Decision making under interval and more general uncertainty: monetary vs. utility approaches Текст научной статьи по специальности «Математика»

Вычислительные технологии

Том 22, № 2, 2017

Decision making under interval and more general uncertainty: monetary vs. utility approaches

V. KREINOVICH

University of Texas at El Paso, El Paso, TX 79968, USA e-mail: vladik@utep.edu

In many situations, our decision results either in a money gain (or loss) and/or in the gain of goods having a money equivalent. A natural idea is to assign a fair price to different alternatives, and then to use these fair prices to select the best alternative. Sometimes, interval uncertainty is present in such situations, which means that we do not know the exact amount of money that we will get for each possible decision, we only know lower and upper bounds on this amount. In this paper, we show how to assign a fair price under interval uncertainty. We also explain how to assign a fair price in the case of more general types of uncertainty such as p-boxes, twin intervals, fuzzy values, etc.

In other situations, the result of a decision is the decision maker's own satisfaction. Then, a more adequate approach is to use utilities — a quantitative way of describing user's preferences. In our paper, after a brief introduction describing what are utilities, how to evaluate them, and how to make decisions based on utilities, we explain how to make decisions in situations with user uncertainty — a realistic situation when a decision maker cannot always decide which alternative is better.

Keywords: decision making, interval uncertainty.

1. Need for decision making under uncertainty

Need for decision making. In many practical situations, we have several alternatives at our disposal, and we need to select one of these alternatives. For example:

• a person saving for retirement needs to find the best way to invest money;

• a company needs to select a location for its new plant;

• a designer must select one of several possible designs for a new airplane;

• a medical doctor needs to select a treatment for a patient.

Need for decision making under uncertainty. Decision making is easier if we know the exact consequences of each alternative selection. Often, however, we only have an incomplete information about consequences of different alternative, and we need to select an alternative under this uncertainty.

What we do in this paper. In this paper, we explain two approaches to decision making under uncertainty: monetary and utility approaches, we explain when each of these approaches is appropriate, how to justify the corresponding heuristic techniques, and how to go beyond these heuristic techniques.

© ICT SB RAS, 2017

Most results about the monetary approach are mentioned in [1], most results about the utility approach are described in [2], the combined description — with a clear separation of the two approaches — is new. Decision making under uncertainty is a vast research area, with thousands of relevant books and papers; we refer readers to papers cited in [1, 2] and to the edited book [3].

2. Decision making under uncertainty: monetary approach and its heuristics

2.1. When monetary approach is appropriate

In many situations, e. g., in financial and economic decision making, the decision results:

• either in a money gain (or loss) and/or

• in the gain of goods that can be exchanged for money or for other goods.

In this case, we select an alternative which the highest exchange value, i.e., the highest price u.

Uncertainty means that we do not know the exact prices. The simplest case is when we only know lower and upper bounds on the price, i. e., we only know that u E [u,u] for given bounds u and u.

2.2. Hurwicz optimism-pessimism heuristic for decision making under interval uncertainty

In the early 1950s, the future Nobel Prize winner L. Hurwicz proposed to select an alternative for which

an • u + (1 — an) • u ^ max.

Here, an E [0,1] described the optimism level of a decision maker:

► an = 1 means optimism;

► an = 0 means pessimism;

► 0 < an < 1 combines optimism and pessimism.

This approach works well in practice. However, this is a semi-heuristic idea.

It is desirable to come up with an approach which can be uniquely determined based on the first principles.

3. Monetary approach to decision making under uncertainty: how to justify the existing heuristics and how to move beyond these heuristics

3.1. Fair price approach: an idea

When we have full information about an object, then we can express our desirability of each possible situation by declaring a price that we are willing to pay to get involved in this situation. Once these prices are set, we simply select the alternative for which the participation price is the highest.

In decision making under uncertainty, it is not easy to come up with a fair price. It would be very useful to develop a regular technique for producing such fair prices. These prices can then be used in decision making, to select an appropriate alternative.

3.2. Case of interval uncertainty

Analysis of the problem. In the ideal case, we know the exact gain u of selecting an alternative. A more realistic case is when we only know the lower bound u and the upper bound u on this gain — and we do not know which values u E [u, u] are more probable or possible and which are not. This situation is known as interval uncertainty. In the rest of the paper, we denote intervals by boldface letters (a, b, ..., Y, Z), and the set of all closed bounded intervals over R is designated as IR.

We want to assign, to each interval [u,u], a number P([u,u]) describing the fair price of this interval.

Conservativeness. Since we know that u < u, we have P([u,u]) < u. Similarly, since we know that u > u, we have u < P([u,u]).

Monotonicity. Let us first consider the case when we keep the lower endpoint u intact, but increase the upper bound. This means that we:

• keep all the previous possibilities, but

• we allow new possibilities, with a higher gain.

In this case, it is reasonable to require that the corresponding price not decrease:

if u = v and u<v then P ([u,u]) < P ([w, w]).

Let us now consider another case, when we dismiss some low-gain alternatives. This should increase (or at least not decrease) the fair price:

if u<v and u = v then P([u,u]) < P([w, w]).

Additivity: idea. Let us consider another requirement on the fair price. This requirement is related to the fact that we can consider two decision processes separately. Alternatively, we can also consider a single decision process in which we select a pair of alternatives:

• the 1st alternative corresponding to the 1st decision, and

• the 2nd alternative corresponding to the 2nd decision. If we are willing to pay

• the amount u to participate in the first process, and

• the amount v to participate in the second decision process,

then we should be willing to pay u + v to participate in both decision processes.

Additivity: case of interval uncertainty. Let us describe what this requirement will look like in the case of interval uncertainty.

In this case, about the gain u from the first alternative, we only know that this (unknown) gain is in [u,u]. About the gain v from the second alternative, we only know that this gain belongs to the interval [v, v}. The overall gain u + v can thus take any value from the interval

[u, u] + [v_, v] =f {u + v : u E [u, u],v E [v_, v]}.

It is easy to check that [u,u] + [v,v] = [u + v,u + v}. Thus, the additivity requirement about the fair prices takes the form

P ([u + v,u + w]) = P ([u,u]) + P ([w,w]).

Fair price under interval uncertainty. Let us see what all these requirement lead to. First, we formulate

Definition 1. By a fair price under interval uncertainty, we mean a real-valued function P([u,u]) for which

(i) u < P([u,u]) < u for all u and u (conservativeness);

(ii) if u = v and u<v, then P([u,u]) < P([w, w]) (monotonicity);

(iii) for all u, u, v, and v, we have additivity

P ([u + v,u + w]) = P ([u,u]) + P ([w,w]). Theorem 1. Each fair price under interval uncertainty has the form

P([u,u]) = anu + (1 — an)u for some an E [0,1].

We thus get a new justification of the Hurwicz optimism-pessimism criterion. Proof: main ideas.

• Due to monotonicity, P([u,u]) = u.

• Due to monotonicity, an ==f P([0,1]) E [0,1].

• For [0,1] = [0,1/n] +... + [0,1/n] (n times), additivity implies that an = nP([0,1/n}), so P([0,1/n]) = aH(1/n).

• For [0,m/n] = [0,1/n] + ... + [0,1/n] (m times), additivity implies P([0,m/n]) = an (m/n).

• For each real number r, for each n, there is an m, such that m/n < r < (m + 1)/n.

• Monotonicity implies

aH(m/n) = P([0,m/n]) < P([0,r]) < P([0, (m +1)/n\) = aH((m +1)/n).

• When n —y w, an (m/n) ^ an r and an ((m + 1)/n) ^ an r, hence P ([0,r]) = an r.

• For [u,u] = [u,y] + [0,u — m], additivity implies that P([u,u]) = u + an(u — u). Q.E.D.

3.3. Case of set-valued uncertainty

Need for set-valued uncertainty. In some cases, in addition to knowing that the actual gain belongs to the interval [u,u] C R, we also know that some values from this interval cannot be possible values of this gain.

For example, if we buy an obscure lottery ticket for a simple prize-or-no-prize lottery from a remote country, we either get the prize or lose the money. In this case, the set of possible values of the gain consists of two values.

Instead of a (bounded) interval of possible values, we can therefore consider a general closed bounded set of possible values on the real axis R.

Fair price under set-valued uncertainty. We want to construct a real-valued function P that assigns, to every bounded closed set S C R, such a real number P(S) that

(i) P([u,u]) = anu + (1 — an)u (conservativeness);

(ii) P(S + S') = P(S) + P(S'), where 5 + = (s + ^ : s E 5, ^ E S'} (additivity). Theorem 2. Each fair price under set uncertainty has the form

P(S) = an sup S + (1 — an) inf S for some an E [0,1].

Proof: idea.

• {s, s} C S C [s, s], where s = inf S and s = sup S;

• thus, [2s, 2s] = {s,s} + [s, s] C 5 + [s, s] C [s, s] + [s, s] = [2s, 2s];

• so S + [s, s] = [2s, 2s], hence P(S) + P([s,s]) = P([2s, 2s]), and

P(S) = P([2s, 2s]) - P([s,s]) = (aH(2s) + (1 - aH)(2s)) - (aHs + (1 - aH)s).

3.4. Case of probabilistic uncertainty

Suppose that for some financial instrument, we know a probability distribution p(x) on the set of possible gains x. What is the fair price P for this instrument?

Due to additivity, the fair price for n copies of this instrument is nP. According to the Large Numbers Theorem, for large n, the average gain tends to the mean value (= the expected value)

" = /xp(x) "*■

Thus, the fair price for n copies of the instrument is close to n^, i. e. nP ~ n^. The larger n, the closer the averages. So, in the limit, we get P = i.e., the fair price is the mean value.

3.5. Case of p-box uncertainty

Probabilistic uncertainty means that for every x, we know the value of the cumulative distribution function F(x) = Prob(^ < x). In practice, we often only have partial information about these values. In this case, for each x, we only know an interval [F(x), F(x)] containing the actual (unknown) value F(x). The interval-valued function [F(x),F(x)] is known as a p-box.

What is the fair price of a p-box? The only information that we have about the cdf is that F(x) E [F(x),F(x)]. For each possible F(x), for large n, n copies of the instrument

are equivalent to n^, with ^ = J xdF(x).

For different F(x) from the p-box, values of ^ for an interval [ where ^ = J x dF(x)

and = J xdF_(x). Thus, the price of a p-box is equal to the price of an interval [ y,,~p\.

We already know that this price is equal to anU + (1 - So, this is a fair price of

a p-box.

3.6. Case of twin intervals

What are twin intervals. Twin, by definition, is an interval of intervals (twice interval), or, in other words, an interval that has interval bounds.

Sometimes, in addition to the interval [x,x], we also have a "most probable" or a "most possible" subinterval [m,m] C [x,x]. For such "twin intervals", addition is naturally defined component-wise:

([x, x], [m, m]) + ([y,y], [n, n}) = ([x + y,x + y], [m + n,m + n]).

Thus, the additivity for additivity requirement about the fair prices takes the form

P([x + y,x + y], [m + n,m + n}) = P([x,x], [m,m]) + P([y,y], [n,n]).

Fair price under twin interval uncertainty. Let us give a formal definition similar to those previously provided for the fair prices under interval and set-valued uncertainty.

Definition 2. By a fair price under twin uncertainty, we mean such a real-valued function P([u,u], [m,m]) that satisfies

(i) u < P([u,u], [m,m]) < u for all u < m < m < u (conservativeness);

(ii) if u < v, m < n, m < n, and u < v, then

P([u,u], [m,m]) < P([vl,v], [n,n]) (monotonicity);

(iii) for all u < m < m < u and v < n < n < v, we have additivity:

P([m + v,u + v], [m + n,m + m]) = P([u, u], [m,m]) + P([vl,v], [n,n]).

Theorem 3. Each fair price under twin uncertainty has the following form, for some aL,au,au E [0,1]:

P([u,u], [m,m]) = m + au(m — m) + ajj(U — m) + a^(u — m). 3.7. Case of fuzzy numbers

Need for fuzzy uncertainty. An expert is often imprecise ("fuzzy") about possible values of this or that quantity of interest. For example, an expert may say that the gain is "small". To describe such information in formal terms, L. Zadeh introduced the notion of fuzzy numbers.

For fuzzy numbers, different values u are possible with different degrees of possibility ^(u) E [0,1]. The value w is a possible value of u + v if:

► for some values u and v satisfying u + v = w,

► u is a possible value of 1st gain, and

► v is a possible value of 2nd gain.

If we interpret "and" as min and "or" ("for some") as max, we get Zadeh's extension principle:

^(w) = max mini ^i(u), .

u,v: u+v=w

The above operation is easiest to describe in terms of a-cuts defined as

u(a) = [u-(a),u+(a)] d=f (u : ^(u) > a}.

Notice that a-cuts of fuzzy numbers are intervals. Then, w(a) = u(a) + v(a) according to the rule for interval addition, i. e.,

w-(a) = u~ (a) + v-(a) and w+(a) = u+(a) + v+(a).

For multiplication, we similarly get

^(w) = max mini ^i(u), ^2(v)}.

u,v: uv=w

In terms of a-cuts of the operands, we have w(a) = u(a) • v(a) according to the rule for interval multiplication, i.e.,

w-(a) = min{ u-(a)v-(a),u-(a)v+(a),u+(a)v-(a),u+(a)v+(a) }

w+(a) = max{ u-(a)v-(a), u- (a)v+ (a) ,u+ (a)v- (a) ,u+ (a)v+ (a)}.

What is the fair price under such fuzzy uncertainty?

Fair price under fuzzy uncertainty. We want to assign, to every fuzzy number s, a real number P(s), so that

(i) if a fuzzy number s is located between u and u, then u < P(s) < u (conservativeness);

(ii) P(u + v) = P(u) + P(v) (additivity);

(iii) if for all a, s-(a) < t-(a) and s+(a) < t+(a), then we have P(s) < P(t) (monotoni-

city);

(iv) if uniformly converges to p, then P(pn) ^ P(p) (continuity). Theorem 4. The fair price under fuzzy uncertainty is equal to

i i P(.) = S„ + f k-(.a) dS-(a) - ! k+ia) dS+ia) ¡or omnc K±(a).

Discussion. We have

J f (x)dg(x) = J f (x)g'(x) dx for g'(x) understood as a generalized function (distribution). Hence, one can write for K±(a)

P(s)= K-(a)s-(a) da + K+(a)s+ (a) da

in the same generalized sense (in the sense of the theory of distributions). Conservativeness means that

J K-(a) da + J K+(a) da = 1. 0 0

Next, we get for the interval [u,u]

P W=( / '" +( / 'da) *

i

Thus, Hurwicz optimism-pessimism coefficient an is equal to f K +(a) da. In this sense, the

0

above formula is a generalization of Hurwicz's formula to the fuzzy case.

4. Non-monetary (utility) approach to decision making under uncertainty: main ideas

4.1. Monetary approach is not always appropriate

In some situations, the result of the decision is the decision maker's own satisfaction; examples include:

• buying a house to live in,

• selecting a movie to watch.

In such situations, monetary approach is not appropriate; for example:

► a small apartment in downtown can be very expensive,

► but many people prefer a cheaper — but more spacious and comfortable — suburban house.

4.2. Non-monetary decision making: traditional utility approach

Analysis of the problem. To make a decision, we must find out the user's preference, and help the user select an alternative which is the best — according to these preferences.

Traditional approach is based on an assumption that for each two alternatives N and A'', a user can tell:

■ whether the first alternative is better for him/her (we will denote this by A" < A');

■ or the second alternative is better (we will denote this by A' < A");

■ or the two given alternatives are of equal value to the user (we will denote this by

A' = A").

The notion of utility. Under the above assumption, we can form a natural numerical scale for describing preferences.

Let us select a very bad alternative A0 and a very good alternative Ai. Then, most other alternatives are better than A0 but worse than Ai. For every probability p E [0,1], we can form a lottery L(p) in which we get Ai with probability p and A0 with probability 1 — p.

When p = 0, this lottery simply coincides with the alternative A0: L(0) = A0. The larger the probability p of the positive outcome increases, the better the result:

p' < p" implies L(pj) < L(p").

Finally, for p = 1, the lottery coincides with the alternative Ai: L(1) = Ai.

Thus, we have a continuous scale of alternatives L(p) that monotonically goes from L(0) = A0 to L(1) = Ai. Due to monotonicity, when p increases, we first have L(p) < A, then we have L(p) > A. The threshold value is called the utility of the alternative A:

u(A) = sup{p : L(p) < A } = inf{p : L(p) > A }.

Then, for every e > 0, we have

L(u(A) — e) < A < L(u(A) + e).

We will describe such (almost) equivalence by =, i.e., we will write that A = L(u(A)).

Fast iterative process for determining u(A). How can we determine the utility value?

Initially, we know the values u = 0 and u =1, such that A = L(u(A)) for a certatin u(A) E [u,u]. In general, once we know an interval [u,u] containing u(A), we compute the midpoint wmid of this interval and compare A with L(umid).

• If A < L(umid), then u(A) < umid, so we know that u E [u, Mmid].

• If L(umid) < A, then wmid < u(A), so u E [«mid,«].

After each iteration, we decrease the width of the interval [u,u] by half. After k iterations, we get an interval of width 2-fc which contains u(A) — i. e., we get u(A) with accuracy 2-fc. The above is a well-known bisection method for localizing roots of scalar functions.

How to make a decision based on utility values. Suppose that we have found the utilities u(A'), u(A"), ..., of the alternatives A', A", ... Which of these alternatives should we choose?

By definition of utility, we have A = L(u(A)) for every alternative A, and L(p') < L(p") if and only if p' < p". We can thus conclude that A' is preferable to A" if and only if u(A') > u(A"). In other words, we should always select an alternative with the largest possible value of utility.

How to estimate utility of an action. For each action, we usually know possible outcomes S1,..., Sn. We can often estimate the probabilities p1,... ,pn of these outcomes. By definition of utility, each situation Si is equivalent to a lottery L(u(Si)) in which we get

• A1 with probability u(Si) and

• A0 with the remaining probability 1 — u(Si).

Thus, the action is equivalent to a complex lottery in which

• first, we select one of the situations Si with probability pi, i.e. P(Si) = pi,

• then, depending on Si, we get A1 with probability P(A1 | Si) = u(Si) and A0 with probability 1 — u(Si).

The probability of getting A1 in this complex lottery is

n n

P(A1) = ^ P(A11 SJP(St) = ^ u(Sl)pl.

i=1 i=1

In the complex lottery, we get

n

► A1 with probability u = Piu($i), and

i=1

► A0 with probability 1 — u.

Overall, we should select the action with the largest value of expected utility

n

u = ^2 Piu(Si).

i=1

Non-uniqueness of utility. The above definition of utility u depends on A0, A1. What if we use different alternatives and N{1

Every A is equivalent to a lottery L(u(A)) in which we get A1 with probability u(A) and A0 with probability 1 — u(A). For simplicity, let us assume that A*0 < A0 < A1 < Nx. Then, A0 = V(u'(A0)) and A1 = L'(u'(A1)). So, A is equivalent to a complex lottery in which:

1) we select A1 with probability u(A) and A0 with probability 1 — u(A);

2) depending on Ai, we get A'1 with probability u'(Ai) and with probability 1 — u'(Ai). In this complex lottery, we get N1 with probability

u'(A) = u(A) (u'(A1) — u'(Ao)) + u'(Ao).

So, in general, utility is defined modulo an (increasing) linear transformation u' = au + b, with a > 0.

Subjective probabilities. In practice, the probabilities Pi of different outcomes are often not known exactly. For each event E, a natural way to estimate its subjective probability is to fix a prize (e.g., $1) and compare:

• the lottery £e in which we get the fixed prize if the event E occurs and 0 if it does not occur, with

• a lottery £(p) in which we get the same amount with probability p.

Here, similarly to the utility case, we get a value ps(E) for which, for every e > 0:

£(ps(E) — e) <lE < £(ps(E) + e). Then, the utility of an action with possible outcomes S\,..., Sn is equal to

n

u = ^ ps(Ei)u(Si).

i= 1

Beyond traditional decision making: towards a more realistic description.

Earlier, we assumed that a user can always decide which of the two alternatives N and A" is better:

■ either N < A'',

■ or A" < N,

■ or A' = A".

In practice, a user is sometimes unable to meaningfully decide between the two alternatives; we will denote this by N || A!'. In mathematical terms, this means that the preference relation is no longer a total (linear) order, it can be a partial order.

From utility to interval-valued utility. Similarly to the traditional decision making approach:

• we select two alternatives A0 < Ai and

• we compare each alternative A which is better than A0 and worse than Ai with lotteries L(p).

Since the preference is a partial order, we have in general:

u(A) = sup{p : L(p) < A} < u(A) = inf{p : L(p) > A}.

For each alternative A, instead of a single value u(A) of the utility, we now have an interval [u(A),u(A)\ such that:

• if p < u(A), then L(p) < A;

• if p > u(A), then A < L(p); and

• if u(A) <p< u(A), then A || L(p).

We will call this interval the utility of the alternative A.

Interval-valued utilities and interval-valued subjective probabilities. To feasibly elicit the values u(A) and u(A), we carry out the following operations:

1) starting with [u,u] = [0,1], bisect an interval such that L(u) < A < L(u) until we find u0 for which A || L(u0);

2) by bisecting an interval [u,uo] for which L(u) < A || L(u0), we find u(A);

3) by bisecting an interval [u0,u] for which L(u0) || A < L(u), we find u(A). Similarly, when we estimate the probability of an event E, we no longer get a single value ps(E). Rather, we get an interval [ps(E),ps(E^ of possible values of probability. By using the bisection method described, we can feasibly elicit the values ps(E) and ps(E).

Decision making under interval uncertainty. For each possible decision d, we know the interval [u(d),u(d)\ of possible values of utility. Which decision shall we select?

A natural idea is to select all decisions d0 that may be optimal, i.e., which are optimal for some function u(d) E [u(d),u(d)].

Checking all possible functions is not feasible. However, it is easy to show that the above

condition is equivalent to an easier-to-check one: u(d0) > maxu(d).

d

The remaining problem is that in practice, we would like to select one decision; which one should be select?

Need for definite decision making. At first glance, if A' || A", it does not matter whether we recommend alternative N or alternative A". Let us show that this is not a good recommendation.

Let A be an alternative about which we know nothing, i. e., for which [u(A),u(A)} = [0,1]. In this case, A is indistinguishable both from a "good" lottery L(0.999) and a "bad" lottery L(0.001). Suppose that we recommend, to the user, that A is equivalent both to L(0.999) and to L(0.001). Then this user will feel comfortable

— first, exchanging L(0.999) with A, and

— then, exchanging A with L(0.001).

So, following our recommendations, the user switches from a very good alternative to a very bad one.

The above argument does not depend on the fact that we assumed complete ignorance about A:

• every time we recommend that the alternative A is "equivalent" both to L(p) and to L(p') (p<p');

• we make the user vulnerable to a similar switch from a better alternative L(jJ) to a worse one L(p).

Thus, there should be only a single value p for which A can be reasonably exchanged with L(p). In precise terms:

■ we start with the utility interval [u(A),u(A)\, and

■ we need to select a single u(A) for which it is reasonable to exchange A with a lottery L(u).

How can we find this value u(A)?

5. Utility approach to decision making under uncertainty: how to justify the existing heuristics and how to move beyond these heuristics

Interval uncertainty: a new justification for the Hurwicz optimism-pessimism criterion. We need to assign, to each interval [u,u], a utility value u(u,u) E [u,u]. Let us denote an == «(0,1).

As we have mentioned earlier, utility is determined modulo a linear transformation u' = au + b. It is therefore reasonable to require that the equivalent utility does not change with re-scaling: for a > 0 and b,

u(au- + b, au+ + b) = au(u-, u+) + b.

In particular, for u- =0, u+ = 1, a = u — u, and b = u, we get

u(u,u) = an (u — u) + u = an u + (1 — an )u.

This is exactly Hurwicz's optimism-pessimism criterion!

Which value an should we choose? An argument in favor of an = 0.5. Let

us take an event E about which we know nothing. For a lottery L+ in which we get Ai if E and A0 otherwise, the utility interval is [0,1]. Thus, the equivalent utility of L+ is an 1 + (1 — )0 = an.

For a lottery L- in which we get A0 if E and Ai otherwise, the utility is [0,1], so equivalent utility is also an.

For a complex lottery L in which we select either L+ or L- with probability 0.5, the equivalent utility is still an. On the other hand, in L, we get Ai with probability 0.5 and A0 with probability 0.5. Thus, L = L(0.5) and hence, u(L) = 0.5. So, we conclude that an = 0.5.

Which action should we choose? Suppose that an action has n possible outcomes Si,..., Sn, with utilities [«(Si),«(Si)], and probabilities [pi,pi]. We know that each alternative is equivalent to a simple lottery with utility Ui = anu(Si) + (1 — an)u(Si). We know that for each i, the z-th event is equivalent to Pi = an Pi + (1 — )pi.

Thus, this action is equivalent to a situation in which we get utility Ui with probability pi.

n

The utility of such a situation is equal to PiUi. So, the equivalent utility of the original

i=i

action is

n

y] (g-HPi + (1 — O.H)pj (aHu(Si) + (1 — aH)u(Si)).

i=i

Observation: the resulting decision depends on the level of detail. Let us

consider a situation in which, with some probability p, we gain a utility u, else we get 0. The expected utility is pu + (1 — p)0 = pu.

Suppose that we only know the intervals [u,u] and [p,p]. The equivalent utility Uk (with k for know) is

Uk = (anP + (1 — aH)p) (aHu + (1 — aH)u). If we only know that utility is from [pu,pu], then

Ud = anpu + (1 — an)pu (d for don't know). Here, additional knowledge decreases utility:

ud — uk = aH(1 — aH)(p — p)(u — u) > 0.

(This is maybe what the Book of Ecclesiastes meant by "For with much wisdom comes much sorrow"?)

Acknowledgement. This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and National Science Foundation grant DUE-0926721.

The author is greatly thankful to the editors, especially to Sergey P. Shary, for their encouragement and help.

References

[1] Lorkowski, J., Kreinovich, V. How much for an interval? A set? A twin set? A p-box? A Kaucher interval? Towards an economics-motivated approach to decision making under uncertainty // Lecture Notes in Computer Science, vol. 9553. Cham; Switzerland: Springer Intern., 2016. P. 66-76. DOI: 10.1007/978-3-319-31769-4.

[2] Kreinovich, V. Decision making under interval uncertainty (and beyond) // Human-Centric Decision-Making Models for Social Sciences. Studies in Computational Intelligence. Series vol. 502 / P. Guo, W. Pedrycz (Eds). Berlin; Heidelberg: Springer Verlag, 2014. P. 163-193.

[3] Human-Centric Decision-Making Models for Social Sciences. Studies in Computational Intelligence. Series vol. 502 / P. Guo, W. Pedrycz (Eds). Berlin; Heidelberg: Springer Verlag, 2014. 418 p. DOI: 10.1007/978-3-642-39307-5.

Received 6 February 2017