Often, we find ourselves in a situation where we have to make some decision , which we may freely choose from a set of available decisions. Usually, we do not choose arbitrarily in : indeed, we wish to make a decision that performs best according to some criterion, i.e., an optimal decision. It is commonly assumed that each decision induces a real-valued gain : in that case, a decision is considered optimal in if it induces the highest gain among all decisions in . This holds for instance if each decision induces a lottery over some set of rewards, and these lotteries form an ordered set satisfying the axioms of von Neumann Morgenstern , or more generally, the axioms of for instance Herstein and Milnor , if we wish to account for unbounded gain.
So, we wish to identify the set of all decisions that induce the highest gain. Since, at this stage, there is no uncertainty regarding the gains , , the solution is simply
Of course, may be empty; however, if the set is a compact subset of —this holds for instance if is finite—then contains at least one element. Secondly, note that even if contains more than one decision, all decisions in induce the same gain ; so, if, in the end, the gain is all that matters, it suffices to identify only one decision in —often, this greatly simplifies the analysis.
However, in many situations, the gains induced by decisions in
are influenced by variables whose values are uncertain. Assuming that these variables can be modelled through a random variablethat takes values in some set (the possibility space), it is customary to consider the gain as a so-called gamble on , that is, we view as a real-valued gain that is a bounded function of , and that is expressed in a fixed state-independent utility scale. So, is a bounded –-mapping, interpreted as an uncertain gain: taking decision , we receive an amount of utility when turns out to be the realisation of . For the sake of simplicity, we shall assume that the outcome of is independent of the decision we take: this is called act-state independence. What decision should we take?
Irrespective of our beliefs about , a decision in is not optimal if its gain gamble is point-wise dominated by a gain gamble for some in , i.e., if there is an in such that for all and for at least one : choosing guarantees a higher gain than choosing , possibly strictly higher, regardless of the realisation of . So, as a first selection, let us remove all decisions from whose gain gambles are point-wise dominated (see Berger [3, Section 1.3.2, Definition 5 ff., p. 10]):
where is understood to be point-wise, and is understood to be the negation of . The decisions in are called admissible, the other decisions in are called inadmissible. Note that we already recover Eq. (1) if there is no uncertainty regarding the gains , i.e., if all are constant functions of . When do admissible decisions exist? The set is non-empty if is a non-empty and weakly compact subset of the set of all gambles on (see Theorem 3 further on). Note that this condition is sufficient, but not necessary.
In what follows, we shall try to answer the following question: given additional information about , how can we further reduce the set of admissible decisions? The paper is structured as follows. Section 2 discusses the classical approach of maximising expected utility, and explains why it is not always a desirable criterion for selecting optimal decisions. Those problems are addressed in Section 3, discussing alternative approaches to deal with uncertainty and optimality, all of which attempt to overcome the issues raised in Section 2, and all of which are known from the literature. Finally, Section 4 compares these alternative approaches, and explains how optimal decisions can be obtained in a computationally efficient way. A few technical results are deferred to the appendix, where we, among other things, generalize a well-known technical condition on the existence of optimal decisions.
2. Maximising Expected Utility?
In practice, beliefs about are often modelled by a (possibly finitely additive) probability measure on a field of subsets of , and one then arrives at a set of optimal decisions by maximising their expected utility with respect to ; see for instance Raiffa and Schlaifer [4, Section 1.1.4, p. 6], Levi [5, Section 4.8, p. 96, ll. 23–26], or Berger [3, Section 1.5.2, Paragraph I, p. 17]. Assuming that the field is sufficiently large such that the gains are measurable with respect to —this means that every is a uniform limit of -simple gambles—the expected utility of the gain gambles is given by:
where we take for instance the Dunford integral on the right hand side; see Dunford [6, p. 443, Sect. 3], and Dunford and Schwartz [7, Part I, Chapter III, Definition 2.17, p. 112]—this linear integral extends the usual textbook integral (see for instance Kallenberg [8, Chapter 1]) to case where is not -additive. Recall that we have assumed act-state independence: is independent of .
As far as it makes sense to rank decisions according to the expected utility of their gain gambles, we should maximise expected utility:
When do optimal solutions exist? The set is guaranteed to be non-empty if is a non-empty and compact subset of the set of all gambles on , with respect to the supremum norm. Actually, this technical condition is sufficient for existence with regard to all of the optimality conditions we shall discuss further on. Therefore, without further ado, we shall assume that is non-empty and compact with respect to the supremum norm. A slightly weaker condition is assumed in Theorem 5, in the appendix of this paper.
Unfortunately, it may happen that our beliefs about cannot be modelled by a probability measure, simply because we have insufficient information to identify the probability of every event in . In such a situation, maximising expected utility usually fails to give an adequate representation of optimality.
For example, let be the unknown outcome of the tossing of a coin; say we only know that the outcome will be either heads or tails (so ), and that the probability of heads lays between and . Consider the decision set and the gain gambles
Clearly, , and
Concluding, if we have no additional information about , but still insist on using a particular (and necessarily arbitrary) , which is only required to satisfy , we find that is not very robust against changes in . This shows that maximising expected utility fails to give an adequate representation of optimality in case of ignorance about the precise value of .
3. Generalising to Imprecise Probabilities
Of course, if we have sufficient information such that can be identified, nothing is wrong with Eq. (3). We shall therefore try to generalise Eq. (3). In doing so, following Walley , we shall assume that our beliefs about are modelled by a real-valued mapping defined on a—possibly only very small—set of gambles, that represents our assessment of the lower expected utility for each gamble in ;111The upper expected utility of a gamble is if and only if the lower expected utility of is . So, for any gamble in , , and therefore, without loss of generality, we can restrict ourselves to lower expected utility. note that can be chosen empty if we are completely ignorant. Essentially, this means that instead of a single probability measure on , we now identify a closed convex set of finitely additive probability measures on , described by the linear inequalities
We choose the domain of the measures sufficiently large such that all gambles of interest, in particular those in and the gain gambles , are measurable with respect to . Without loss of generality, we can assume to be the power set of , although in practice, it may be more convenient to choose a smaller field.
For a given -measurable gamble , not necessarily in , we may also derive a lower expected utility by minimising subject to the above constraints, and an upper expected utility by maximising over the above constraints. In case and
are finite, this simply amounts to solving a linear program.
In the literature, is called a credal set (see for instance Giron and Rios , and Levi [5, Section 4.2, pp. 76–78], for more comments on this model), and is called a lower prevision (because they generalise the previsions, which are fair prices, of De Finetti [11, Vol. I, Section 3.1, pp. 69–75]).
The mapping obtained, corresponds exactly to the so-called natural extension of (to the set of -measurable gambles), where is interpreted as a supremum buying price for (see Walley [9, Section 3.4.1, p. 136]). In this interpretation, for any , we are willing to pay any utility prior to observation of , if we are guaranteed to receive once turns out to be the outcome of . The natural extension then corresponds to the highest price we can obtain for an arbitrary gamble , taken into account the assessed prices for . Specifically,
where varies over , over , , …, vary over , and , …, over .
It may happen that is empty, in which case is undefined (the supremum in Eq. (5) will always be ). This occurs exactly when incurs a sure loss as a lower prevision, that is, if we can find a finite collection of gambles , …, in such that , which means that we are willing to pay more for this collection than we can ever gain from it, which makes no sense of course.
Finally, it may happen that does not coincide with on . This points to a form of incoherence in : this situation occurs exactly when we can find a finite collection of gambles , , …, and non-negative real numbers , …, , such that
This means that we can construct a price for , using the assessed prices for , which is strictly higher than . In this sense, corrects , as is apparent from Eq. (5).
Although the belief model described above is not the most general we may think of, it is sufficiently general to model both expected utility and complete ignorance: these two extremes are obtained by taking either equal to a singleton, or equal to the set of all finitely additive probability measures on (i.e., ). It also allows us to demonstrate the differences between different ways to make decisions with imprecise probabilities on the example we presented before.
In that example, the given information can be modelled by, say, a lower prevision on , defined by and , where is the gamble defined by and . For this , the set corresponds exactly to the set of all probability measures on , such that . We also easily find for any gamble on that
3.1. -Maximin and -Maximax
As a very simple way to generalise Eq. (3), we could take the lower expected utility as a replacement for the expected utility (see for instance Gilboa and Schmeidler , or Berger [3, Section 4.7.6, pp. 215–223]):
this criterion is called -maximin, and amounts to worst-case optimisation: we take a decision that maximises the worst expected gain. For example, if we consider the decision as a game against nature, who is assumed to choose a distribution in aimed at minimizing our expected gain, then the -maximin solution is the best we can do. Applied on the example of Section 2, we find as a solution .
In case , i.e., in case of complete ignorance about , it holds that . Hence, in that case, -maximin coincides with maximin (see Berger [3, Eq. (4.96), p. 216]), ranking decisions by the minimal (or infimum, to be more precise) value of their gain gambles.
Some authors consider best-case optimisation, taking a decision that maximises the best expected gain (see for instance Satia and Lave ). In our example, the “-maximax” solution is .
Eq. (3) is essentially the result of pair-wise preferences based on expected utility: defining the strict partial order on as whenever , or equivalently, whenever , we can simply write
where the operator selects the -maximal, i.e., the -undominated elements from a set with strict partial order .
Using the supremum buying price interpretation, it is easy to derive pair-wise preferences from : define as whenever . Indeed, means that we are disposed to pay a strictly positive price in order to take decision instead of , which clearly indicates strict preference of over (see Walley [9, Sections 3.9.1–3.9.3, pp. 160–162]). Since is a strict partial order, we arrive at
Note that Walley [9, Sections 3.9.2, p. 161] has a slightly different definition: instead of working from the set of admissible decisions as in Eq. (7), Walley starts with ranking if or ( and ), and then selects those decisions from that are maximal with respect to this strict partial order. Using Theorem 3 from the appendix, it is easy to show that Walley’s definition of maximality coincides with the one given in Eq. (7) whenever the set is weakly compact. This is something we usually assume to ensure the existence of admissible elements; in particular, weak compactness is assumed in Theorem 5 (see appendix). The benefit of Eq. (7) over Walley’s definition is that Eq. (7) is easier to manage in the proofs in the appendix.
3.3. Interval Dominance
Another robustification of is the strict partial ordering defined by whenever ; this means that the interval is completely on the right hand side of the interval . The above ordering is therefore called interval dominance (see Zaffalon, Wesnes, and Petrini [14, Section 2.3.3, pp. 68–69] for a brief discussion and references).
The resulting notion is weaker than maximality: applied on the example of Section 2, , which is strictly larger than .
In the example of Section 2, we have shown that may not be very robust against changes in . Robustifying against changes of in , we arrive at
this provides another way to generalise Eq. (3). The above criterion selects those admissible decisions in that maximize expected utility with respect to at least one in ; i.e., they select the E-admissible (see Good [15, p. 114, ll. 8–9], or Levi [5, Section 4.8, p. 96, ll. 8–20]) decisions among the admissible ones. We find for the example.
In case is defined on and for all , then every E-admissible decision is also admissible, and hence, in that case, gives us exactly the set of E-admissible options.
4. Which Is the Right One?
Evidently, it is hard to pinpoint the right choice. Instead, let us ask ourselves: what properties do we want our notion of optimality to satisfy? Let us summarise a few important guidelines.
Clearly, whatever notion of optimality, it seems reasonable to exclude inadmissible decisions. For ease of exposition, let’s assume that the inadmissible decisions have already been removed from , i.e., ; this implies in particular that gives us the set of E-admissible decisions.
Now note that, in general, the following implications hold:
as is also demonstrated by our example. A proof is given in the appendix, Theorem 1.
E-admissibility, maximality, and interval dominance have the nice property that the more determinate our beliefs (i.e., the smaller ), the smaller the set of optimal decisions. In contradistinction, -maximin and -maximax lack this property, and usually only select a single decision, even in case of complete ignorance. However, if we are only interested in the most pessimistic (or most optimistic) solution, disregarding other reasonable solutions, then -maximin (or -maximax) seems appropriate. Utkin and Augustin  have collected a number of nice algorithms for finding -maximin and -maximax solutions, and even mixtures of these two. Seidenfeld  has compared -maximin to E-admissibility, and argued against -maximin in sequential decision problems.
If we do not settle for -maximin (or -maximax), should we choose E-admissibility, maximality, or interval dominance? As already mentioned, interval dominance is weaker than maximality, so in general we will end up with a larger (and arguably too large) set of optimal options. Assuming the non-admissible decisions have been weeded, a decision is not optimal in with respect to interval dominance if and only if
Thus, if has elements, interval dominance requires us to calculate natural extensions, and make comparisons, whereas for maximality, by Eq. (7), we must calculate natural extensions, and perform comparisons—roughly speaking, each natural extension is a linear program in (size of ) variables and (size of ) constraints, or vice versa if we solve the dual program. So, comparing maximality and interval dominance, we face a tradeoff between computational speed and number of optimal options.
However, this also means that interval dominance is a means to speed up the calculation of maximal and E-admissible decisions: because every maximal decision is also interval dominant, we can invoke interval dominance as a first computationally efficient step in eliminating non-optimal decisions, if we eventually opt for maximality or E-admissibility. Indeed, eliminating those decisions that satisfy Eq. (10), we will also eliminate those decisions that are neither maximal, nor E-admissible.
Regarding sequential decision problems, we note that dynamic programming techniques cannot be used when using interval dominance (see De Cooman and Troffaes ), and therefore, since dynamic programming yields an exponential speedup, maximality and E-admissibility are certainly preferred over interval dominance once dynamics enter the picture.
This leaves E-admissibility and maximality. They are quite similar: they coincide on all decision sets that contain two decisions. In case we consider larger decision sets, they coincide if the set of gain gambles is convex (for instance, if we consider randomised decisions). As already mentioned, E-admissibility is stronger than maximality, and also has some other advantages over maximality. For instance, , so, choosing decision with probability and decision with probability is preferred to decision . Therefore, we should perhaps not consider decision as optimal.
E-admissibility is not vulnerable to such argument, since no E-admissible decision can be dominated by randomized decisions: if for some it holds that for all , then also
for any convex combination of gain gambles, and hence, it also holds that
which means that no convex combination can dominate with respect to .
A powerful algorithm for calculating E-admissible options has been recently suggested by Utkin and Augustin [16, pp. 356–357], and independently by Kikuti, Cozman, and de Campos [19, Sec. 3.4]. If has elements, finding all (pure) E-admissible options requires us to solve linear programs in variables and constraints.
As we already noted, through convexification of the decision set, maximality and E-admissibility coincide. Utkin and Augustin’s algorithm can also cope with this case, but now one has to consider in the worst case linear programs, and usually several less: the worst case only obtains if all options are E-admissible. For instance, if there are only E-admissible pure options, one has to consider only at most of those linear programs, and again, usually less.
In conclusion, the decision criterion to settle for in a particular application, depends at least on the goals of the decision maker (what properties should optimality satisfy?), and possibly also on the size and structure of the problem if computational issues arise.
I especially want to thank Teddy Seidenfeld for the many instructive discussions about maximality versus E-admissibility. I also wish to thank two anonymous referees for their helpful comments. This paper has been supported by the Belgian American Educational Foundation. The scientific responsibility rests with its author.
Appendix A Proofs
This appendix is dedicated to proving the connections between the various optimality criteria, and existence results mentioned throughout the paper. In the whole appendix, we assume the following:
Recall, denotes some set of decisions, and every decision induces a gain gamble , where is the set of all gambles (bounded – mappings).
denotes a lower prevision, defined on a subset of . With we denote a field on such that all gain gambles and gambles in are measurable with respect to , i.e., are a uniform limit of -simple gambles. could be for instance the power set of .
is assumed to avoid sure loss, and is its natural extension to the set of all -measurable gambles. is the credal set representing , as defined in Section 3. We will make deliberate use of the properties of natural extension (for instance, superadditivity: , and hence also ). We refer to Walley [9, Sec. 2.6, p. 76, and Sec. 3.1.2, p. 123] for an overview and proof of these properties.
We use the symbol for an arbitrary finitely additive probability measure on , and denotes the Dunford integral with respect to . This integral is defined on (at least) the set of all -measurable gambles.
a.1. Connections between Decision Criteria
The following relations hold.
Suppose that is -maximax in : maximises in . Since is the upper envelope of , and is weak-* compact (see Walley [9, Sec. 3.6]), there is a in such that . But, , for every because is -maximax. Thus, belongs to .
Suppose that is maximal. Then, again by Eq. (7), for all in . But, , hence, also for all in , which means that belongs to .
Finally, suppose that is -maximin: maximises in . But then for all in ; must be maximal. ∎
We first prove a technical but very useful lemma about the existence of optimal elements with respect to preorders; it’s an abstraction of a result proved by De Cooman and Troffaes . Let’s start with a few definitions.
A preorder is simply a reflexive and transitive relation.
Let be any set, and let be any preorder on . An element of a subset of is called -maximal in if, for all in , implies . The set of -maximal elements is denoted by
For any in , we also define the up-set of relative to as
Let be a Hausdorff topological space. Let be any preorder on such that for any in , the set is closed. Then, for any non-empty compact subset of , the following statements hold.
For every in , the set is non-empty and compact.
The set of -maximal elements of is non-empty.
For every in , there is a -maximal element of such that .
(i). Since is reflexive, it follows that , so is non-empty. Is it compact? Clearly, , so is the intersection of a compact set and a closed set, and therefore must be compact too.
(ii). Let be any subset of the non-empty compact set that is linearly ordered with respect to . If we can show that has an upper bound in with respect to , then we can infer from a version of Zorn’s lemma [20, (AC7), p. 144] (which also holds for preorders) that has a -maximal element. Let then be an arbitrary finite subset of . We can assume without loss of generality that , and consequently . This implies that the intersection of these up-sets is non-empty: the collection of compact and hence closed ( is Hausdorff) subsets of has the finite intersection property. Consequently, since is compact, the intersection is non-empty as well, and this is the set of upper bounds of in with respect to . So, by Zorn’s lemma, has a -maximal element: is non-empty.
(iii). Combine (i) and (ii) to show that the non-empty compact set has a maximal element with respect to . It is then a trivial step to prove that is also -maximal in : we must show that for any in , if , then . But, if , then also since by construction. Hence, , and since is -maximal in , it follows that . ∎
The weak topology on is simply the topology of point-wise convergence. That is, a net in converges weakly to in if for all .
If is a non-empty and weakly compact set, then contains at least one admissible decision, and even more, for every decision in , there is an admissible decision in such that .
It is easy to derive from Eq. (2) that
Hence, a decision is admissible in exactly when its gain gamble is -maximal in . We must show that has -maximal elements.
By Lemma 2, it suffices to prove that, for every , the set is closed with respect to the topology of point-wise convergence.
Let be a net in , and suppose that converges point-wise to : for every , . But, since for every , it must also hold that . Hence, . We have shown that every converging net in converges to a point in . Thus, is closed. This establishes the theorem. ∎
Let’s now introduce a slightly stronger topology on . This topology has no particular name in the literature, so let’s just call it the -topology. It is determined by the following convergence.
Say that a net in -converges to in , if
for all (point-wise convergence), and
(convergence in -norm).
This convergence induces a topology on : it turns
into a locally convex topological vector space, which also happens to be Hausdorff. A topological basis atconsists for instance of the convex sets
for , and for all . It has more open sets and more closed sets than the weak topology, but it has less compact sets than the weak topology. On the other hand, this topology is weaker than the supremum norm topology, so it has fewer open and closed sets, and more compact sets, compared to the supremum norm topology. Note that in case is finite, it reduces to the weak topology, which is in that case also equivalent to the supremum norm topology.
Note that , , and for all , are -continuous, simply because
If is non-empty and compact with respect to the -topology, then the following statements hold.
is non-empty for all .
(i). Introduce the following order on : say that whenever . Let’s first show that, for all , the set is -closed.
Let be a net in , and suppose that -converges to . Since the integral is -continuous, it follows that . Concluding, belongs to . We have established that every converging net in converges to a point in . Thus, is -closed.
By Lemma 2, it follows that has at least one -maximal element , that is, maximises in . Since any -compact set is also weakly compact, there is a -maximal element in such that , by Theorem 3. But then, , and hence, also maximises in . Because is -maximal in , it also maximises in . This establishes that belongs to : this set is non-empty.
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