Towards decision making under general uncertainty

Pownuk А.; Kosheleva О.; Kreinovich V.

UDC 519.816 DOI: 10.25513/2222-8772.2017.4.110-119

TOWARDS DECISION MAKING UNDER GENERAL

UNCERTAINTY

Andrzej Pownuk

Ph.D. (Phys.-Math.), Instructor, e-mail: [email protected] Olga Kosheleva Ph.D. (Phys.-Math.), Associate Professor, e-mail: [email protected]

Vladik Kreinovich Ph.D. (Phys.-Math.), Professor, e-mail: [email protected]

University of Texas at El Paso, El Paso, Texas 79968, USA

Abstract. There exist techniques for decision making under specific types of uncertainty, such as probabilistic, fuzzy, etc. Each of the corresponding ways of describing uncertainty has its advantages and limitations. As a result, new techniques for describing uncertainty appear all the time. Instead of trying to extend the existing decision making idea to each of these new techniques one by one, we attempt to develop a general approach that would cover all possible uncertainty techniques.

Keywords: decision making, general uncertainty, fuzzy uncertainty, probabilistic uncertainty.

1. Formulation of the Problem

Need for decision making under uncertainty. The ultimate goal of science and engineering is to make decisions, i.e., to select the most appropriate action.

Situations when we have full information about possible consequences of each action are rare. Usually, there is some uncertainty. It is therefore important to make decisions under uncertainty.

There are many different techniques for describing uncertainty. There are many different techniques for describing uncertainty: probabilistic, fuzzy (see, e.g., [4,10,12]), possibilistic, interval-valued or, more generally, type-2 fuzzy (see, e.g., [6,7]), complex-valued fuzzy [2], etc. For many of these techniques, there are known methods for decision making under the corresponding uncertainty.

All the current techniques for describing uncertainty have their advantages and their limitations. Because of the known limitations, new - more adequate - techniques for describing uncertainty appear all the time. For each of these techniques, we need to understand how to make decisions under the corresponding uncertainty.

A problem that we try to solve in this paper. At present, this understanding mostly comes technique-by-technique. A natural question is: can we develop a general framework that would allow us to make decision under general uncertainty?

The main objective of this paper is to develop such a general formalism.

Towards a precise formulation of the problem. Let us start with a monetary problem. Suppose that we need to make a financial decision, such as investing a given amount of money in a certain financial instrument (such as shares or bonds).

If we knew the exact consequences of this action, then we would know exactly how much money we will have after a certain period of time. This happens, e.g., if we simply place the given amount in a saving account with a known interest rate.

In most situations, however, we are uncertain of the possible financial consequences of this action. In other words, for each investment scheme, there are several possible consequences, with monetary amounts xi,...,xn. By using an appropriate uncertainty technique, we can describe our degree of certainty that the i-th alternative is possible by the corresponding value ^j. Depending on the formalism for describing uncertainty,

• a value ^ can be a number - e.g., when we use probabilistic or fuzzy uncertainty,

it can be an interval - when we use interval-valued fuzzy, it can be a complex number - if we use complex-valued fuzzy,

• it can be a fuzzy set - if we use type-2 fuzzy techniques, etc.

For another investment scheme, we can have n' different possible consequences, with monetary values xl,... ,x'n, and degrees of certainty .

To make a decision, we need to compare this investment, in particular, with situations like placing money in a saving account, in which we simply get a fixed amount of money after the same period of time.

If this fixed amount of money is too small, then investing in an uncertain financial instrument is clearly better.

If this fixed amount of money is sufficiently large, then getting this fixed amount of money is clearly better than investing in an uncertain financial instrument.

There should be a threshold value of the fixed amount at which we go from the instrument being preferable to a fixed amount being preferable. This threshold fixed amount of money is thus equivalent, to the user, to the investment in an uncertain instrument.

So, for each uncertain investment, in which we get:

• the amount xl with degree of possibility

• the amount x2 with degree of possibility ... ,

• amount xn with degree of possibility

we have an equivalent amount of money. We will denote this equivalent amount of money by f (xi,... ,xn,Vi,... ,^n).

Our goal is to find out how this equivalent amount of money depends on the values x, and Once we know the equivalent amount of money corresponding to each uncertain investment, we will be able to select the best of the possible investments: namely, it is natural to select the investment for which the corresponding equivalent amount of money is the largest possible.

What about non-financial decision making situations? It is known (see, e.g., [3,5,8,11]) that decisions of a rational person can be described as optimizing a certain quantity called utility.

Thus, in general, we have the following problem: for each uncertain situation, in which we get:

• utility xi with degree of possibility

• utility x2 with degree of possibility ... ,

• utility xn with degree of possibility

we have an equivalent utility value. We will denote this equivalent utility value by

f(xi, ...,Xn,^i,...

Our goal is thus to find out how this equivalent utility value depends on the values xi and Once we know the equivalent utility value corresponding to each possible decision, we will be able to select the best of the possible decisions: namely, it is natural to select the decision for which the corresponding equivalent utility value is the largest possible.

Comment. In the following text, to make our thinking as understandable as possible, we will most talk about financial situations - since it is easier to think about money than about abstract utilities. However, our reasoning is applicable to utilities as well.

2. Analysis of the Problem

First reasonable assumption: additivity. We are interested in finding a function f (x1,..., xn, ,..., ^n) of 2n variables.

Suppose that the money that we get from the investment comes in two consequent payments. In the i-th alternative, we first get the amount xi, and then -almost immediately - we also get the amount yi.

We can consider the resulting investment in two different ways. First, we can simply ignore the fact that the money comes in two installments, and just take into account that in each alternative i, we get the amount xi + yi. This way, the equivalent amount of money is equal to

f (xi + yi,... ,xn + yn,Vi,... ,Vn).

Alternatively, we can treat both installments separately:

• in the first installment, we get xi with uncertainty

• in the second installment, we get yi with uncertainty Thus:

• the first installment is worth the amount f (xi,... ,xn, ..., ^n), and

• the second installment is worth the amount f (yi,... ,yn,^i,..., ^n).

The overall benefit is the sum of the amounts corresponding to both installments. So, in this way of description, the overall money value of the original investment is equal to the sum of the money values of the two installments:

f (xi, ...,x„,^i,...,^n) + f (yi,... ,yn,^ i,... ,^n).

The equivalent benefit of the investment should not depend on the way we compute it, so the two estimates should be equal:

f (xi + yi,... ,xn + ... ,ßn) =

= f (xi, ...,Xn,^i,...,^n) + f (Vi, ...,Vn,ßi,... ,ßn).

Functions satisfying this property are known as additive. Thus, we can say that for each combination of values ... ,^n, the dependence on xi,... ,xn is additive.

Second reasonable assumption: bounds. No matter what happens, we get at

least minxi and at most maxxj. Thus, the equivalent benefit of an investment

i i

cannot be smaller than min xi and cannot be larger than maxxi:

ii

minxi ^ f (xi,..., xn, ^i,... , ^n) ^ maxxi.

What we can conclude form the first two assumptions. It is known (see, e.g., [1]) that every bounded additive function is linear, i.e., that we have

n

f (xi, Ci(^i, . . . ,^n) ■ Xi.

i=1

So, instead of a function of 2n variables, we now have a simpler task for finding n functions ci(^i,..., ^n) of n variables.

Nothing should depend on the ordering of the alternatives. The ordering of the alternatives is arbitrary, so nothing should change if we change this ordering. For example, if we swap the first and the second alternatives, then instead of

Cl(^i,^2, . . .) ■ Xi + C2(^i,^2, . . .) ■ X2 + ...

we should have

C2(^2,^i, . . .) ■ Xi + Ci(^2,^i, . . .) ■ X2 + ...

These two expression must coincide, so the coefficients at x1 must coincide, and we must have

C2(^2,^1, ...) = ci (^1 ,^2,...). In general, we should thus have

ci, . . . , ^n) = Cl^ . . . ^i—1, ^i+1, . . . , ^n).

Thus, the above expression should have the form

n

f (Xi, ...,Xn)^ Ci(^i,^i, . . . 1,^i+1, . . . ,^n) ■ Xi.

i=1

Now, the problem is to find a single function c1(^1,... ,^n) of n variables.

Combining alternatives with the same outcomes. Based on the above formula, the value c1(^1,^2,... ,^n) corresponds to f(1,0,...,0), i.e., to a situation when we have:

• the value 1 with degree of possibility ,

• the value 0 with degree of possibility , ... ,

• the value 0 with degree of possibility

In alternatives 2 through n, we have the same outcome 0, so it makes sense to consider them as a single alternative. To find the degree of possibility of this combined alternatives, we need to apply some "or"-operation © to the degrees of possibility ... of individual alternatives.

For probabilities, this combination operation is simply the sum a © b = a + b, for fuzzy, it is a t-conorm, etc. In general, the degree of certainty of the combined alternative is equal to © ... © Thus, the equivalent value of this situation is equal to ci© ... © ^n). So, we have

ci(^i,^2,... ,^n) = ci(^i,^2 © ... © ^n), and the above expression for the equivalent benefit takes the following form

n

f (xi, . . . ,xn) = ^ Ci © ^i-i © ^i+i © ... © ^n) ■ xi.

i=i

Now, the problem is to find a single function ci(^i,^2) of two variables. Let us simplify this problem even further.

Yet another reasonable requirement. Let us consider a situation in which we have three alternatives, i.e., in which, we get:

• the amount xi with degree of possibility

• the amount x2 with degree of possibility and

• the amount x3 with degree of possibility

According to the above formula, for this situation, the equivalent benefit is equal to

Ci(^i,^2 © ^3) ■ xi + Ci(^2,^i © ^3) ■ x2 + Ci(^3,^i © ^2) ■ x2.

On the other hand, we can consider an auxiliary situation A in which we get:

• the amount xi with degree of possibility and

• the amount x2 with the degree of possibility This situation is equivalent to the amount

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xa = Ci ) ■ xi + Ci(^2,^i) ■ x2,

and the degree of possibility of this auxiliary situation can be obtained by applying the corresponding "or"-operation to the degrees and and is thus, equal to ^a = ^i © ^2.

By replacing the first two alternatives in the original 3-alternative situation with the equivalent alternative, we get the equivalent situation, in which we get:

• the value xA with degree of possibility and

• the value x3 with degree of possibility

For this equivalent situation, the equivalent amount is equal to

ci(^A,^3) ■ xa + ci(^3,^a) ■ x3.

Substituting the expressions for xA and into this formula, we conclude that the equivalent amount is equal to

Cl(ui © ^2,^3) ■ (ci(^i,^2) ■ xi + Ci(^2,^2) ■ x2) + Ci(^3,^i © ^2) ■ x3 =

= Ci(^i © ^2,^3) ■ Ci(^i,^2) ■ xi + Ci(^i © ^2,^3) ■ Ci(^2,^2) ■ x2 +

+Ci(^3,^i © ^2) ■ x3.

We get two expressions for the same equivalent amount. These expressions must coincide. This means, in particular, that the coefficients at xi at both expressions must coincide, i.e., that we should have

What can we extract from this requirement. Let us consider an auxiliary function c(a,b) = ci(a,b © a), where b © a is an inverse to ©, i.e., the value for which a © (b © a) = b.

By definition of the new operation ©, we have

b = (a © b) © b.

Thus, we have

c(a, a © b) = ci((a © b) © a) = ci(a, b). In other words, for every a and b, we have

ci(a, b) = c(a, a © b).

Substituting this expression for ci(a,b) into the above formula, we conclude that

The left-hand side depends only on two values x == and z == © u2 © and

def

does not depend on the value y = © :

c(x,z) = c(y,z) ■ c(x,y). Thus, if we fix some value y0, we conclude that

c(x, z) = g(z) ■ h(x),

where we denoted g(z) =f c(y0,z) and h(x) =f c(x,y0).

Describing ci(a,b) in terms of the auxiliary function c(a,b), we can transform the expression for the equivalent monetary value to

n

i=i

Substituting the expression c(x,z) = g(z) ■ h(x) into this formula, we conclude that the equivalent monetary value takes the form

■ g ■ xi,

i=i

def

n)

where we denoted g = g(^i © ... © ^n).

For the case when xi — x2 — ... — xn, the boundedness requirement implies that the equivalent value is equal to xi. Thus, we have

n

xi = ^ h(^i) ■ g ■ xi.

i=i

Dividing both sides by xi, we conclude that

n

1 = g '^2 h(^i)

i=i

and hence, that

1

g = -n-.

E h(^i)

i=i

So, the equivalent monetary value is equal to the following expression:

n

E h(^i) ■ xi

i=i_

n '

E h(^i)

i=i

So, now we are down to a single unknown function h(u). 3. Conclusions

General conclusion. We need to decide between several actions. For each action, we know the possible outcomes xi,... ,xn, and for each of these possible outcomes i, we know the degree of possibility of this outcome. The above analysis shows that the benefit of each action can then be described by the following formula

n

E h(^i) ■ xi

i=i

E h(^i)

i=i

for an appropriate function h(u).

How can we find the function h(u)? If we have two alternatives with the same outcome xi = x2, then we can:

either treat them separately, leading to the terms

h(^i) ■ g(^i © ^2 © . . .) ■ xi + h(^2) ■ g(^i © ^2 © . . .) ■ xi + ...

• or treat them as a single alternative xi, with degree of possibility © thus leading to the term

Probabilistic case. In this case, the values are probabilities, and as we have mentioned, we have © = + So, the above condition takes the form

+ ^2) = h(ui) + h(^2).

Thus, in the probabilistic case, the function h(u) must be additive.

The higher probability, the more importance should be given to the corresponding alternative, so the function h(u) should be monotonic. It is known (see, e.g., [1]) that every monotonic additive function is linear, so we must have h(u) = c ■ ^ for some constant Thus, the above formula for the equivalent amount takes the form

n

C ' ^i ' xi

i=i_

n .

EC ■ ^i

i=i

n

For probabilities, = 1. So, dividing both the numerator and the denominator

i=i

by C, we conclude that the equivalent benefit has the form

n

^ ^ ^i ■ xi.

i=i

This is exactly the formula for the expected utility that appears when we consider the decision of rational agents under probabilistic uncertainty [3,5,8,11].

Fuzzy case. In the fuzzy case, a © b is a t-conorm. It is known (see, e.g., [9]) that every t-comrm can be approximated, with arbitrary accuracy, by an Archimedean t-conorm, i.e., by a function of the type G-i(G(a) + G(b)), where G(a) is a strictly increasing continuous function and G-i denotes the inverse function. Thus, from

G(a) + G(b) = G(c).

The requirement that

h(a) + h(b) = h(c). In other words, if G(a) + G(b) = G(c), then

h(a) + h(b) = h(c).

If we denote A = G(a), B = G(b), and C = h(c), then a = G-i(A), b = G-i(B), c = G-i(C), and the above requirement takes the following form: if A + B = C, then

h(G-i(A)) + h(G-i(B)) = h(G-i(C)).

So, for the auxiliary function H(A) == h(G-i(A)), we have A + B = C implying that H(C) = H(A) + H(B), i.e., that H(A + B) = H(A) + H(B). The function H(A) is monotonic and additive, hence H(A) = k ■ A for some constant k.

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So, H(A) = h(G-i(A)) = k ■ A. Substituting A = G(a) into this formula, we conclude that

h(G-i(G(a)) = h(a) = k ■ G(a). Thus, in the fuzzy case, the equivalent monetary value of each action is equal to

n

E k ■ G(^i) ■ xi

i=i_

n .

E k ■ G(^i)

i=i

Dividing both the numerator and the denominator by the constant k, we get the final formula

n

E G(^j) ■ xi i=i_

n ,

E G(^i)

i=i

where G(a) is a "generating" function of the t-conorm, i.e., a function for which the t-conorm has the form

G-i(G(a) + G(b)).

Fuzzy case: example. For example, for the algebraic sum t-conorm

we have

and thus,

so we have G(a) = — ln(1 — a).

Thus, the formula for the equivalent amount takes the form

n

Eln(1 — ^i) ■ xi

i=i_

n .

Eln(1 — 0i)

i=i

Acknowledgments

This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, and by an award "UTEP and Prudential Actuarial Science Academy and Pipeline Initiative" from Prudential Foundation.

References

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10. Nguyen H.T., Walker E.A. A First Course in Fuzzy Logic. Chapman and Hall/CRC, Boca Raton, Florida, 2006.

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О ПРИНЯТИИ РЕШЕНИЙ ПРИ ОБЩЕЙ НЕОПРЕДЕЛЁННОСТИ

А. ^внук

к.ф.-м.н., ст. преподаватель, e-mail: [email protected] О. Кошелева к.ф.-м.н., доцент, e-mail: [email protected] В. Крейнович

к.ф.-м.н., профессор, e-mail: [email protected]

Техасский университет в Эль Пасо, США

Аннотация. Существуют методы принятия решений при некоторых типах неопределённости, такие как вероятностные, нечёткие и т. д. Каждый из соответствующих способов описания неопределённости имеет свои преимущества и ограничения. В результате все время появляются новые методы описания неопределённости. Вместо того чтобы пытаться последовательно расширять существующую идею принятия решений на каждый из этих новых методов, мы пытаемся разработать общий подход, который будет охватывать все возможные методы неопределённости.

Ключевые слова: принятие решений, общая неопределённость, нечёткая неопределённость, вероятностная неопределённость.

Дата поступления в редакцию: 21.03.2017

Towards decision making under general uncertainty Текст научной статьи по специальности «Математика»

Аннотация научной статьи по математике, автор научной работы — Pownuk А., Kosheleva О., Kreinovich V.

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