their conditioning events are disjoint. If they are moreover also coherent, then they are called separately coherent.

Theorem 1.2

Given a partition c01-math-0272 of c01-math-0273 , a coherent c01-math-0274 -marginal c01-math-0275 , and separately coherent conditional sets of desirable gambles c01-math-0276 , c01-math-0277 , then their combination c01-math-0278 , with c01-math-0279 , is coherent as well.

Proof.

We need to prove that c01-math-0280 avoids partial loss. Assume it does not, then there will be a finite subset c01-math-0281 of c01-math-0282 , a nonzero gamble c01-math-0283 , and nonzero gambles c01-math-0284 , c01-math-0285 , such that c01-math-0286 . Coming from the c01-math-0287 -marginal c01-math-0288 , c01-math-0289 is constant on elements of c01-math-0290 ; because c01-math-0291 is coherent, it will be positive on at least one event c01-math-0292 . Being contingent gambles, the c01-math-0293 , c01-math-0294 , have disjoint support. This means that c01-math-0295 , contradicting separate coherence.

Combining separately coherent conditional uncertainty models and a coherent marginal one is called marginal extension.

In the proof above, we use the additivity criterion to combine a finite number of contingent gambles. Some authors specifically strengthen this [672, § 6.8, App. F]:

1.15

equation

where c01-math-0297 is an (infinite) class of disjont events of c01-math-0298 . If this extra criterion is required for all possible partitions of c01-math-0299 , then c01-math-0300 is called fully conglomerable.

Let us look at an example to illustrate the issue targeted with conglomerability.

Example 1.3

Consider c01-math-0301 , the set of nonzero integers, as the possibility space. Let the gamble c01-math-0302 be defined by c01-math-0303 and c01-math-0304 and define c01-math-0305 for all c01-math-0306 . Then c01-math-0307 is compatible with the assessment c01-math-0308 , but not with c01-math-0309 itself.

Apart from marginal extension, other types of extensions can be found in the literature, such as the epistemic irrelevance extension [482] and the exchangeable natural extension [214, 212, 213]. These extensions correspond to combinations under an additional structural assessment. Such structural assessments are often based on statements of indifference between pairs of gambles.

To make our understanding of desirability more precise, and e.g., to properly introduce the concept of indifference, we next look into the relationship with partial preference orders.

1.4 Partial preference orders

For purposes of decision making, it is useful to be able to compare gambles; e.g., each may correspond to a certain action and we wish to determine which should be preferred.

Desirability as presented in Section 1.2 is compatible with two types of partial preference orders, which specify pairwise comparisons of gambles. Each casts a different light on the phrase ‘if we accept ownership of it when offered’ in the definition of a desirable gamble. And in each case, this added clarification translates into a slight strengthening of one of the coherence criteria. So after this section, we end up with two variants of desirability, each of which describes coherent sets of desirable gambles that are, as uncertainty models, equivalent to a partial preference order of the corresponding type. Only one is used further on.

Much of the material in this chapter remains mathematically unaffected in the essentials by the strengthened criteria. We point out any minor modification necessary to preceding material.

1.4.1 Strict preference

The first partial preference order we look at, the strict one, is also the most straightforward one. A gamble c01-math-0310 is (strictly) preferred to c01-math-0311 —formally, c01-math-0312 —if we are eager to exchange c01-math-0313 for c01-math-0314 . This concept is linked to desirability by positing that a gamble is desirable if and only if it is preferred to status quo, i.e., the zero gamble. So, formally, the relationship between the strict preference relation c01-math-0315 on c01-math-0316 and a set of desirable gambles c01-math-0317 becomes:

1.16 equation

A strict preference relation c01-math-0319 is coherent when the associated set of desirable gambles is. This can be expressed using the following rationality criteria:

1.17 equation

1.18

equation

1.19

equation

1.20 equation

where c01-math-0323 , c01-math-0324 , and c01-math-0325 can be any gamble on c01-math-0326 . Irreflexivity makes the partial preference strict; adding transitivity, which is the equivalent of additivity for sets of desirable gambles, makes it a preorder. Mixture independence encompasses both positive scaling (take c01-math-0327 ) and the first equivalence of Equation (1.16), the so-called cancellation property (take c01-math-0328 c01-math-0328 ). Monotonicity corresponds to accepting partial gains and together with irreflexivity and transitivity entails avoiding partial loss.

Irreflexivity precludes the zero gamble from belonging to the associated set of desirable gambles. Strengthening the rationality criteria of Section 1.2.1 accordingly is done by modifying avoiding partial loss, Criterion (1.5), to

1.21

equation

where the last version can be used because together with additivity it entails the first two. Also, any coherent set of desirable gambles containing part of the kernel of a gamble space transformation is incompatible with that transformation.

1.4.2 Nonstrict preference

A gamble c01-math-0330 is nonstrictly preferred to c01-math-0331 —formally, c01-math-0332 —if we are willing, i.e., not adverse, to exchange c01-math-0333 for c01-math-0334 . This concept is linked to desirability by positing that a gamble is desirable if and only if it is nonstrictly preferred to status quo, i.e., the zero gamble. So, formally, the relationship between the nonstrict preference relation c01-math-0335 on c01-math-0336 and a set of desirable gambles c01-math-0337 becomes:

1.22 equation

Analogously to what we saw before, a nonstrict preference relation c01-math-0339 is coherent when the associated set of desirable gambles is. This can be expressed using the following rationality criteria:

1.23 equation

1.24 equation

1.25

equation

1.26 equation

where c01-math-0344 , c01-math-0345 , and c01-math-0346 can be any gamble on c01-math-0347 . Reflexivity makes the partial preference nonstrict. Transitivity, mixture independence, and monotonicity have a similar function as for the strict variant.

Now, reflexivity forces the zero gamble to belong to the associated set of desirable gambles. Strengthening the rationality criteria of Section 1.2.1 accordingly is now done by modifying accepting partial gain, Criterion (1.4), to

1.27

equation

Formula (1.9) for natural extension has to be modified accordingly, i.e., c01-math-0349 needs to be replaced by c01-math-0350 .

There are two interesting symmetric relations that can be derived from a nonstrict preference order. The first is the equivalence relation called indifference: we are indifferent between two gambles c01-math-0351 and c01-math-0352 if we both nonstrictly prefer c01-math-0353 over c01-math-0354 and vice-versa:

1.28 equation

A nonstrict preference relation is compatible with a gamble space transformation if it entails indifference between the kernel gambles and the zero gamble. The second symmetric relation is incomparability: two gambles c01-math-0356 and c01-math-0357 are incomparable if we nonstrictly prefer neither over the other:

1.29 equation

In keeping with the intuition provided by its name, incomparability is not an equivalence relation, as it is not reflexive.

We illustrate these two concepts in Figure 1.10: the zero gamble and three equivalence classes of the indifference relation are ordered according to the relative nonstrict preference of their elements. Although the gambles in c01-math-0359 are nonstrictly preferred to those in c01-math-0360 , the gambles in both are incomparable with those in c01-math-0361 ; this highlights that incomparability is often intransitive.

c01f010

Figure 1.10 An illustration of nonstrict preference, indifference, and incomparability.

1.4.3 Nonstrict preferences implied by strict ones

Incomparability and indifference are very useful concepts: the former allows us to highlight where the information available is lacking and the latter allows us to straightforwardly specify many common structural—e.g., symmetry—assessments. However, to define them, we need a reflexive and thus nonstrict preference relation. Therefore, we give an example of how we can associate a nonstrict preference relation c01-math-0362 to a strict one c01-math-0363 , or equivalently, associate a set of nonstrictly desirable gambles c01-math-0364 to a set of strictly desirable gambles c01-math-0365 .

In general, there is no unique way of doing this and one must realize that it means adding commitments of some kind. Of course the resulting relation must satisfy the coherence criteria for nonstrict preference orders—reflexivity, specifically. That could be achieved simply by letting c01-math-0366 be equal to c01-math-0367 , but this approach would make indifference a vacuous concept: c01-math-0368 implies c01-math-0369 and therefore the same would hold for c01-math-0370 whenever c01-math-0371 , so there could only be the trivial indifference between a gamble and itself.

A nontrivial approach needs to be justified by giving an interpretation to the difference between strict and nonstrict preference that can be made mathematically precise. We draw inspiration from the difference between strict and nonstrict vector orderings (relative to c01-math-0372 ; cf. Section 1.2.5): c01-math-0373 if and only if c01-math-0374 for all gambles c01-math-0375 . For our preference relations, this becomes: a gamble c01-math-0376 is nonstrictly preferred to a gamble c01-math-0377 if and only if adding any gamble c01-math-0378 to c01-math-0379 makes c01-math-0380 strictly preferred to c01-math-0381 . Formally:

1.30 equation

If adopted, this rule is essentially an additional rationality axiom.

Proof that c01-math-0383 satisfies (

1.23)-(1.26)

Reflexivity is immediate, because c01-math-0384 . Given c01-math-0385 and c01-math-0386 , additivity implies c01-math-0387 , so transitivity holds. To prove that mix-independence holds, i.e.,

c01-math-0388

, realize that positive scaling implies c01-math-0389 ; then the equivalence's right-hand side can be written as c01-math-0390 , so that we see both sides only formally differ, in their arbitrary scaling factor. Monotonicity's first right-hand conjunct follows from c01-math-0391 and additivity; the second, c01-math-0392 , follows from the fact that c01-math-0393 , but c01-math-0394 .

An immediate consequence of Definition (1.30) is

1.31 equation

The incomparability and indifference relations associated to c01-math-0396 are respectively denoted by c01-math-0397 and c01-math-0398 . (De Cooman & Quaeghebeur [213] use an indifference relation thus defined.)

Let us clarify the material in this section by giving a somewhat more extensive illustration in Figure 1.11. We show a number of cases, i.e., sets of strictly desirable gambles (in grey). These are convex cones and are completely determined by a number of desirable and nondesirable extreme rays, i.e., cone lines starting in the zero gamble whose elements' (non)desirability is not imposed by convexity. Extreme rays whose elements are neither desirable in the strict preference sense nor in the associated nonstrict preference sense, are drawn dashed. Those whose elements are only nonstrictly desirable are drawn dotted. The ones with elements that are strictly—and thus also nonstrictly—desirable remain undecorated.

c01f011

Figure 1.11 An illustration of the possible different preference relations between gambles.

In the top row, the strongest relation between the gambles c01-math-0399 and c01-math-0400 shown in each of the cases, as implied by the given set of strictly desirable gambles is: c01-math-0401 , c01-math-0402 , and c01-math-0403 . For the first two cases on the bottom row, we have c01-math-0404 and c01-math-0405 ; we see that the cone's border structure can have an important effect. For the bottom rightmost case, we again have c01-math-0406 ; monotonicity leaves us no other choice.

In the first four cases of the illustration, the set c01-math-0407 is the closure of c01-math-0408 . The last two cases, however, remind us that this is not always so: the former because of Equation (1.30), the latter also, but more directly because of monotonicity.

1.4.4 Strict preferences implied by nonstrict ones

In decision making, strict preferences are especially useful, as they allow us to really choose one gamble over another. So it can be useful to be able to associate a strict preference relation c01-math-0409 to a nonstrict one c01-math-0410 , or equivalently, associate a set of strictly desirable gambles c01-math-0411 to a set of nonstrictly desirable gambles c01-math-0412 .

Also here we want to use the distinction between strict and nonstrict preference established in Equation (1.29). This essentially means that we need to find the set c01-math-0413 which has c01-math-0414 as its associated set of nonstrictly desirable gambles.

However, as can be seen in the first and last first-row cases in the illustration of Figure 1.11, there is not always a unique such c01-math-0415 . Moreover, within the current framework, not all sets of nonstrictly desirable gambles allowed by Criteria (1.23)-(1.26) even have such an associated set of strictly desirable gambles (cf. example below).

Although this is a topic about which little research has been done, a sketch of how the framework might be extended may be interesting for some readers. The leftmost drawing in Figure 1.12 gives an example of a situation incompatible with Equation (1.29): here both the set of strict and nonstrictly desirable gambles have open faces.

c01f012

Figure 1.12 An illustration of an extension of desirability involving lexicographic utility.

One way to make sense of it is by interpreting it as a partial view of a more complex uncertainty model in which the payoffs of gambles are defined with infinitesimal precision. For finite possibility spaces this can be implemented using—two-tier, for the illustration of Figure 1.12—lexicographic utility: In such a context)(, a lexicographic gamble c01-math-0416 can be written as c01-math-0417 , where c01-math-0418 is an infinitesimal quantity and c01-math-0419 and c01-math-0420 are, respectively, real-valued gambles for tier zero and one. The uncertainty model consists of a set of desirable lexicographical gambles c01-math-0421 and the original situation is a view of lexicographic gambles that are constant over the tiers, i.e., of the type c01-math-0422 , with c01-math-0423 some real-valued gamble. In this wider context, compatibility with (1.29) is preserved.

Another way could consist of defining separate but connected strict and nonstrict preference orders, and not derive one from the other.

1.5 Maximally committal sets of strictly desirable gambles

At the end of Section 1.2.3, we saw that coherent sets of desirable gambles form a partial order under inclusion. This carries over to coherent extensions of assessments that avoid partial loss. The least committal extension is their intersection and can be calculated using the natural extension (cf. Theorem 1.1).

In this short section (based on [213, § 2.4]), we are going to investigate the other extreme and look at the maximally committal—or maximal—coherent sets of desirable gambles, at maximally committal—or maximal—extensions of assessments, and their relationship with the least committal one. We from now on work with a strict preference interpretation of desirability, augmented with Rule (1.29). This means that avoiding partial loss is replaced by avoiding nonpositivity: the zero gamble is not desirable. This is crucial to obtaining interesting results; a similar analysis would not be as straightforward or even possible under a nonstrict preference interpretation.

A maximal coherent set of desirable gambles c01-math-0424 is one that is not included in any other coherent set of desirable gambles. In other words, when adding any gamble c01-math-0425 in c01-math-0426 to c01-math-0427 , this would result in an assessment that incurs nonpositivity. The set of maximal coherent sets of desirable gambles on c01-math-0428 is denoted by c01-math-0429 .

What do these maximal coherent sets of desirable gambles look like? The following proposition provides a nice characterization:

Proposition 1.4

The set c01-math-0430 in c01-math-0431 is maximal if and only if c01-math-0432 for all nonzero gambles c01-math-0433 on c01-math-0434 .

Proof.

Coherence, avoiding nonpositivity to be precise, makes c01-math-0435 a necessity for all nonzero gambles c01-math-0436 on c01-math-0437 . So we have to prove that maximality of c01-math-0438 is equivalent to c01-math-0439 for all nonzero gambles c01-math-0440 .

First necessity; assume c01-math-0441 is maximal but nevertheless c01-math-0442 for some nonzero gamble c01-math-0443 . Then both c01-math-0444 and c01-math-0445 incur nonpositivity, which, because c01-math-0446 is coherent and thus a cone excluding the zero gamble, means both c01-math-0447 and c01-math-0448 . Or, in other words, c01-math-0449 , a contradiction.

Now sufficiency; if c01-math-0450 for all nonzero gambles c01-math-0451 , then any set that strictly includes c01-math-0452 incurs nonpositivity.

We see that maximal sets of desirable gambles are halfspaces in a very concrete sense. They are neither open nor closed. An illustration with two cases is given in Figure 1.13.

c01f013

Figure 1.13 Illustration of maximal sets of desirable gambles.

Now, how can we use these maximal sets of desirable gambles to our advantage? Given an assessment c01-math-0453 , both nonpositivity avoidance and the least committal extension can be characterized using the set c01-math-0454 of maximal coherent sets of desirable gambles. This is done in the following two results:

Theorem 1.3

An assessment c01-math-0455 avoids nonpositivity if and only if c01-math-0456 .

Proof sketch

First sufficiency; assume there is a c01-math-0457 in c01-math-0458 such that c01-math-0459 . Then c01-math-0460 and thus c01-math-0461 .

Now necessity; assume c01-math-0462 avoids nonpositivity. In case c01-math-0463 is finite, one can then construct a maximal set of desirable gambles that includes c01-math-0464 in a finite number of steps by each time enlarging the assessment with a gamble from outside its natural extension and the negation thereof [156, Theorem 12]. In case c01-math-0465 is infinite, a nonconstructive approach can be applied in which infinite such chains of sets of desirable gambles that include c01-math-0466 are employed [213, Theorem 3].

Corollary 1.1

The least committal extension of an assessment c01-math-0467 that avoids nonpositivity, i.e., its natural extension c01-math-0468 , is the intersection c01-math-0469 of the maximal sets of desirable gambles that include c01-math-0470 .

Proof.

That the least committal extension is a subset follows from avoiding nonpositivity. Assume it is a strict subset and take c01-math-0471 in c01-math-0472 . Then c01-math-0473 avoids nonpositivity and thus c01-math-0474 is a coherent extension of c01-math-0475 , but it is not included in any element of c01-math-0476 , a contradiction.

To close off this section, we present a result that shows derived—e.g., conditional or marginal—sets of desirable gambles of maximal models are maximal as well.

Proposition 1.5

If c01-math-0477 is maximal, then c01-math-0478 is maximal for a linear increasing injective transformation c01-math-0479 from c01-math-0480 to c01-math-0481 with increasing inverse.

Proof.

Proposition 1.2 ensures that c01-math-0482 is coherent. Now assume that the derived set of desirable gambles c01-math-0483 is nonmaximal, i.e., that there is some gamble c01-math-0484 on c01-math-0485 such that c01-math-0486 , or c01-math-0487 . Then, because of c01-math-0488 's self-conjugacy, c01-math-0489 , or c01-math-0490 , a contradiction.

1.6 Relationships with other, nonequivalent models

We have studied sets of desirable gambles and investigated their connection to partial preference orders, which gave rise to strict and nonstrict variants. In the preceding section we settled on the strict variant. Sets of desirable gambles and partial preference orders are equivalent uncertainty models: one can be expressed in terms of the other and vice versa. In this section, we investigate their connection with other, commonly used, but nonequivalent models.

Given an assessment denoted c01-math-0491 , where c01-math-0492 is a dummy variable that stands for an uncertainty model generating the assessment, we will use the notational conventions c01-math-0493 and c01-math-0494 .

1.6.1 Linear previsions

Linear previsions are positive, linear, normed functionals and popular uncertainty models in classical probability theory. A linear prevision provides fair prices for gambles, i.e., it is a real-valued function on c01-math-0495 . (We imbue the set of real-valuedfunctionals on c01-math-0496 with the topology of pointwise convergence.) They are mathematically equivalent, as uncertainty models, to (finitely additive) probability measures and, on finite possibility spaces, to probability mass functions (cf. Section 2.2.2).

The set of all linear previsions is denoted by c01-math-0497 . It is a closed convex set with extreme points that correspond to the c01-math-0498 -valued finitely additive probabilities [672, § 3.6.7]. For finite c01-math-0499 it is a simplex, with the degenerate previsions as its extreme points. There is a degenerate prevision c01-math-0500 for each elementary event c01-math-0501 of c01-math-0502 . Its defining property is that c01-math-0503 for every gamble c01-math-0504 .

Fair prices of indicators are called probabilities, and the following notational shorthand is used later on for every event c01-math-0505 : c01-math-0506 .

Given a linear prevision c01-math-0507 as an assessment, what gambles are desirable? We take a gamble to be strictly desirable whenever its fair price is positive. This results in the following rule for going from linear previsions to sets of desirable gambles:

1.32

equation

Due to linearity, the space of gambles with fair price zero will be a linear subspace of c01-math-0509 and therefore the corresponding set of desirable gambles will be a halfspace. Therefore the assessment c01-math-0510 is an open halfspace, which because of linearity and positivity avoids partial loss. When c01-math-0511 coincides with c01-math-0512 , the border gambles on the delimiting hyperplane are nonstrictly, but not strictly desirable. This is illustrated on the left in Figure 1.14. Due to coherence, enforced by the natural extension, however, this is not always the case; for example, the partial loss incurring gamble that is negative everywhere except in the elementary event c01-math-0513 , where it is zero, will have zero as its fair price under c01-math-0514 . The illustration on the right in Figure 1.14 depicts c01-math-0515 .

c01f014

Figure 1.14 Illustration of sets of desirable gambles corresponding to linear previsions.

Linear previsions are continuous functions, which explains why the associated assessment, the inverse image of c01-math-0516 , is an open set. Coherent sets of desirable gambles need not be open (or closed), and we know that in particular the maximal ones in c01-math-0517 , also halfspaces, are not. So linear previsions cannot express preferences as finely as coherent halfspaces of desirable gambles can. If we wish to define a correspondence between the latter and the former, we will end up with an uncertainty model that is less committal, effectively replacing the possibly nonopen halfspace with the usually open natural extension of its interior.

1.6.2 Credal sets

Very popular imprecise-probabilistic models are credal sets, or sets of linear previsions. Credal sets can, for example, arise in situations when a linear prevision cannot be precisely elicited in the sense that only bounds on fair prices for gambles can be obtained, or also when differing linear previsions, each provided by a different expert, are pooled.

Given a credal set c01-math-0518 as an assessment, what gambles are desirable? We take a gamble to be strictly desirable when it is so under any linear prevision in the credal set. This results in the following rule for going from credal sets to sets of desirable gambles:

1.33

equation

Each element in the credal set results in a linear constraint, i.e., restricts the desirable gambles to lie in a specific open halfspace. Because linear previsions are convex combinations of degenerate previsions, this rule will give the same result whether c01-math-0520 or its convex hull is used; so the convex hull's border structure is uniquely important.

Let us look at an illustration of this rule for a possibility space c01-math-0521 in Figure 1.15.

c01f015

Figure 1.15 An illustration of the rule for obtaining the set of desirable gambles c01-math-0522 corresponding to a credal set c01-math-0523 .

On the left, we show a convex credal set c01-math-0524 , the light grey filled polytope, within the simplex spanned by the degenerate previsions, the grey triangle. The border structure is shown as follows: Open border segments are indicated by dashed lines; closed ones are undecorated. The endpoints of border segments are indicated by dots, filled darkly when part of the credal set and white otherwise. For each, we give the corresponding linear prevision's probability mass function: i.e., the coefficient vector c01-math-0525 of its convex combination in terms of the respective degenerate previsions c01-math-0526 , c01-math-0527 , and c01-math-0528 .

In the middle, we show, in the plane of gambles with sum one, a sufficient subset of the constraints corresponding to c01-math-0529 ; the stubs indicate the delimited halfspace. All but one correspond to border segment endpoints: open halfspaces c01-math-0530 for endpoints in the credal set—indicated with dashed lines—and closed ones c01-math-0531 otherwise—indicated with full lines. The constraint c01-math-0532 does not correspond to such an endpoint. All nondepicted constraints are redundant, as well as c01-math-0533 , which is made redundant by c01-math-0534 and c01-math-0535 . Together, the constraints define the assessment c01-math-0536 .

On the right, the resulting set of desirable gambles c01-math-0537 is shown. Note how included (excluded) exposed border segment endpoints of c01-math-0538 map to open (closed) faces of c01-math-0539 and how nonopen (open) faces of c01-math-0540 map to excluded (included) border rays of c01-math-0541 . The nonopen to excluded mapping makes it clear that not all of c01-math-0542 's border structure can be preserved: the resulting model is less committal.

Now, given a coherent set of strictly desirable gambles c01-math-0545 , what is the credal set that we should associate with it? The natural counterpart to the idea expressed in Equation (1.32) would be the set of linear previsions c01-math-0546 in c01-math-0547 such that c01-math-0548 for all gambles c01-math-0549 in c01-math-0550 . However, given a set c01-math-0551 in c01-math-0552 for some c01-math-0553 in c01-math-0554 , then we know no linear prevision c01-math-0555 in c01-math-0556 will satisfy the requirement, not even c01-math-0557 . Namely, c01-math-0558 then includes border gambles c01-math-0559 of the open set c01-math-0560 such that c01-math-0561 , and except for c01-math-0562 , all border gambles of c01-math-0563 are included in some element of c01-math-0564 . Therefore, to deal with the fact that previsions cannot express such border structure and to allow us to associate a credal set with any coherent set of desirable gambles, we must replace the strict inequality with a nonstrict one:

1.34 equation

The resulting credal set has a simple structure:

Proposition 1.6

The credal set c01-math-0566 associated to a coherent set of desirable gambles c01-math-0567 is closed and convex.

Proof.

The set c01-math-0568 of all linear previsions is closed and convex. Considering for a moment previsions just as linear functionals on c01-math-0569 , i.e., not restricted to c01-math-0570 , then for any c01-math-0571 in c01-math-0572 , c01-math-0573 is a linear constraint, i.e., it specifies a closed half-space. So c01-math-0574 is an intersection of closed convex sets, which is closed and convex.

Let us look at an illustration of this rule in Figure 1.16, again for a possibility space c01-math-0575 . The meaning of the picture elements is the same as before. On the left, we show a set of desirable gambles c01-math-0576 , or rather, its intersection with the sum-one plane. In the middle, we show the necessary and sufficient set of linear constraints in prevision-space corresponding to this set of desirable gambles: each of the constraints corresponds to an extreme point of the closure of c01-math-0577 ; their intersection is shaded grey. On the right, we show the resulting credal set c01-math-0578 .

c01f016

Figure 1.16 An illustration of the rule for obtaining the credal set c01-math-0543 corresponding to a set of desirable gambles c01-math-0544 .

1.6.3 To lower and upper previsions

We have seen that linear previsions correspond to a very specific type of sets of desirable gambles, halfspaces, although they cannot determine the border structure of the halfspace. Linear previsions specify fair prices for gambles. We now show how this idea can be generalized, i.e., how we can associate a pair of super and sublinear previsions to any set of desirable gambles, although again they will not determine border structure. These are respectively called lower and upper previsions, and defined in terms of, respectively, buying and selling prices for gambles.

The association is made by making comparisons with constant gambles, which we denote by their constant value. An illustration with a constant gamble c01-math-0579 is given on the left in Figure 1.17. The reason is that the set of constant gambles can be linearly ordered—trivially so. So they facilitate the connection with the price scale.

c01f017

Figure 1.17 An illustration of the association of lower and upper previsions and probabilities c01-math-0580 and c01-math-0581 with sets of desirable gambles.

Consider a coherent set of strictly desirable gambles c01-math-0582 and the corresponding strict partial preference relation c01-math-0583 . The corresponding lower prevision c01-math-0584 —interpreted to specify supremum acceptable buying prices—and upper prevision c01-math-0585 —specifying infimum acceptable selling prices—are defined as follows for every gamble c01-math-0586 on c01-math-0587 :

1.35

equation

1.36

equation

An illustration is given in the middle in Figure 1.17. These definitions make lower and upper previsions conjugate: c01-math-0590 . This means that both previsions are equivalent uncertainty models and either can be used on its own; we work with lower previsions.

Linear previsions are self-conjugate lower—and thus upper—previsions: coinciding acceptable buying and selling prices are fair prices.

Similarly as for linear previsions, lower and upper previsions of indicators are called lower and upper probabilities. The following notational shorthands are used: c01-math-0591 and c01-math-0592 . An illustration is given on the right in Figure 1.17.

Conceptually, desirability is relatively simple: generating convex cones from initial assessments and slicing those to obtain conditional and marginal models. This conceptual framework can also be exploited to derive results in the theory of lower previsions. For that, we need to be able to associate a set of desirable gambles with a given lower prevision. To facilitate this, we first look at a number of simplified variants of sets of desirable gambles.

1.6.4 Simplified variants of desirability

The previous sections have made it clear that the complicated border structures coherent sets of desirable gambles can possess cannot be expected to be preserved when translating them to other uncertainty models. In this section, we are going to have a look at variants of desirability where these issues are simplified away.

The first variant comes as a pair, with one member of the pair corresponding to a strict partial preference order, and the second to a nonstrict one: A coherent set of almost desirable gambles c01-math-0593 must satisfy positive scaling, additivity, accepting sure gain, and avoiding sure loss—i.e., Criteria (1.1), (1.2), (1.6), and (1.7)—and moreover be closed [672, §§ 3.7.3–6]. So it is a closed convex cone including the positive orthant, the zero gamble, but no gamble in the uniformly negative orthant. A coherent set of surely desirable gambles c01-math-0594 must satisfy the same criteria and moreover be open. So it is an open convex cone containing the uniformly positive orthant and no gamble of the negative orthant. Both models are equally expressive, but respectively correspond to a nonstrict and strict partial preference order (with modified monotonicity conditions): almost preference c01-math-0595 and sure preference c01-math-0596 . In case one is given, the other can be defined in terms of it using the following correspondences: c01-math-0597 and c01-math-0598 , so their relationship is different from Equation (1.29).

Working with fully open or closed convex cones simplifies matters greatly. However, in order to be able to do so we have had to replace avoiding partial loss and accepting partial gains by their sure variants. In case one does not wish to do this and remain within the class of coherent sets of strictly and nonstrictly desirable gambles—i.e., those that satisfy Criteria (1.1), (1.2), (1.4), and (1.5)—the following variant can