When uncertainty is described by probabilities, decision making is usually done by maximising expected utility. Except in degenerate cases, this leads to a unique optimal decision. If, however, the probability measure is only partially specified—for example by lower and upper bounds on the probabilities of specific events—this method no longer works. Essentially, the problem is that two different probability measures that are both compatible with the given bounds may lead to different optimal decisions. In this context, several generalisations of maximising expected utility have been proposed; see for an nice overview.
A common feature of many such generalisations is that they yield set-valued choices: when presented with a set of options, they generally return a subset of them. If this turns out to be a singleton, then we have a unique optimal decision, as before. If, however, it contains multiple options, this means that they are incomparable and that our uncertainty model does not allow us to choose between them. Obtaining a single decision then requires a more informative uncertainty model, or perhaps a secondary decision criterion, as the information present in the uncertainty model does not allow us to single out an optimal option. Set-valued choice is also a typical feature of decision criteria based on other uncertainty models that generalise the probabilistic ones to allow for imprecision and indecision, such as lower previsions and sets of desirable gambles.
Choice functions provide an elegant unifying mathematical framework for studying such set-valued choice. They map option sets to option sets: for any given set of options, they return the corresponding set-valued choice. Hence, when working with choice functions, it is immaterial whether there is some underlying decision criterion. The primitive objects of this framework are simply the set-valued choices themselves, and the choice function that represents all these choices, serves as an uncertainty model in and by itself.
A major advantage of working with choice functions is that they allow us to impose axioms on choices, aimed at characterising what it means for choices to be rational and internally consistent; see for example the seminal work by Seidenfeld et al. . Here, we undertake a similar mission, yet approach it from a different angle. Rather than think of choice in an intuitive manner, we provide it with a concrete interpretation in terms of attitudes towards gambling, borrowing ideas from the theory of sets of desirable gambles [2, 8, 1, 3]. From this interpretation alone, and nothing more, we develop a theory of coherent choice that includes a full set of axioms, a representation in terms of sets of desirable gambles, and a natural extension theorem.
In order to facilitate the reading, proofs and intermediate results have been relegated to the Appendix.
2 Choice Functions
A choice function is a set-valued operator on sets of options. In particular, for any set of options , the corresponding value of is a subset of . The options themselves are typically actions amongst which a subject wishes to choose. As is customary in decision theory, every action has a corresponding reward that depends on the state of a variable , about which the subject is typically uncertain. Hence, the reward is uncertain too. The purpose of a choice function is to represent our subject’s choices between such uncertain rewards.
Let us make this more concrete.
First of all, the variable takes values in some set of states .
The reward that corresponds to a given option is then a function on .
We will assume that this reward can be expressed in terms of a real-valued linear utility scale, allowing us to identify every option with a real-valued function on .111 A more general approach, which takes options to be elements of an arbitrary vector space, encompasses the horse lottery approach, and was explored by Van Camp
A more general approach, which takes options to be elements of an arbitrary vector space, encompasses the horse lottery approach, and was explored by Van Camp. Our results here can be easily extended to this more general framework. We take these functions to be bounded and call them gambles. We use to denote the set of all such gambles and also let
Option sets can now be identified with subsets of , which we call gamble sets. We restrict our attention here to finite gamble sets and will use to denote the set of all such finite subsets of , including the empty set.
Definition 1 (Choice function)
A choice function is a map from to such that for every .
Gambles in that do not belong to are said to be rejected. This leads to an alternative representation in terms of so-called rejection functions.
Definition 2 (Rejection function)
The rejection function corresponding to a choice function is a map from to , defined by for all .
Since a choice function is completely determined by its rejection function, any interpretation for rejection functions automatically implies an interpretation for choice functions. This allows us to focus on the former.
Our interpretation for rejection functions now goes as follows. Consider a subject whose uncertainty about is represented by a rejection function , or equivalently, by a choice function . Then for a given gamble set , the statement that a gamble is rejected from —that is, that —is taken to mean that there is at least one gamble in that our subject strictly prefers over .
This interpretation is of course still meaningless, because we have not yet explained the meaning of strict preference. Fortunately, that problem has already been solved elsewhere: strict preference between elements of has an elegant interpretation in terms of desirability [8, 4], and it is this interpretation that we intend to borrow here. To allow us to do so, we first provide a brief introduction to the theory of sets of desirable gambles.
3 Sets of Desirable Gambles
A gamble is said to be desirable if our subject strictly prefers it over the zero gamble, meaning that rather than not gamble at all, she strictly prefers to commit to the gamble where, after the true value of the uncertain variable has been determined, she will receive the (possibly negative) reward .
A set of desirable gambles is then a subset of , whose interpretation will be that it consists of gambles that our subject considers desirable. The set of all sets of desirable gambles is denoted by . In order for a set of desirable gambles to represent a rational subject’s beliefs, it should satisfy a number of rationality, or coherence, criteria.
Axioms 1 and 2 follow immediately from the meaning of desirability: zero cannot be strictly preferred to itself, and any gamble that is never negative but sometimes positive should be strictly preferred to the zero gamble. Axiom 3 is implied by the assumed linearity of our utility scale.
Every coherent set of desirable gambles induces a binary preference order —a strict vector ordering—on , defined by , for all . The intuition behind this definition is that a subject strictly prefers the uncertain reward over if she strictly prefers trading for over not trading at all, or equivalently, if she strictly prefers the net uncertain reward over the zero gamble. The preference order fully characterises : one can easily see that if and only if . Hence, sets of desirable gambles are completely determined by binary strict preferences between gambles.
4 Sets of Desirable Gamble Sets
Let us now go back to our interpretation for choice functions, which is that a gamble in is rejected from if and only if there is some gamble in that our subject strictly prefers over . We will from now on interpret this preference in terms of desirability: we take it to mean that is desirable. In this way, we arrive at the following interpretation for a choice function . Consider any and , then
In other words, if we let , then according to our interpretation, the statement that is rejected from is taken to mean that contains at least one desirable gamble.
A crucial observation here is that this interpretation does not require our subject to specify a set of desirable gambles. Instead, all that is needed is for her to specify those gamble sets that to her contain at least one desirable gamble. We call such gamble sets desirable gamble sets and collect them in a set of desirable gamble sets . As can be seen from Equation (1), such a set of desirable gamble sets completely determines a choice function and its rejection function :
The study of choice functions can therefore be reduced to the study of sets of desirable gamble sets. We will from now on work directly with the latter. We will use the collective term choice models for choice functions, rejection functions, and sets of desirable gamble sets.
Let denote the set of all sets of desirable gamble sets , and consider any such . The first question to address is when to call coherent: which properties should we impose on a set of desirable gamble sets in order for it to reflect a rational subject’s beliefs? We propose the following axiomatisation, using as a shorthand notation for ‘, and ’.
Definition 4 (Coherence)
A set of desirable gamble sets is called coherent if it satisfies the following axioms:
, for all ;
, for all ;
if and if, for all and , , then
and , for all .
We denote the set of all coherent sets of desirable gamble sets by .
Since a desirable gamble set is by definition a set of gambles that contains at least one desirable gamble, Axioms 1 and 5 are immediate. The other three axioms follow from the principles of desirability that also lie at the basis of Axioms 1–3: the zero gamble is not desirable, the elements of are all desirable, and any finite positive linear combination of desirable gambles is again desirable. Axioms 2 and 3 follow naturally from the first two of these principles. The argument for Axiom 4 is more subtle; it goes as follows. Since and are two desirable gamble sets, there must be at least one desirable gamble and one desirable gamble . Since for these two gambles, the positive linear combination is again desirable, we know that at least one of the elements of is a desirable gamble. Hence, it must be a desirable gamble set.
5 The Binary Case
Because we interpret them in terms of desirability, one might be inclined to think that sets of desirable gamble sets are simply an alternative representation for sets of desirable gambles. However, this is not the case: we will see that sets of desirable gamble sets constitute a much more general uncertainty framework than sets of desirable gambles. What lies behind this added generality is that it need not be known which gambles are actually desirable. For example, within the framework of sets of desirable gamble sets, it is possible to express the belief that at least one of the gambles or is desirable while remaining undecided about which of them actually is; in order to express this belief, it suffices to state that . This is impossible within the framework of sets of desirable gambles.
Any set of desirable gamble sets determines a unique set of desirable gambles based on its binary choices only, given by
That choice models typically represent more than just binary choice is reflected in the fact that different can have the same . Nevertheless, there are sets of desirable gamble sets that are completely characterised by a set of desirable gambles, in the sense that there is a (necessarily unique) set of desirable gambles such that , with
It follows from the discussion at the end of Section 3 that such sets of desirable gamble sets are completely determined by binary preferences between gambles. We therefore call them, and their corresponding choice functions, binary. For any such binary set of desirable gamble sets , the unique set of desirable gambles such that is given by .
Consider any set of desirable gamble sets . Then is binary if and only if .
The coherence of a binary set of desirable gamble sets is completely determined by the coherence of its corresponding set of desirable gambles.
Consider any binary set of desirable gamble sets and let be its corresponding set of desirable gambles. Then is coherent if and only if is.
6 Representation in Terms of Sets of Desirable Gambles
That there are sets of desirable gamble sets that are completely determined by a set of desirable gambles is nice, but such binary choice models are typically not what we are interested in here, because then we could just as well use sets of desirable gambles to represent choice. It is the non-binary coherent choice models that we have in our sights here. But it turns out that our axioms lead to a representation result that allows us to still use sets of desirable gambles, or rather, sets of them, to completely characterise any coherent choice model.
Theorem 6.1 (Representation)
Every coherent set of desirable gamble sets is dominated by at least one binary set of desirable gamble sets: . Moreover, .
This powerful representation result allows us to incorporate a number of other axiomatisations  as special cases in a straightforward manner, because the binary models satisfy the required axioms, and these axioms are preserved under taking arbitrary non-empty intersections.
7 Natural Extension
In many practical situations, a subject will typically not specify a full-fledged coherent set of desirable gamble sets, but will only provide some partial assessment , consisting of a number of gamble sets for which she is comfortable about assessing that they contain at least one desirable gamble. We now want to extend this assessment to a coherent set of desirable gamble sets in a manner that is as conservative—or uninformative—as possible. This is the essence of conservative inference.
We say that a set of desirable gamble sets is less informative than (or rather, at most as informative as) a set of desirable gamble sets , when : a subject whose beliefs are represented by has more (or rather, at least as many) desirable gamble sets—sets of gambles that definitely contain a desirable gamble—than a subject with beliefs represented by . The resulting partially ordered set is a complete lattice with intersection as infimum and union as supremum. The following theorem, whose proof is trivial, identifies an interesting substructure.
Let be an arbitrary non-empty family of sets of desirable gamble sets, with intersection . If is coherent for all , then so is . This implies that is a complete meet-semilattice.
This result is important, as it allows us to a extend a partially specified set of desirable gamble sets to the most conservative coherent one that includes it. This leads to the conservative inference procedure we will call natural extension.
Definition 5 (Consistency and natural extension)
For any assessment , let . We call the assessment consistent if , and we then call the natural extension of .
In other words: an assessment is consistent if it can be extended to some coherent rejection function, and then its natural extension is the least informative such coherent rejection function.
Our final result provides a more ‘constructive’ expression for this natural extension and a simpler criterion for consistency. In order to state it, we need to introduce the set and two operators on—transformations of—. The first is denoted by , and defined by
so contains all gamble sets in , all versions of with some of their non-positive options removed, and all supersets of such sets. The second is denoted by , and defined for all by
where we used the notations and for -tuples of options and real numbers , , so and . We also used as a shorthand for ‘ for all and ’.
Theorem 7.2 (Natural extension)
Consider any assessment . Then is consistent if and only if and . Moreover, if is consistent, then .
Our representation result shows that binary choice is capable of representing general coherent choice functions, provided we extend its language with a ‘disjunction’ of desirability statements—as is implicit in our interpretation—, next to the ‘conjunction’ and ‘negation’ that are already implicit in the language of sets of desirable gambles—see  for a clear exposition of the latter claim.
In addition, we have found recently that by adding a convexity axiom, and working with more general vector spaces of options to allow for the incorporation of horse lotteries, our interpretation and corresponding axiomatisation allows for a representation in terms of lexicographic sets of desirable gambles , and therefore encompasses the one by Seidenfeld et al.  (without archimedeanity). We will report on these findings in more detail elsewhere.
Future work will address (i) dealing with the consequences of merging our accept-reject statement framework  with the choice function approach to decision making; (ii) discussing the implications of our axiomatisation and representation for conditioning, independence, and indifference (exchangeability); and (iii) expanding our natural extension results to deal with the computational and algorithmic aspects of conservative inference with coherent choice functions.
This work owes a large intellectual debt to Teddy Seidenfeld, who introduced us to the topic of choice functions. His insistence that we ought to pay more attention to non-binary choice if we wanted to take imprecise probabilities seriously, is what eventually led to this work.
The discussion in Arthur Van Camp’s PhD thesis  was the direct inspiration for our work here, and we would like to thank Arthur for providing a pair of strong shoulders to stand on.
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. We have both contributed equally to this paper. But since a paper must have a first author, we decided it should be the one who took the first significant steps: Jasper, in this case.
-  De Cooman, G., Quaeghebeur, E.: Exchangeability and sets of desirable gambles. International Journal of Approximate Reasoning 53(3), 363–395 (2012), special issue in honour of Henry E. Kyburg, Jr.
-  Couso, I., Moral, S.: Sets of desirable gambles: Conditioning, representation, and precise probabilities. International Journal Of Approximate Reasoning 52(7) (2011)
De Cooman, G., Miranda, E.: Irrelevance and independence for sets of desirable gambles. Journal of Artificial Intelligence Research 45, 601–640 (2012)
-  Quaeghebeur, E., De Cooman, G., Hermans, F.: Accept & reject statement-based uncertainty models. International Journal of Approximate Reasoning 57, 69–102 (2015)
-  Seidenfeld, T., Schervish, M.J., Kadane, J.B.: Coherent choice functions under uncertainty. Synthese 172(1), 157–176 (2010)
-  Troffaes, M.C.M.: Decision making under uncertainty using imprecise probabilities. International Journal of Approximate Reasoning 45(1), 17–29 (2007)
-  Van Camp, A.: Choice Functions as a Tool to Model Uncertainty. Ph.D. thesis, Ghent University, Faculty of Engineering and Architecture (January 2018)
-  Walley, P.: Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24, 125–148 (2000)
Appendix 0.A Proofs and intermediate results
In this appendix, besides the operators that were introduced in the main text, we also require two additional ones:
Applying them in sequence has the same effect as applying the operator .
Consider any set of desirable gamble sets . Then
Consider any , which means that there is some such that . Then . Hence, if we let , then . Since and implies that , this allows us to conclude that .
Conversely, consider any , which means that there is some such that . Then since , there is some such that . Hence, we find that , which, since , implies that . ∎
Proof (Proposition 1)
If , then is trivially binary. So let us assume that is binary. We will prove that . Since is binary, there is a set of desirable gambles such that . For any , this implies that
Hence, we find that , which indeed implies that . ∎
Consider any coherent set of desirable gambles . Then is a coherent set of desirable gamble sets.
For Axiom 1, observe that Equation (2) trivially implies that . For Axiom 2, observe that implies that because we know from the coherence of [Axiom 1] that . For Axiom 3, observe that is equivalent to for all , and take into account the coherence of [Axiom 2]. For Axiom 4, consider any , and let for any particular choice of the for all and . Then and , so we can fix any and . The coherence of [Axiom 3] then implies that , and therefore also , whence indeed . And, finally, that satisfies Axiom 5 is an immediate consequence of its definition (2). ∎
Consider any coherent set of desirable gamble sets . Then is a coherent set of desirable gambles, and .
We first prove that is coherent, or equivalently, that it satisfies Axioms 1–3. For Axiom 1, observe that implies that . Since satisfies Axiom 2, this implies that , contradicting Axiom 1. For Axiom 2, observe that for any , is equivalent to , and take into account Axiom 3. And, finally, for Axiom 3, observe that implies that , and that Axiom 4 then implies that , or equivalently, that , for any choice of .
For the last statement, consider any , meaning that . Consider any , then on the one hand , so . But since on the other hand also , we see that , and therefore Axiom 5 guarantees that . ∎
Proof (Proposition 2)
Next, suppose that is coherent. Lemma 3 then implies that is coherent as well. ∎
We will call a coherent set of desirable gamble sets maximal, if it is not dominated by any other coherent set of desirable gamble sets, and we collect all maximal coherent sets of desirable gamble sets in the set : for any ,
Any coherent set of desirable gamble sets is dominated by some maximal coherent set of desirable gamble sets: .
It is clearly enough to establish that the partially ordered set has a maximal element, and we use Zorn’s Lemma to that effect. So consider any chain in , then we must prove that has an upper bound in . Since is clearly an upper bound, we are done if we can prove that is coherent.
For Axiom 1, simply observe that since belongs to no element of [since they are all coherent], it cannot belong to their union .
For Axiom 2, consider any . Then there is some such that , and since is coherent, this implies that .
For Axiom 3, consider any , then we know that [since is coherent], and therefore also , since .
For Axiom 4, consider any and, for all and , choose . Since , we know that there are such that and . Since is a chain, we can assume without loss of generality that , and therefore . Since is coherent, it follows that .
And finally, for Axiom 5, consider any and any such that . Then we know that there is some such that . Since is coherent, this implies that also . ∎
A coherent set of desirable gamble sets is binary if and only if
First assume that is binary. We then know from Proposition 1 that , implying that , for all . Consider any such that . Then there is some such that . But then , so we can consider an element . Since clearly , we see that and therefore, that .
Next assume that Equation (3) holds. Because of Proposition 1, it suffices to show that . We infer from Lemma 3 that is a coherent set of desirable gambles, and that . Assume ex absurdo that , so there is some such that , or equivalently, such that . But then we must have that , because otherwise with and therefore , a contradiction. But then it follows from Equation (3) that there is some such that . Since it follows from that also , we see that also . We can now repeat the same argument with instead of to find that it must be that , so there is some such that and . Repeating the same argument over and over again will eventually lead to a contradiction with . Hence it must be that . ∎
Immediate consequence of Lemma 5. ∎
Consider any coherent set of desirable gamble sets , then .
That is an immediate consequence of the definition of the operator. To prove that , consider any , which means that there is some such that . We need to prove that . Since satisfies Axiom 5, it suffices to prove that .
If , then . Therefore, without loss of generality, we may assume that . For any , Lemma 6 implies that we may replace by and still be guaranteed that the resulting set belongs to . Hence, we can replace all elements of with and still be guaranteed that the result belongs to . Applying Axiom 2 now guarantees that, indeed, . ∎
To prove that satisfies Axiom 1 and does not contain , assume ex absurdo that or . We then find that there is some such that either or . In both cases, it follows that . If , this contradicts our assumption that satisfies Axiom 1. If , it follows from Lemma 6 that we can replace every by and still be guaranteed that the resulting gamble set belongs to , again contradicting our assumptions.
To prove that satisfies Axiom 3, simply observe that the operator never removes gamble sets from a set of desirable gamble sets, so the gamble sets , , which belong to by Axiom 3, will also belong to the larger .
To prove that satisfies Axiom 4, consider any , meaning that there are such that and . For any and , we choose , and let
Then we have to prove that . Since satisfies Axiom 4, we infer from that
belongs to as well. Furthermore, since and imply that and , we see that
Hence, . Since , this implies that, indeed, .
Finally, to prove that satisfies Axiom 5, consider any and any such that . We need to prove that . That implies that there is some such that . Let , then and therefore also , because satisfies Axiom 5. We now infer from that
Since , this allows us to conclude that , and therefore, since , that, indeed, . ∎
Consider a coherent set of desirable gamble sets and any such that and for all . Choose any and let
Then is a coherent set of desirable gamble sets that is a superset of and contains , and furthermore and .
To prove that , assume ex absurdo that . Since , we can pick any element , and then and therefore by Axiom 5, contradicting the assumptions. To prove that , assume ex absurdo that , then we infer that also [use Proposition 3 and the coherence of ], contradicting the assumptions. To prove that , it suffices to notice that , whence also . Similarly, since is clearly a superset of , the same is true for .
It only remains to prove, therefore, that is coherent. To this end, we intend to show that the set of desirable gamble sets satisfies Axioms 1, 3, 4 and 5 and that . The coherence of will then be an immediate consequence of Proposition 4.
For Axiom 5, consider any and any such that , then we must prove that also . Since , we know that there is some and, for all , some choice of , such that
For every , we now choose some real such that and such that, for all , . Since and , and are finite, this is clearly always possible. Let
and, for each , let be the unique element of for which , and let . We then see that
Furthermore, since and , it follows from the coherence of and Axiom 5 that . Hence, indeed, .
For Axiom 4, consider any and, for all and , any choice of . Then we must prove that
Since , there are and, for all and , some choices of and , such that
Now fix any and , and let , with and . Then
We consider two cases. If , we let