Decision making under interval uncertainty: toward (somewhat) more convincing justi ﬁ cations for Hurwicz optimism-pessimism approach

Purpose – In real life, we only know the consequences of each possible action with some uncertainty. A typical example is interval uncertainty, when we only know the lower and upperbounds on the expected gain. A usual way to compare such interval-valued alternatives is to use the optimism – pessimism criterion developed by Nobelist Leo Hurwicz. In this approach, a weighted combination of the worst-case and the best-casegains is maximized. There existseveral justi ﬁ cations for this criterion;however,some oftheassumptions behind these justi ﬁ cations are not 100% convincing. The purpose of this paper is to ﬁ nd a more convincing explanation. Design/methodology/approach – The authorsused utility approach to decision-making. Findings – The authors proposednew, hopefullymore convincing,justi ﬁ cations for Hurwicz ’ s approach. Originality/value – This is a new, more intuitive explanation of Hurwicz ’ s approach to decision-making underinterval uncertainty.

1. Formulation of the problem Need to make decisions under interval uncertainty. In many real-life situations, we need to make a decision, i.e. we need to select one of the possible alternatives. For example, we want to select the best investment strategy, we need to decide whether to accept a new job offer, etc.
In the ideal world, we should know the exact consequence of each possible alternative. In such an ideal case, we select an alternative which is the best for us. For example, if the goal of the investment is to save for retirement, then we should select the investment strategy that will bring us the larger amount of money by the expected retirement date.
In real world, there is uncertainty. We can rarely predict the exact consequences of each action. In the simplest case, instead of knowing the exact amount of money m resulting from each alternative, we only know that this amount will be somewhere between the values m and m. In other words, we do not know the exact value m, but instead we only know the interval m; m ½ that contains the actual (not yet known) value m. Such a situation is known as the situation of interval uncertainty. If we know intervals corresponding to different alternatives, which alternative should we select?
In other cases, in addition to the bounds m and m, we also have some information about which values from the corresponding interval are more probable and which are less probable. In other words, we have some informationusually partialabout the actual probability distribution on the interval m; m ½ . Sometimes, we know the exact probability distribution. In this case, we can, e.g., select the alternative for which the expected gain is the largestof, if we want to be cautious, e.g. the alternative for which the gain guaranteed with a certain probability (e.g. 80%) is the largest.
In practice, we rarely know the exact probability distribution. Even if we know that the distribution is, e.g. Gaussian, we still do not know the exact values of the corresponding parametersfrom the observations, we can only determine parameters with some uncertainty. For different possible combinations of these parameters, the expected gainor whatever else characteristic we usemay take different values. Thus, for each alternative, instead of the exact value m of the corresponding objective function (such as expected gain), we have a whole interval m; m ½ of possible values of this objective function. So, we face the exact same problem as in the simplest possible casewe need to select an alternative in a situation when for each alternative, we only know the interval of possible values of the objective function.
How decisions under interval uncertainty are currently made. As we have mentioned earlier, decision-making under interval uncertainty is an important practical problem. Not surprisingly, methods for solving this problem have been known for many decades. Usually, practitioners use a solution proposed in the early 1950s by the future Nobelist Leo Hurwicz; see, e.g. Hurwicz (1951), Luce and Raiffa (1989), Kreinovich (2014). According to this solution, a decision-maker should: first, select a parameter a from the interval [0,1]; and then select an alternative for which the following combination attains the largest possible value: This idea is known as the optimism-pessimism criterion, and the selected value a is known as the optimism parameter. The reason for these terms is straightforward: Hurwicz optimismpessimism approach If a = 1, this means that the decision-maker simply selects the alternative with the largest possible value of m. In other words, the decision-maker completely ignores the possibility that the outcome of each alternative can be smaller than in the best possible case and bases his/her decision exclusively on comparing these best possible consequences of different actions. This is clearly an extreme case of an optimist. Vice versa, if a = 0, this means that the decision-maker simply selects the alternative with the largest possible value of m. In other words, the decision-maker completely ignores the possibility that the outcome of each alternative can be better that in the worst possible case and bases his/her decision exclusively on comparing these worst possible consequences of different actions. This is clearly an extreme case of a pessimist.
Both these situations are extreme. In real life, most people take into account both good and bad possibilities, i.e. in Hurwicz terms, they make decisions based on some intermediate value awhich is larger than the pessimist's 0 but smaller than the optimist's 1.
How can we explain the current approach to decision-making under uncertainty. There exist reasonable explanations for Hurwicz criteria, both: for the case when the outcome of each alternative is simply monetary; and for the case when the outcome is not monetaryin this case, decision theory helps us describe the user's preferences in terms of special values known as utilities; see, e.g. Fishburn (1969), Luce and Raiffa (1989), Raiffa (1997), Nguyen et al. (2009), Kreinovich (2014 and references therein.
Remaining problem and what we do in this paper. In both monetary and utility cases, to derive Hurwicz's formula, we need to make certain assumptions: (1) some of these assumptions are more reasonable; and (2) some of these assumptions are slightly less convincing.
Natural questions are as follows: Do we need these somewhat less convincing assumptions? Can we avoid them altogetherand, if not, can we replace them with somewhat more convincing assumptions?
These are the question that we will analyzeand answerin this paper. Structure of this paper. We will start this paper with the easier-to-describe and easierto-analyze case of monetary alternatives. First, in Section 2, we describe the usual assumptions leading to the Hurwicz criterion, explain how the Hurwicz criterion can be derived from these assumptions (in this, we largely follow (Kreinovich, 2017)) and why some of these assumptions may not sound fully convincing. Then, in Section 3, we present newhopefully more convincingassumptions and show how Hurwicz criterion can be derived from the new assumptions.
Then, we deal with the utility case. In Section 4, we briefly remind the readers who are not familiar with all the technical details of decision theory, what is utility and what are the properties of utility. In Section 5, we describe the usual assumptions leading to Hurwicz criterion for the utility case (they are somewhat different from the monetary case), explain how Hurwicz criterion can be derived from these assumptions (in this, we also largely follow (Kreinovich, 2017)) and why some of these assumptions may not sound fully convincing.
Finally, in Section 6, we show that the Hurwicz criterion can be derived from the (hopefully) more convincing assumptions in the utility case as well.
2. Monetary case: usual derivation of the Hurwicz criterion and the limitations of this derivation To make a decision, we need to have an exact numerical equivalent for each interval. We want to be able to compare different alternatives with interval uncertainty. In particular, for each interval-valued alternative m; m ½ and for each alternative with a known exact monetary value m, we need to be able to decide the following: whether the exact-valued alternative is better; or whether the interval-valued alternative is better.
Of course, if m < m, then no matter what is the actual value from the interval m; m ½ , this value will be larger than m. Thus, in this case, the interval alternative is clearly better. We will denote this by m < m; m ½ . Similarly, if m > m, then no matter what is the actual value from the interval m; m ½ , this value will be smaller than m. Thus, in this case, the interval alternative is clearly worse: One can show that because of this, there is a threshold value separating the two cases, namely, the value.
Let us denote this threshold valuedepending on m and mby f m; m ð Þ. By definition, for every « > 0, we have: In particular, this property holds for an arbitrarily small « , including such small « that no one will notice the difference between the value m and the values m -« and m þ «. So, from the practical viewpoint, we can say that the interval m; m ½ is equivalent to the monetary value f m; m ð Þ. We will denote this equivalence by following: From this viewpoint, all we need to do to describe decision-making under interval uncertainty is to describe the corresponding function f m; m ð Þ. The numerical value f m; m ð Þshould always be between m and m. As we have mentioned earlier, for every value m < m, we have m < m; m ½ , the set fm : m < m; m ½ g contains all the values from the set À1; m ð Þ . Thus, its supremum f m; m ð Þhas to be greater than or equal to all the values m < m, in particular, than all the values m ¼ m À 1=n. So, we must have: for all n. In the limit n ! 1, we conclude that m # f m; m ð Þ.

Hurwicz optimismpessimism approach
Similarly, as for every value m > m, we have m; m ½ > m, the set: contains all the values from the set m; 1 ð Þ. Thus, its infimum f m; m ð Þhas to be smaller than or equal to all the values m > min particular, than all the values m ¼ m þ 1=n. So, we must have f m; m ð Þ< m þ 1=n for all m. In the limit n ! 1, we conclude that f m; m ð Þ# m. Based on the two previous examples, we should always have m # f m; m ð Þ# m. Let us prepare for the usual derivation of Hurwicz criterion. To explain the usual derivation of Hurwicz criterion from several assumptions, let us first provide the usual motivation for these assumptions.
Monotonicity. Let us assume that we start with an interval m; m ½ , and then we: delete all the lowest-value options, i.e. options for which m # m 0 for some m 0 > m; and/or add several higher-value options, with m > m, e.g. all the values from m to some larger value m 0 > m.

After this, we get a clearly better interval
Additivity. Suppose that we have two situations: (1) in the first situation, we can get any value from a to a; and (2) in the second situation, we can get any value from b to b.
By definition of the function f m; m ð Þ, we are willing to pay the value f a; a ð Þto participate in the first situation and the value f b; b to participate in the second situation. What if we consider these two choices as a single situation? In this case, the smallest possible value that we get overallin both situationsis when we get the smallest possible value a in the first situation and the smallest possible value b in the second situation. In this case, the overall value is a þ b.
Similarly, the largest possible value that we get overallin both situationsis when we get the largest possible value a in the first situation and the largest possible value b in the second situation. In this case, the overall value is a þ b.
Thus, when we consider these two choices as a single situation, the interval of possible monetary gains has the form a þ b; a þ b h i . So, the equivalent monetary value of the two choices treated as a single situation is f a þ b; a þ b . It is reasonable to require that the price that we pay for two choices sold together should be equal to the sum of the prices that we pay for two choices taken separately, i.e. that . This property is known as additivity. The usual derivation of Hurwicz criterion. Now, we are ready to describe the usual derivation of Hurwicz criterion.
Proposition 1. For a value function f m; m ð Þ, the following two conditions are equivalent to each other: the value function is monotonic and additive; and the value function has the Hurwicz form.
Proof. It is easy to prove that a Hurwicz-form value function is monotonic and additive. Vice versa, let us assume that a value function f m; m ð Þis monotonic and additive. Let us denote a ¼ def f 0; 1 ð Þ. Because of additivity, for every natural number n, we have: Similarly, for every m and n, we have: For every real number r, we have m/n # r # (m þ 1)/n, where m ¼ def br Á nc. Thus, owing to monotonicity, we have f(0,m/n) # f(0,r) # f(0,(m þ 1)/n), i.e. a · (m/n) # f(0,r) # a · (m þ 1)/n. Here, 0 # rm/n # 1/n, so in the limit n ! 1, we have m/n ! r and (m þ 1)/n ! r. Thus, the above inequality leads to f(0,r) = a · r.
In particular, for every m # m, we have f 0; m À m ð Þ¼ a Á m À m ð Þ . By the property of a value function, we have m # f m; m ð Þ # m, i.e. f m; m ð Þ ¼ m. Thus, owing to additivity: One can easily check that this is indeed the Hurwicz expression.
Limitations. The previously mentioned motivations are reasonably reasonable, but they may not be 100% convincing.
Indeed, we argued that if the worst-case scenario is possible for each of the two situations, then it is possible that we have the worst-case scenario in both situations. This may sound reasonable, but it is not in full agreement with common sense. Indeed, e.g. when we fly from point A to point B, we understand: that there may an unexpected delay at the airport A; that a plane may have a problem in flight and we will have to get back; and that there may a problem at the airport B and we will get stuck on the plane.

Hurwicz optimismpessimism approach
But, we honestly do not believe that all these low-probable disasters will happen at the samethis only happens in comedies describing lovable losers who always get into trouble. We can raise another issue about the additivity requirement that additivity assumes that for the combination of two items, we always pay the same price as for the two items separately. Sometimes, this is true, but often, this is not true: there are discounts if you buy several items (or several objects of the same type) at the same time.
What should we do? As the arguments that we used previously to justify the assumptions are not 100% convincing, maybe we can find somewhat more convincing arguments in favor of Hurwicz formula or, alternatively, maybe these more convincing arguments can lead us to a different formula?
This is what we will analyze in Section 3. In mathematical terms, this property is known as shift-invariance.
Discussion. At first glance shift-invariant is very similar to additivity. Indeed, it can be viewed as a particular case of additivity, in which the first interval is simply the interval [m, m] consisting of a single number m.
But good news is that both previously mentioned objections to general additivity do not apply here. Indeed, we are not talking about a combination of rare events, so the first objection is not applicable. The second objection is also not applicable, as although we may expect a discount if we buy two big bottles of milk, no one expects a discount if we buy a bottle of milk and a fixed amount of money (e.g. when we ask to change a big banknote when paying).
Need for additional assumptions. If we limit ourselves only to shift-invariance, we will get too many possibilities in addition to Hurwicz formula: specifically, one can see that we can have a more general expression: where F(z) is a monotonic function defined for all z ! 0 for which F(z) # z for all ze.g. F(z) = z/(1 þ z) (by the way, it is possible to show that the above expression is the most general form of a monotonic shift-invariant value function).
To narrow down the class of possible value functions, we need to make additional reasonable assumptions. We will describe one such assumption right away.
A new assumptiontransitivity. Let us start with the same interval [0,1] with which we started the proof of Proposition 1. Similarly to this proof, let us denote the value f(0, 1) corresponding to this interval by a.
What can we conclude that from the fact that f(0,1) = a? Well, owing to shift invariance, we can conclude that for every x, we have f(x,1 þ x) = a þ x. From the mathematical viewpoint, this is all that we can conclude. However, from the common sense viewpoint, we can make yet another conclusion.
Indeed, e.g. for each x from the interval [0,1], the alternative corresponding to the interval [x,1 þ x] is equivalent to getting a monetary amount a þ x: [x,1 þ x] : a þ x. If we do not know which of these intervals the alternative corresponds to, but we know that it corresponds to one of these alternatives, this means that the actual gain can take any value from the union of these intervals. Each of these intervals is equivalent to the value a þ x, thus, the union of these intervals is equivalent to the set of all possible values a þ x when x [ [0,1]: Let us estimate the left-hand side and the right-hand side of this equality. The smallest possible value in the left-hand side is when we take the smallest value from the interval [x,1 þ x], i.e. the value xfor the smallest possible value x from the interval [0,1] (i.e., for the value x = 0). Thus, the smallest possible value in the lefthand side is equal to 0. The largest possible value in the left-hand side is when we take the largest value from the interval [x,1 þ x], i.e. the value 1 þ xfor the largest possible value x from the interval [0,1] (i.e., for the value x = 1). Thus, the largest possible value in the lefthand side is equal to1 þ 1 = 2.
So, the left-hand side of the previously mentioned equality is the interval [0,2]. Similarly: The smallest possible value in the right-hand side is when we take the smallest possible value x from the interval [0,1], i.e. the value x = 0. Thus, the smallest possible value in the right-hand side is equal to a þ 0 = a. The largest possible value in the right-hand side is when we take the largest possible value x from the interval [0,1], i.e. the value x = 1. Thus, the smallest possible value in the right-hand side is equal to a þ 1.
So, the left-hand side of the above equality is the interval [a, 1 þ a]. Thus, the previously mentioned equivalent takes the form [0,2] : [a,1 þ a]. Good news is that we already knownas a particular case of shift-invariancethat the interval [a, 1 þ a] is equivalent to the value a þ a = 2a. Thus, by transitivity of equivalence, we conclude that the interval [0,2] is equivalent to 2a, i.e. that f(0,2) = 2a. Then, by shift-invariance, we will get f(x,2þx) = 2a þ x for each x.
Instead of stacking intervals of width 1, we could similarly stack intervals of a different width w.
New derivation of Hurwicz formula. It turns out that this way, we can indeed get a new derivation of Hurwicz formula. Let us describe all this in precise terms.

Hurwicz optimismpessimism approach
We say that a value function is transitive if for each w and for all m # m, we have f '; ' À Á ¼ f r; r ð Þ, where: Comment: In this definition, we only described transitivity for the case when all combined intervals have the exact same width. Our main motivation for this restriction is that, as we will show, only such transitivity is neededand in derivations, it is always desirable to avoid unnecessarily general assumptions and to limit ourselves only to weakest possible assumptionsweakest possible among those that will lead to the desired derivation. There is another reason for this limitation: as we how later in this section, if we generalize this property too much, then there will be no realistic value function at all that would satisfy thus generalized property.
Proposition 2. For a value function f m; m ð Þ, the following two conditions are equivalent to each other: The value function is monotonic, shift-invariant and transitive. The value function has the Hurwicz form.
f(0, r) = a · r for all real numbers r. From this formula, in that proof, we used, in effect, shiftinvariance to prove that the Hurwicz formula is indeed true for all m # m. As we still assume shift-invariance, this means that we have a derivation of the Hurwicz formula in this case as well.
The proposition is proven. Discussion: we cannot generalize the transitivity property too much. Let us show that the transitivity assumption cannot be realistically generalized too much, to cases when united intervals have different widths.
Definition 3 On the other hand, here, f(0,a) = a · a, so: thus f r; r ð Þ ¼ a Á a ¼ a 2 . Thus, the generalized transitivity is satisfied only when a = a 2 , i.e. when either a = 0 or a = 1.
The proposition is proven.
4. What is utility and what are the properties of utility: a brief reminder What is utility. To apply computer-based number-oriented tools for making decisions in a nonmonetary case, we need to describe the user's preferences in numerical terms. In decision Hurwicz optimismpessimism approach theory (Fishburn, 1969;Luce and Raiffa, 1989;Raiffa, 1997;Nguyen et al., 2009;Kreinovich, 2014), this is done as follows. Let us select the two extreme alternatives: a very bad alternative A À which is worse than anything that we will actually encounter; and a very good alternative A þ which is better than anything that we will actually encounter.
For each real number p from the interval [0, 1], we can form a lotterywe will denote this lottery by L(p)in which: we get the very good alternative A þ with probability p; and we get the very bad alternative A À with the remaining probability 1 À p.
To find how valuable is each alternative A for the decision-maker, we ask him/her to compare the alternative A with lotteries L(p) corresponding to different probabilities p. Here: when p is small, close to 0, the lottery L(p) is similar to the very bad alternative A À and is, thus, worse than A; we will denote this by A -< A; and when p is close to 1, the lottery L(p) is similar to the very good alternative A þ and is, thus, better than A: A < L(p).
Also, the smaller the probability p of getting a very good alternative, the worse the lottery L(p). Thus: if L(p) < A and p 0 < p, then L(p 0 ) < A; and if A < L(p) and p < p 0 , then A < L(p 0 ).
Thus, similarly to the monetary case, there exists a threshold value: supfp : L p ð Þ < Ag ¼ inffp : A < L p ð Þg; We will denote this threshold value by u(A). This threshold value is known as the utility of the alternative A.
Similarly to the monetary case, for every « > 0, we have L(u(A) -« ) A < L(u(A) þ « ). This is true for arbitrarily small « , in particular, for the values « for which the difference in probabilities among u(A) -«, u(A), and u(A) þ « are practically unnoticeable. So, we can conclude that from the practical viewpoint, the alternative A is equivalent to the lottery L(u(A)). We will denote this equivalence by A : L(u(A)).
Utility is defined modulo a linear transformation. The numerical value of the utility u(A) depends not only on the alternative A, it also depends on which pair How will that change the numerical value of utility?
If an alternative A has utility u 0 (A) with respect to the pair (A 0 -,A 0 þ ), this means that this alternative is equivalent to the lottery L 0 (u 0 (A)), in which: we get A 0 þ with probability u 0 (A); and we get A 0 -with the remaining probability1u 0 (A).
As A -< A 0 -< A þ , we can find a utility value u(A 0 -) for which the alternative A 0 -is equivalent to the lottery L(u(A 0 -)), in which: we select A þ with probability u(A 0 -); and we select A À with probability 1u(A 0 -).
Similarly, we have A 0 þ : L(u(A 0 þ )). Thus, the original alternative A is equivalent to a twostage lottery, in which: first, we select either A 0 þ (with probability u(A 0 )) or A 0 -(with probability 1u(A 0 )); then, we select either A þ or A À with probabilities depending on what we selected on the first stage: if we selected A 0 þ on the first stage, then we select A þ with probability u(A 0 þ ) and A À with probability1u(A 0 þ ), and if we selected A 0 -on the first stage, then we select A þ with probability u(A 0 -) and A À with probability 1u(A 0 -).
As a result of this two-stage lottery, we get either A þ or A À , and the probability of selecting A þ is equal to: By definition, this probability is the utility u(A) of the alternative A with respect to the pair (A -, A þ ), thus: The right-hand side is a linear expression in terms of u 0 (A). So, we conclude that utilities corresponding to different pairs can be obtained from each other by a linear transformation. In other words, the numerical value of the utility is defined modulo a generic linear transformationjust like the numerical value of time and temperature, where the corresponding linear transformations mean selecting a different starting point and/or a different measuring unit. 5. Utility case: usual derivation of the Hurwicz criterion and the limitations of this derivation Formulation of the problem. As we have mentioned earlier, in many practical situations, we do not know the exact consequence of each action, and thus, we do not know the exact value of the corresponding utility. Instead, for such situations, we only know the interval u; u ½ of possible utility values. According to the general idea of utility, to describe the decision-maker's preferences for such interval-valued situations, we must assign, to each such interval, an appropriate utility value. Similarly to the monetary case, we will denote this utility value by f u; u ð Þ, and we will call the corresponding function a value function. Clearly, we must have u # f u; u ð Þ# u, and clearly, if u and/or u increase, the intervalvalued alternative becomes better, i.e. the value function should be monotonic.
What are other natural properties of the value function?
We cannot reuse assumptions from the monetary case. We cannot simply use the same properties as in the monetary case. For example, additivity makes no sense: it makes perfect sense to add dollar amounts; but it makes no sense to add probabilities (and utilities, as we have explained, are probabilities).
So, we need alternative assumptions.
Hurwicz optimismpessimism approach So, we need additional assumptionsassumptions which are more convincing that scaleinvariance.
What we propose. What we propose is the previously described transitivity property. The arguments in favor of this property apply verbatim to the utility case. And we already knowfrom Proposition 2that if we require shift-invariance and transitivity, then the only value functions we get are Hurwicz ones.
Thus, indeed, we get a new, (hopefully) more convincing, derivation of the Hurwicz criterion in the utility case as well.