Using Fuzzy Functions to Select an Optimal Action in Decision Theory
Richard Al´o , Andre de Korvin and Franc¸ois Modave University Houston Down Town One Main Street, Houston, Tx, 77002 CS Dept. University of Texas at El Paso, 500 W. University Ave. El Paso, Tx, 79968-0518 email: alor , dekorvina @dt.uh.edu,
[email protected] Abstract In decision under uncertainty, we aim at finding a scoring procedure to determine an optimal decision without prior knowledge on the actual state of the world. It is well known that the probability approach to this problem leads to paradoxes, that are related to independence properties required on the preferences. This naturally leads us to drop additivity and therefore, replace probabilities with fuzzy measures. The aim of this paper is to present several benefit functions to identify optimal actions, in the fuzzy measure and fuzzy logic perspective.
1 Introduction One of the goals of decision making is to come up with a strategy to determine an optimal course of action. In decision under uncertainty, the decision maker does not know the a priori true state of the world, and in many cases, does not even have a probability available on the set of states of the world . A possible approach to solve such problems is to assume some rationality axioms (that are understandable from a decision maker’s behavior perspective) on the preference relation defined over the actions, and derive the existence of a probability measure and a utility function such that, if and are two actions and means is preferred to ,
"!# $%& ''(
(1)
where the integrals represent the subjective expected utility of the actions. We encounter two problems here. To have the additivity of the probability measure, we require an independence hypothesis, the sure-thing principle [14], that is unlikely to be satisfied in most practical applications, as the Ellsberg’s paradox [8] shows. Therefore, the addivity property needs to be weakened, leading to the same type of representation with a fuzzy expected utility with respect to a fuzzy measure. Second, Savage’s theorem requires the states of the world to be an infinite set, which is not convenient in practice. Through the concept of lotteries (finite support probability distributions), Anscombe and Aumann [1] allows an additive representation of the preferences in the finite setting, that still requires an independence hypothesis which appears to be too strong. These lotteries require the existence of a benefit function which is always assumed to be accurately known. A first contribution of this paper is to allow this function to be fuzzy valued. Second, we can relaxed the hypotheses even more by allowing uncertainty over the lotteries, and we propose two criteria to select an optimal act in this situation (either maximizing an extended Choquet integral or maximizing an entropy function). Eventually, we present a selection criterion for set-valued actions.
1
2 Fuzzy Integration and Decision Making First, we recall some definitions of fuzzy measures and the Choquet integral and basics of decision under uncertainty. For more details on fuzzy integration, see for example [5] ,[4] and for application to decision making, see [10].
2.1 Fuzzy integration is a set and
In this section,
is a set of subsets of
such that ,
Definition 1 A fuzzy measure (or non-additive measure) on
.
is a set function
such that
" !# $ % '&)(* 021 + -, . / 3 54768 (96 ;:=A$ BC < = for every finite family
O
of subsets of . It is well known that if H"I$
*657
is a belief function (see [10]) then -(
that is the inferior envelop of lebesgue integrals with respect to a family of probabilities yielding a belief function, is a Choquet integral. 3
If is a function from integral is
into fuzzy subsets of
that is
X+
1
*
"I$
57 *
* where is defined as in the previous section. Therefore, if we assume best act >2 is to maximize the extended Choquet integral
with respect to equivalent to
where
& is a belief function, a strategy to pick the
HJ J ]
+
+1,.
is a belief function generated by ( * 57 * +1,. 3
, then a natural extension of the Choquet
. We can show that the above expression is
4.2 Maximizing the Entropy
54 e>
An alternate way to pick the best act is to choose the appropriate distribution ( 2 . Indeed, each ( has a density * * given by as is a finite set. The idea is to pick ( 2 in such a way that its density function ( maximizes the entropy
8?4 O O3 O
TN
&
&
F\ > R RR 6 P Q QW O Q O O O O T O O O 0I T O O 96 V 6/
* over ( . * The construction of is very natural using the following algorithm: while do
< : = ?/ACB D Let be the largest subset of such that 8
O
end while * We then define
for
.
4.3 Maximizing without Belief functions
O
O
eO H JJHJ < O
The above results are valid when the fuzzy measure over the states of the world is a belief function. Hence, we need to come up with a strategy to deal with the case where is a general fuzzy measure. In this section, we assume that the probability over is not known, and we have a set of families , " to choose from. Each family represents a family of possibile probability distributions over , for an expert . Once the distribution has been chosen based on expert , an appropriate distribution within is chosen using the results presented in the above section, or using possible additional information. Thus, we have a two-step selection mechanism. First, we need to choose the “right” family of distributions , then choose an appropriate distribution within the selected family. The choice of the family of distributions is based on expert credibility. We assume, we have a preference relation (a weak-order, that is a reflexive, connected and transitive binary relation) over the set of experts, generating a preference relation over the family where F
O
indicates that
O
OO
O
9
O
= O O # =
is preferred to F . Of course, we have that F
4
F
O
as a larger set gives more freedom to select an optimal distribution in the second stage of the selection mechanism. We will also make the following assumption:
d
48
d
We then follow [12] to select an optimal act. Let us denote by consideration. We define a function over by 7 7 48
and let us denote Let
for
O
U f
by
d d
8
8
8
4 U > `F D E I I I N N I ! ` % N U 4 6 ` % '>
and
such that
$ :
$ :
H$
and otherwise. Then, following [12],
)
5
!
in the following way. We have that
. Hence, there exists a negative-valued function $ on
Then, we define
the set of families of distributions over , under
denote all the distributuions over . Then we can define
for 57
is a preference relation on . Hence, we have a (constructive, following lemma 3 in [12]) way to select an element in , and the above section, provided a way to select within the chosen element.
5 Set-valued functions
In this section, we will deal with the case were for an act and a state , we have a family of possible distributions over the set of outcomes . We denote this family by . That, we assume the acts to be set-valued. We should first note that despite an apparent complexity, assuming set-valued functions is very natural. In general, in decision making, we assume, we can choose the granularity of the description of the problem, to avoid dealing with set-valued function. Though, set-valued analysis is well developped as [2] shows. In fact, it is very easy to relate set-valued analysis with fuzzy sets. One way to find the best set-valued act would be to use optimization technique from set-valued analysis (e.g finding the Tangent cone), but we will see how we can adapt more traditional techniques, in line with what we have presented so far to deal with this particular case. We could define a preference relation over the distributions at state as follows: +1,. +-,/. '
HJ
J
where the right preference s defined by comparing two fuzzy sets. but, we would like these preferences to be complete and transitive, hence, we choose an other approach. To avoid technical details, we will assume that the space of possible benefits is finite, but this is not restrictive at all here.
where
I@! ! J JHJ !
4 I H JJHJ > O O N O
. Then, we define our maximizing set
5
by
O
O
Hence, the membership of in is the fraction to which is the maximal element of membership of in is . ' if and only if We now define by ' +-,/. +1,.
H J ! O
where
X
3
O
O HJ
and naturally, the
4 >
represents the fraction of falling into . We now make the following assumptions, over the preference relation that we have defined. ' then the preference attached to the set falls in between the (1). We assume that if ' preference attached to and ' that is '
4 > 4 8)> 4 d > 4 )> d4 d/+ > 4 \ \+ > 4 + >
4 > 4 8+ >
assuming that non of the preferences are trivial. We refer to this property as the weak dominance principle. ' ' ' (2). If and E is equal to ' E is equal to the empty set, then ' ' 8 8
We will refer to this property as the weak independence property. We can see easily that it is an extension of the sure-thing principle of Savage. See also [7] for a discussion of a similar independence property in the single-valued case, but in a qualitative setting (i.e Sugeno integrals). ' and ' the greatest and lowest element of . Then, if we have the weak Let us denoted by dominance principle and the weak independence property, we have that ' ' ' following [3]. It is assumed that the order generated on the ' is a linear order. Let us now define the convex combination of and by: ' '
&U
'9
4 &U 'U ' >
` '9 & T &U ` & T ' J H J MI HJ 4 J '>
where is a coefficient depending on the decision maker. A value close to indicates a very optimistic behavior from the decision maker whereas a value close to indicates a pessimistic behavior as the decision maker would be expecting the worst possible case. This expression can be rewritten: ' ' ' where
`
+-,/.
+ , 57,/. + ,
H"I$
Therefore, is a utility function defined on ' and its value is a distribution on the set of outcomes given state a natural way to compare two acts and is to maximize ,/.
If ( is a distribution over , we pick
)>
. Thus, for a
in order to maximize * ,/.
We can make the following remark. Let us define the notion of positive responsivness (PR) by ' then ' ' 8 ' for all in then and if ' ' ' 8
4 8)> $4 > 4 )> 4 8)> $4 > 4 )> 6
4 8 '
4 d8 + > 4 + > & / .HJJHJ < & < & & T N & < PI
' Then, if we modify the conclusion of the weak dominace principle to we can 8 8 show that id there is at least 6 distributions, there does not exist and ordering on satisfying (PR) and (WIND) (see [11]). If the number of distributions is at least 3, there does not exist a binary relation on ' satisfying (WDP) and (WIND). ' Let us say that ' for " are tied and similarly for for " with respect to subsethood. Then, using the function
'
/ eJHJHJ
`
&
NO
< PI
'
F
'
can be used in lieu of the previous function, in order to avoid the drawbacks presented in [11].
6 Conclusions In this paper, we have presented several strategies to pick the best act in a decision under uncertainty problem in various situations. First, we have generalized a result given by one of the author to allow fuzzy-valued benefit functions. Then, we have shown how it was possible to introduce a second-order uncertainty over the lotteries, and have proposed two criteria to find the best act. One was to define an extended Choquet integral, the other was to maximize an Entropy function. This result was the extended when we do not have a belief function. Eventually, we have given a selection mechanism of the best act when the value of an act is set-valued, under mild assumptions.
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