1 3 Probability weighted means as surrogates for stochastic dominance 4 in decision making 5 6 7 Ronald R. Yager Q1 a,b,⇑ , Naif Alajlan c 8 a Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, United States 9 b Visiting Distinguished Scientist, King Saud University, Riyadh, Saudi Arabia 10 c ALISR Laboratory, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia 11 13 article info 14 Article history: 15 Received 18 February 2014 16 Received in revised form 14 April 2014 17 Accepted 15 April 2014 18 Available online xxxx 19 Keywords: 20 Uncertainty 21 Decision making 22 Stochastic dominance 23 Means 24 OWA operators 25 26 abstract 27 We discuss the role of stochastic dominance as tool for comparing uncertain payoff alternatives. How- 28 ever, we note the fact that this is a very strong condition and in most cases a stochastic dominance rela- 29 tionship does not exist between alternatives. This requires us to consider the use of surrogates for 30 stochastic dominance to compare alternatives. Here we consider a class of surrogates that are called 31 Probability Weighted Means (PWM). These surrogates are numeric values associated with an uncertain 32 alternative and as such comparisons can be based on these values. The PWM are consistent with stochas- 33 tic dominance in the sense that if alternative A stochastically dominated alternative B then its PWM value 34 is larger. We look at a number of different examples of probability weighted means. 35 Ó 2014 Elsevier B.V. All rights reserved. 36 37 38 39 1. Introduction 40 Decision making in situations in which there is a probabilistic 41 uncertainty associated with the payoff that results from the selec- 42 tion of an alternative is a very common task. Here each alternative 43 is characterized by an uncertain payoff profile, a probability distri- 44 bution over possible payoffs. A crucial problem here is the selec- 45 tion of a preferred alternative from a set of possible alternatives. 46 While the objective is clear, select the alternative that gives the 47 biggest payoff, the comparison of these uncertainty profiles with 48 regard to this objective is difficult. One well-regarded method for 49 comparing two uncertainty profiles is via the idea of stochastic 50 dominance [1–3]. Essentially alternative A stochastically domi- 51 nates alternative B if for any payoff value x alternative A has a 52 higher probability of resulting in a payoff greater then or equal x 53 then does alternative B. While providing an intuitively reasonable 54 paradigm for deciding which of two alternatives is preferred, sto- 55 chastic dominance is a strong condition and generally a stochastic 56 dominance relationship between two alternatives does not exist, 57 neither one stochastically dominates the other. In order to provide 58 operational decision tools we look for surrogates for stochastic 59 dominance. These surrogates associate with each alternative a 60 numeric value, the larger the value the more preferred, and hence 61 always allows comparison between alternatives. An important fea- 62 ture of these surrogates is their consistency with stochastic domi- 63 nance in the sense that if A stochastically dominates B then the 64 surrogate value of A is larger then the surrogate value of B. Here 65 we consider a class of surrogates that we refer to as Probability 66 Weighted Means (PWM). 67 2. Stochastic dominance and decision making 68 In decision making under uncertainty we are faced with the 69 problem of selecting a preferred alternative from among a collec- 70 tion of alternatives based upon each alternative’s payoff profile. 71 Assume A i is a decision alternative consisting of a set of possible 72 payoffs that can result from the selection of this alternative, C ij 73 for j = 1 to n i , and an associated uncertainty profile over this set 74 of payoffs. Here we shall assume all the C ij are numeric values. 75 An important example of uncertainty profile associated with a 76 decision alternative is a probabilistic uncertainty profile. Here each 77 C ij has a probability p ij > 0. 78 A fundamental task here is to decide if alternative A 1 is pre- 79 ferred to alternative A 2 , A 1 ‘‘>’’ A 2 . One commonly used method 80 for comparing alternatives is based upon the idea of stochastic 81 dominance [1–10] with the understanding that if alternative A 1 82 stochastically dominates alternative A 2 then A 1 is the preferred 83 alternative. We say that A 1 stochastically dominates A 2 if for all http://dx.doi.org/10.1016/j.knosys.2014.04.024 0950-7051/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding Q2 author at: Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, United States. E-mail addresses: [email protected](R.R. Yager Q1 ), [email protected](N. Alajlan). Knowledge-Based Systems xxx (2014) xxx–xxx Contents lists available at ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys KNOSYS 2817 No. of Pages 7, Model 5G 2 May 2014 Please cite this article in press as: R.R. Yager Q1 , N. Alajlan, Probability weighted means as surrogates for stochastic dominance in decision making, Knowl. Based Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
Please cite this article in press as: R.R. Yager, N. Alajlan, Probability weighted means as surrogates for stochastic dominance in decision making,Based Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
Ronald R. Yager a,b,⇑, Naif Alajlan c
a Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, United Statesb Visiting Distinguished Scientist, King Saud University, Riyadh, Saudi Arabiac ALISR Laboratory, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia
a r t i c l e i n f o a b s t r a c t
27282930313233343536
Article history:Received 18 February 2014Received in revised form 14 April 2014Accepted 15 April 2014Available online xxxx
We discuss the role of stochastic dominance as tool for comparing uncertain payoff alternatives. How-ever, we note the fact that this is a very strong condition and in most cases a stochastic dominance rela-tionship does not exist between alternatives. This requires us to consider the use of surrogates forstochastic dominance to compare alternatives. Here we consider a class of surrogates that are calledProbability Weighted Means (PWM). These surrogates are numeric values associated with an uncertainalternative and as such comparisons can be based on these values. The PWM are consistent with stochas-tic dominance in the sense that if alternative A stochastically dominated alternative B then its PWM valueis larger. We look at a number of different examples of probability weighted means.
� 2014 Elsevier B.V. All rights reserved.
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1. Introduction
Decision making in situations in which there is a probabilisticuncertainty associated with the payoff that results from the selec-tion of an alternative is a very common task. Here each alternativeis characterized by an uncertain payoff profile, a probability distri-bution over possible payoffs. A crucial problem here is the selec-tion of a preferred alternative from a set of possible alternatives.While the objective is clear, select the alternative that gives thebiggest payoff, the comparison of these uncertainty profiles withregard to this objective is difficult. One well-regarded method forcomparing two uncertainty profiles is via the idea of stochasticdominance [1–3]. Essentially alternative A stochastically domi-nates alternative B if for any payoff value x alternative A has ahigher probability of resulting in a payoff greater then or equal xthen does alternative B. While providing an intuitively reasonableparadigm for deciding which of two alternatives is preferred, sto-chastic dominance is a strong condition and generally a stochasticdominance relationship between two alternatives does not exist,neither one stochastically dominates the other. In order to provideoperational decision tools we look for surrogates for stochasticdominance. These surrogates associate with each alternative a
numeric value, the larger the value the more preferred, and hencealways allows comparison between alternatives. An important fea-ture of these surrogates is their consistency with stochastic domi-nance in the sense that if A stochastically dominates B then thesurrogate value of A is larger then the surrogate value of B. Herewe consider a class of surrogates that we refer to as ProbabilityWeighted Means (PWM).
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2. Stochastic dominance and decision making
In decision making under uncertainty we are faced with theproblem of selecting a preferred alternative from among a collec-tion of alternatives based upon each alternative’s payoff profile.Assume Ai is a decision alternative consisting of a set of possiblepayoffs that can result from the selection of this alternative, Cij
for j = 1 to ni, and an associated uncertainty profile over this setof payoffs. Here we shall assume all the Cij are numeric values.An important example of uncertainty profile associated with adecision alternative is a probabilistic uncertainty profile. Here eachCij has a probability pij > 0.
A fundamental task here is to decide if alternative A1 is pre-ferred to alternative A2, A1 ‘‘>’’ A2. One commonly used methodfor comparing alternatives is based upon the idea of stochasticdominance [1–10] with the understanding that if alternative A1
stochastically dominates alternative A2 then A1 is the preferredalternative. We say that A1 stochastically dominates A2 if for all
2 R.R. YagerQ1 , N. Alajlan / Knowledge-Based Systems xxx (2014) xxx–xxx
KNOSYS 2817 No. of Pages 7, Model 5G
2 May 2014
Q1
values C, Prob(A1 > C) P Prob(A2 > C) and there exists at least onevalue C⁄ such that Prob(A1 > C⁄) > Prob(A2 > C⁄). The intuition ofstochastic dominance is essentially that we prefer alternatives thathave larger probability of resulting in bigger payoffs.
We observe that
ProbðA1 > CÞ ¼X
j;C1j>C
p1j and ProbðA2 > CÞ ¼X
j;C2j>C
p2j
We further observe that since 1 ¼P
j;Cij6Cpij þP
j;Cij>Cpij then ifProb(A1 > C) P Prob(A2 > C) it follows that
1�X
j;C1j6C
p1j P 1�X
j;C2j6C
p2j
and henceP
j;C2j6Cp2j PP
j;C1j>Cp1j. Recalling thatP
j;Cij6Cpij is thecumulative distribution function, Fi(C), for Ai, Fi(C) = Prob(Aj 6 C).Thus alternative A1 stochastically dominates A2 if F1(C) 6 F2(C) forall C and for at least one C⁄ we have F1(C⁄) < F2(C⁄).
Formally we recall that a cumulative distribution function (CDF)associates with a probability distribution a mapping Fi:R ? [0,1]with the properties
(1) Monotonicity: Fi(a) P Fi(b) for a > b(2) Fi(b) = 1 for all b P Maxj¼1 to ni
½Cij�(3) Fi(a) = 0 for all a < Minj¼1 to ni
½Cij�
In the following for notational convenience we shall use ‘‘<’’ todenote any relationship G1(x) 6 G2(x) for all x and their exists atleast one x⁄ such that G1(x⁄) < G2(x⁄). Using this notation we havejust indicated A1 stochastically dominates A2 if F1(x) ‘‘<’’ F2(x) forall x 2 R and that is if Prob(A1 6 x) ‘‘<’’ Prob(A2 6 x) for all x.
In the following, without loss of generality, we shall use the fol-lowing more convenient structure to investigate the issues of inter-est. Let C = {Cj/j = 1 to n} be a collection of relevant numeric payoffsin a decision problem. A decision alternative Ai consists of a prob-ability distribution such that pij is the probability of obtaining pay-off Cj if we choose Ai. We note that if pij = 0 then Cj is not a possiblepayoff under Ai. Furthermore we shall assume the Cj have beenindexed in ascending order Cj+1 > Cj. Using this notation we see itthat a cumulative distribution function is expressible as
PðAi 6 xÞ ¼ FiðxÞ ¼X
js:t:Cj6x
pij
We see that for any x < C1 we have Fi(x) = 0 and for any x P Cn wehave Fi(x) = 1.
We emphasize here that while C is a finite subset of real num-bers, Fi(x) is defined over the whole real line.
As we have earlier indicated we say that alternative A1 stochas-tically dominates A2, if F1(x) ‘‘<’’ F2(x) for all x, F1(x) 6 F2(x) for all xand F1(x) < F2(x) for at least one x. We shall refer to this as A1 > SDA2.In Fig. 1 we show a typical example of stochastic dominance. HereF1 stochastically dominates F2.
1
F1(x)
F2 (x)
Fig. 1. Illustration of stochastic dominance.
Please cite this article in press as: R.R. Yager, N. Alajlan, Probability weighted mBased Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
As we have indicated if Aj > SDAk then alternative Aj is preferredto Ak. Assume A = {A1, . . . ,Aq} are a collection of alternatives thenstochastic dominance can be viewed as a binary relationship onthe space A [11]. Viewed as a binary relationship we see that it istransitive
A1>SDA2 and A2>SDA3 A1 >SDA3
From this transitivity it follows that if this relationship is complete,for each pair Aj and Ak either Aj > SDAk or Ak > SDAj, then we caninduce a linear ordering over the space A with respect to our pref-erence of the alternative.
However, one important problem often arises with this agenda.The property of completeness is often lacking with respect to spaceA. That is, there often exists pairs of alternatives, Aj and Ak suchthat neither Aj > SDAk nor Ak > SD Aj. This lack of completeness is aresult of the fact that while stochastic dominance is a clear indica-tion of preference between alternatives it is a relatively strongrequirement and often does not exist between pairs of alternatives.
Because of this difficulty we must look for surrogates to sto-chastic dominance as way of comparing alternatives.
3. Probability weighted means
We shall now look at some properties that follow from stochas-tic dominance, F1(x) ‘‘<’’ F2(x) for all x.
We first recall the concept of median associated with a proba-bility distribution. Given a probability distribution P over the orderspace, C = {C1, . . . , Cn} and its associated cumulative distributionfunction (CDF), Fi, then the median is the element Cmed such that
FiðCmed�1Þ < 0:5 6 FiðCmedÞ
It is essentially the payoff where the CDF transitions from less the0.5 to at least 0.5.
Assume A1 and A2 are two decision alternatives such thatA1 > SDA2, then F1(x)6 F2(x) for all x. Assume the median of A1
occurs at Cmed(1), that is F1(Cmed(1)) P 0.5 then it is clear that F2-
(Cmed(1)) P 0.5. From this it follows that Cmed(2), the median of A2
cannot occur at a value greater than Cmed(1). This allows us to con-clude the following.
OBSERVATION: Assume A1 and A2 are such that A1 > SDA2 thenMed(A1) P Med(A2)
In Fig. 2 we clearly illustrate this relationshipWe now shall look at the relationship between the expected
values of alternative and their relationship with respect to stochas-tic dominance. First we shall look at the relationship between analternative’s expected value and its associated CDF. Consider a gen-eric alternative A with probability pj associated with payoff Cj
where we have indexed the payoffs in increasing order, Cj+1 > Cj.We recall its expected value is EðAÞ ¼
Pnj¼1pjCj and FðCjÞ ¼
ProbðA 6 CjÞ ¼Pj
i¼1Pi. Here we shall by convention letFðC0Þ ¼ 0 ¼
P0i¼1pi. We also note the F(Cn) = 1.
1
0.5
F2
Med(F2 ) Med(F1)
F1
Fig. 2. Median calculation in case of stochastic dominance.
eans as surrogates for stochastic dominance in decision making, Knowl.
j¼1 FðCjÞðCj�1 � CjÞSince Ci > Ci�1 then we express this as
EðAÞ ¼ Cn �Xn�1
j¼1
FðCjÞðCj � Cj�1Þ
where Cj � Cj�1 > 0. So we see that the smaller the values of F(Cj) thebigger the expected value. Thus the smaller the CDF, the larger theexpected value. However we also previously indicated that stochasticdominance prefers smaller CDFs , thus we see that stochastic domi-nance is related to larger expected values. This relationship betweenstochastic dominance and expected value will provide a basis forusing the expected value as surrogate for stochastic dominance.
We now look at the relationship between the expected values oftwo alternatives that are in a stochastic dominance relationship.Here we assume A1 > SDA2, that is F1(x) ‘‘<’’ F2(x). We recallEðA1Þ ¼
Pnj¼1p1jCj and EðA2Þ ¼
Pnj¼1p2jCj. As we have just shown
these can be expressed in terms of the associated CDF functions
EðA1Þ ¼ Cn �Xn�1
j¼1
F1ðCjÞðCj � Cj�1Þ
EðA2Þ ¼ Cn �Xn�1
j¼1
F2ðCjÞðCj � Cj�1Þ
We now observe since F1(Cj) 6 F2(Cj) for all j and there exists at leastone Cj⁄ so that F1(Cj⁄) < F2(Cj⁄) we see that E(A1) > E(Ai). Thus we canconclude that if A1 > SDA2 then the expected value of A1 is larger thenthat of A2.
Another measure associated with a probability distribution isthe median. Assume X = {C1, . . . , Cn} and the indexing is such thatCi > Cj if i > j. Let A be an alternative having pj as the probabilityof Cj. We recall that Med(A) is the value CK such thatXK�1
j¼1
pj < 0:5 6XK
j¼1
pj
Consider now another decision alternative B that has the probabil-ities qj instead of pj. Thus for B, qj is the probability of getting Cj. Fur-thermore assume A > SDB. Thus for each i we have
Pij¼1qj P
Pij¼1pj.
Assume MedðBÞ ¼ CKB . This means thatPKB�1
j¼1 qj < 0:5. From the sto-chastic dominance relationship between A and B it follows thatPKB�1
j¼1 pj < 0:5. From this it follows that MedðAÞ ¼ CKA P CKB . Hencewe see that if A > SDB, then Med(A) P Med(B). Please note here thatthis is only P not the strict > as in the case of the expected value.
The expected value and the median are two examples of a moregeneral concept associated with a probability distribution on anumeric space. We shall refer to this as a probability weightedmean, PWM.
Definition 1. Assume a decision alternative A is described by acollection of n pairs, A = {(aj,pj) for j = 1 to n} where aj is a numericpayoff and pj is its associated probability. Here each pj 2 [0,1] andtheir sum is one. We say that R(A) is a probability weighted mean,PWM, if
(1) Minj[aj] 6 R(A) 6Maxj[aj]
Please cite this article in press as: R.R. Yager, N. Alajlan, Probability weighted mBased Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
(2) If pk = 1 then R(A) = ak
(3) If pk = 0 then R(A) is indifferent to the value of ak
(4) If A and B are two alternative such that A > SDB thenR(A) > R(B)
Condition 3 is equivalent to requiring that if eA is obtained fromA by removing A(ak,pk) then RðAÞ ¼ RðeAÞ
We shall allow for a weaker version of condition 4, 4W. We shallR(A) is 1 weak PWM if R satisfies the following condition 4Winstead of 44) If A and B are two alternative such that A > SDB thenR(A) P R(B).
We note that all regular PWM, ones satisfying 4 are always aweak PWM but not all weak PWM are regular PWM.
Before investigating this definition we make some observationsabout some of the concepts used.
If A = {(aj,pj)) for i = 1 to n} and we form A+ by adding one pair(an+1,pj+1) such that the pj+1 = 0 then CDF(A) = CDF(A+), FA(x) = FA+(x)for all x. The implication here is that adding any number of zeroprobability payoffs to an alternative does not affect its CDF.
Observation: Assume A and B are two decision alternativessuch that A > SDB. If A+ is obtained from A by adding a number ofzero probability components and B+ is obtained from B by addinga number, not necessarily the same, zero probability componentsthen A+ > SDB+. Thus we see adding zero probability componentsdoes not effect the stochastic dominance relationship.
If R(A) is an operation that has zero probability indifference, sat-isfies property three of a PWM, then if A+ is obtained from A by add-ing any number of zero probability components then R(A) = R(A+).
In the following we define an operation that we shall call nor-malization. Assume A1 = {(p1j,c1j), for j = 1 to n1} and A2 = {(p2j,c2j),for j = 1 to n2} are two decision alternative’s uncertainty profileshere pij is the probability associated with the payoff of cij. We shalldenote C1 as the set of all c1j and denote C2 is the set of all c2j. Let{a1, . . . , an} be the union of all payoff, fa1; . . . ; ang ¼ fc11; c12; . . . ;
c1n1g [ fc21; . . . ; c2n2g ¼ C1;[C2. We now define the normalized ver-sions of A1 and A2 as bA1 ¼ fðaj; pjÞ, for j = 1 to n} and bA2 ¼ fðaj; qjÞ,for j = 1 to n} such that
if aj = c1k 2 C1 then pj = p1k
if aj R C1 then pj = 0if aj = c2k 2 C2 then qj = p2k
aj R C2 then qj = 0
Essentially normalization is attained by adding to each of thedecision alternative’s payoff profile, with zero probability, the pay-offs in {a1, . . . ,an} that do not originally appear in it.
Thus we see that if A1 and A2 are two decision alternative pro-files and if bA1 and bA2 are obtained by a normalized process thenCDFðbA1Þ ¼ CDFðA1Þ and CDFðbA2Þ ¼ CDFðA2Þ. Thus any relationshipwith respect to stochastic dominance between A1 and A2 also holdsbetween bA1 and bA2. Furthermore if R is any operator this is zeroprobability indifferent then RðA1Þ ¼ RðbA1Þ and RðA2Þ ¼ RðbA2Þ. Thusthe relationship between RðbA1Þ and RðbA2Þ is the same as thatbetween R(A1) and R(A2).
Thus here we see that if A1 and A2 are two decision alternativeswe can always apply a normalization procedure without effectingissues related to investigation of properties related to the PWM.In particular, this allows us, in the following to assume all relevantdecision alternatives are defined with respect to the same set ofpossible payoff, (a1, . . . ,an) without jeopardizing the validity ofany of the results obtained.
4. Basic probability weighted means
We now provide a result that is fundamental to all that follows.
eans as surrogates for stochastic dominance in decision making, Knowl.
4 R.R. YagerQ1 , N. Alajlan / Knowledge-Based Systems xxx (2014) xxx–xxx
KNOSYS 2817 No. of Pages 7, Model 5G
2 May 2014
Q1
Theorem 1. Assume wj and vj for j = 1 to n are two collection ofweights, wj and vj 2 [0,1] and
Pnj¼1v j ¼
Pnj¼1wj ¼ 1. Let a1, . . . , an be
a set of numeric values so that ai+1 6 ai, there are in descending order.If for each i = 1 to n we have
Pij¼1wj P
Pij¼1v j then
Pnk¼1wkak PPn
k¼1vkak.
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Proof 1. Let Tk ¼Pk
j¼1wj and let Sk ¼Pk
j¼1wj we observe thatT0 = S0 = 0 and Tn = Sn = 1.
We also observe that
wk ¼ Tk � Tk�1 and vk ¼ Sk � Sk�1
From this we obtain
Xn
k¼1
wkak ¼Xn
k¼1
ðTk � Tk�1Þak ¼ ðT1 � T0Þa1 þ ðT2 � T1Þa2
þ ðT3 � T2Þa3 þ . . . ðTn � Tn�1Þan
Xn
k¼1
wkak ¼ �T0a1 þXn�1
k¼1
ðak � ak�1ÞTk þ Tnan
Since T0 = 0 and Tn = 1 then
Xn
k¼1
wkak ¼Xn�1
k¼1
ðak � akþ1ÞTk þ an
Similarly we have
Xn
k¼1
vkak ¼Xn�1
k¼1
ðak � akþ1ÞSk þ an
Since ak � ak+1 P 0 and Tk P Sk our result follows
Xn
k¼1
wkak PXn
k¼1
vkak: �
389
390391
Corollary 1. Again assume ai+16 ai and ifPn
j¼kþ1wj 6Pn
j¼kþ1v jfor allk then again we have
Pnk¼1wkak P
Pnk¼1vkak.
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Proof 2. SincePn
j¼1v j ¼Pn
j¼1wj ¼ 1, then ifPn
j¼kþ1wj 6Pn
j¼kþ1v j itfollows that
Pkj¼1wj P
Pkj¼1v j which was the condition in the pre-
ceding theorem. h
We now shall prove some properties about the PWM, the prob-ability weighted mean operator. We first show that any PWM, R, ismonotonic with respect to the payoffs. Assume A = {(ai,pi), for i = 1to n} and B = {(bi,pi), for i = 1 to n} are two alternative payoff pro-files having the same probabilities but with bi P ai for all i. Thusthe payoffs for B are at least as large as those in A.
Consider now the respective CDFs, FA and FB. For any x let Ax bethe subset of the indices of tuples (ai,pi) so that ai 6 x and let Bx
be the subset of the indices of tuples (bi,pi) so that bi 6 x. Fromthe fact that ai 6 bi then for any i 2 Bx we also have i 2 Ax, essen-tially, we have Bx # Ax. Consider now the associated CDFs,FAðxÞ ¼
Pi2Ax
pi and FBðxÞ ¼P
i2Bxpi. Since Bx # Ax then FA(x) P FB(x)
and hence B > SDA.If R is any PWM then from property 4 if B > SDA then R(B) P
R(A). Thus we can conclude that R is monotonic with response tothe payoffs, if A = {(ai,pi)} and B = ((bi,pi)} bi P ai, then R(B) P R(A).Thus monotonicity is a reasonable property to require of a PWM.
We now show another reasonable property of a PWM. Shiftingof probability from smaller valued payoffs to high valued payoffscan never decrease the value of R, it should increase it or leave itthe same.
Please cite this article in press as: R.R. Yager, N. Alajlan, Probability weighted mBased Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
Assume A = {(aj,pj), j = 1 to n} and B = {(aj,qj), j = 1 to n} are twodecision alternatives, uncertainty profiles. Without loss of general-ity assume the indexing is such that aj+1 > aj, the index is in ascend-ing order. Let i and k be two indexes such that i > k, ai > ak. Assumethe pj and qj are as follows
qj ¼ pj for j–i; k
qi ¼ pi þ D
qk ¼ pk � D
Thus we obtained B from A by moving probability from a smallervalued payoff to a bigger valued payoff. Let us look of the CDF forthese profiles FAðxÞ ¼
Pj;aj6xpj and FBðxÞ ¼
Pj;aj6xqj.
We see that for x < ak we have FAðxÞ ¼Pk�1
j¼1 pj andFBðxÞ ¼
Pk�1j¼1 qj and here for all j in this range pj = qj and hence
FA(x) = FB(x) for x < ak. For ak 6 x < ai we have for k 6 r < i
FAðxÞ ¼Xr
j¼1
pj ¼Xk�1
j¼1
pj þ pk þXr
j¼kþ1
pj
FBðxÞ ¼Xr
j¼1
qj ¼Xk�1
j¼1
pj þ pk � DþXr
j¼kþ1
pj
and hence FB(x) < FA(x) in the rangeFinally in the range x P ai we see FAðxÞ ¼
Prj¼1pj where r P i
and
FBðxÞ ¼Xr
j¼1
qj ¼Xr
j ¼ 1j–i; k
qj þ qk þ qi ¼Xr
j ¼ 1j–i; k
pj þ pk � Dþ pi þ D
¼Xr
j¼1
pj ¼ FAðxÞ
From this we see that FB(x) 6 FA(x) all x and for at least one x,FB(x)6 FA(x) and hence B > SDA and then for any R, R(B) P R(A).
Even the boundary condition 2, which says that if pk = 1 thenR(A) = aK can be seen as kind of stochastic dominance. Considerthe case of an alternative with pk = 1. In the case the associatedCDF has
FðxÞ ¼ 0 for x < ak
FðxÞ ¼ 1 for x P ak
If we have two alternative uncertainty profiles A1 and A2 havingrespectively pk1
¼ 1 and pk2¼ 1 the RðA1Þ ¼ ak1 and RðA2Þ ¼ ak2 . If
k2 P k1, then ak2 P ak1 . Furthermore, here F1(x) = 0 for x < ak1 andF1(x) = 1 for x P ak1 while F2(x) = 0 for x < ak2 and F1(x) = 1 forx P ak2 . Thus here we see that F1(x) ‘‘>’’ F2(x) and hence A2 > SDA1.
As we see in the following the usual expected value is an exam-ple of a PWM. Here with A = {(aj,pj) for j = 1 to n} we defineRðAÞ ¼
Pnj¼1pjaj. Here it is well known that Minj½aj� 6
Pnj¼1pjaj 6
Maxj½aj�. If pk = 1 and all other pj = 0 then R(A) = ak. If, without lossof generally, if we assume that pn = 0 then RðAÞ ¼
Pnj¼1pjaj ¼Pn�1
j¼1 pjaj. Finally we have already shown that if A and B are twoalternatives so that A > SDB then R(A) > R(A) if R is the expectedvalue.
Let us now look at the median. We recall that if A = {(aj,pj), j = ton} and if without loss of generality we assume, aj+1 P aj then
Med(A) = ai wherePi�1
j¼1pj < 0:5 6Pi
j¼1pj. Let us see if this satisfiesour four conditions. First since Med(A) is one of the aj then
Minj½aj� 6 MedðAÞ 6Maxj½aj�
We now see that if pk = 1 and all other pK = 0 thenPk�1
j¼1 pj ¼ 0 andPkj¼1pj ¼ 1 and hence
Pk�1j¼1 pj < 0:5 6
Pkj¼1pj and thus Med(A) = ak.
We further see that if pi = 0 then we havePi�1
j¼1pj ¼Pj
j¼1pj and hence
eans as surrogates for stochastic dominance in decision making, Knowl.
R.R. YagerQ1 , N. Alajlan / Knowledge-Based Systems xxx (2014) xxx–xxx 5
KNOSYS 2817 No. of Pages 7, Model 5G
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ai can’t be the median. Finally the requirement of consistency withrespect to stochastic dominance.
Assume A = {(aj,pj), j = 1 to n} and B = {(aj,qj), i = to n). In the fol-lowing without loss of generality we shall assume that aj are inascending order aj+1 > aj. Assume A > SDB, that is FA(x) 6 FB(x), thenfor any x,
Pi;ai6xpi 6
Pi;ai6xqi. Assume Med(A) = ak, thus
Xk�1
j¼1
pj < 0:5 6Xk
j¼1
pj ¼ FAðakÞ
From the fact that FA(x) 6 FB(x) we see thatPk
j¼1qj ¼ FB (ak) PFA(ak) P 0.5. Thus we see Med(B) 6 ak. Here then if A > SDB and Ris the median then we see that R(A) P R(B). We emphasize in thiscase of median we only have R(A) P R(B) when A > SDB this meansthat the Median is only a weak PWM.
We now look at some other examples of weighted aggregationand see if they satisfy our definition of a PWM. Consider first, thatwe define R so that R(A) is the maximal payoff having a non-zeroprobability. Again assume A = {(aj,pj), j = 1 to n} where aj < aj+1,the aj are in ascending order. Let us see if this satisfies our fourconditions.
(1) Since R(A) 2 {a1, . . . , an} then Minj(aj) 6 R(A) 6Maxj[aj](2) If pK = 1 then all pj = 0 for j – K are zero and hence the only
choice for R(A) is ak
(3) Since we require then R(A) be a payoff with non-zero prob-ability that the value of any element with zero probabilitydoes not effect the outcome.
(4) Assume A = {(aj,pj), j = 1 to n} and B = {(aj,qj),j = 1 to n} withthe aj indexed in ascending order. Assume we have A > SDB,
in this casePk
j¼1pj 6Pk
j¼1qj. Let k⁄ be the smallest index
such thatPk�
j¼1qj ¼ 1. From this it follows that RðBÞ ¼ ak� .
Let k+ be the smallest index so thatPkþ
j¼1qj ¼ 1. From this itfollows RðAÞ ¼ akþ .
We now show that k+ P k⁄. Assume k+ < k⁄ then the1 ¼
Pkþ
j¼1pi PPkþ
j¼1qi < 1 which contradicts our assumption thatA > SDB. Here then we have akþ P ak� . Thus R defined so that it isthe maximal payoff with non-zero probability is a weak PWM.
Consider now the definition R so that R(A) is the minimal play-off having a non-zero probability. Again assume A = {(aj,pj), j = 1 ton} where the aj are indexed in ascending order. Since the proof ofthe first three conditions are similar to the case of max we shalljust focus on the fourth condition showing that if A > SDB thenR(A) P R(B). Let B = {(aj,qj), j = 1 to n}. Again under stochastic dom-
inance,Pk
j¼1pj 6Pk
j¼1qj for all k. Let k⁄ be the smallest index so thatPk�
j¼1qj > 0, that isP1�k�
j¼1 qj ¼ 0 and hence RðBÞ ¼ ak� . From the sto-
chastic dominance conditionP1�k�
j¼1 pj ¼ 0 andPk�
j¼1Pj 6Pk�
j¼1qj.
Thus if k+ is the smallest index for whichPkþ
j¼1Pj > 0 then k+ P k+
and hence akþ P ak� . Thus in this case we have R(A) P R(B) andhence the use of the minimal payoff with a non-zero probabilityis also only a weak PWM.
We now shall consider a class of PWM, which generalizes thepreceding, that we shall refer to as transitional type PWM. AssumeA = {(aj,pj), j = 1 to n} where the aj are indexed in ascending order.Let a 2 (0,1] be a parameter. We shall say define R(A) = ak where kis such that
Xk�1
j¼1
pj < a 6Xk
j¼1
pj
Thus here R(A) equals the ordered payoff which transitions the sumof probabilities of the outcomes from less than a to equal or greater
Please cite this article in press as: R.R. Yager, N. Alajlan, Probability weighted mBased Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
then a Let us see that this satisfies four conditions that make R aPWM.
(1) Clearly since R(A) 2 {a1, . . . , an) then Minj[aj] 6R(A)6Maxj[aj]
(2) If pk = 1 then it is the only index that can transition the sumfrom zero to at least a, that is since
Pk�1j¼1 pj ¼ 0 and
Pkj¼1pj =1
and hence R(A) = ak.(3) If Pk = 0 then
Pk�1j¼1 pj ¼
Pkj¼1Pj and hence R(A)–ak.
(4) Again assume B = {(aj,qj), j = i to q} and assume A>SDB;Pij¼1qj P
Pij¼1pj for all i. Assume RðBÞ ¼ ak� that is k⁄ is the
smallest index for whichPk�
j¼1qj P a. Then if k+ is the small-est index for
Pk�
j¼1Pj P a it must be the case that k+ P k⁄. Inthis situation RðAÞ ¼ akþ and since akþ P ak� we haveR(A) P R(B). Here we see that this type of aggregation aweak PWM.
Actually some of earlier examples are examples of a transitionalPWM. This true for the Median, a = 0.5 for the Max, a = 1 and forthe Min a = 2? 0.
We now consider another related form for R that we refer as ana � b type R. Here we assume 0 < a 6 b 6 1. Again assume A = {(aj, -pj), j = 1 to n)} with the aj indexed in ascending order. Here weobtain R(A) using the following procedure
(1) Let k1 be such thatPk1�1
j¼1 pj < a 6Pk1
j¼1pj
(2) Let k2 be such thatPk2�1
j¼1 pj < b 6Pk2
j¼1pj
We see k1 and k2 are the respective transition indices for aand b and k1 6 k2.
(3) Assume I = {k/k1 6 k 6 k2 and k is an integer} it is the set ofindices between k1 and k2. Let V ¼
Pj¼k2j¼k1
pk and let v j ¼pj
V
for j = k1 to k2 we then define
eans a
RðAÞ ¼Xk2
j¼K1
v jaj
Here we are essentially discounting the smaller arguments witha total probability and the larger arguments with 1 � b portion ofthe probability and then taking an average of the remaining argu-ments. We note that if a = 0 and b = 1 then we get the usualexpected value and if b = a then we get the a transition type PWM.
5. Further examples of probability weighted means
Let us look at the probability weighted OWA operator discussedin [12]. We first recall that an OWA operator of dimension n is amapping OWA(a1; . . . ; anÞ ¼
Pnj¼1wjbj where the wj are the collec-
tion of weights such that wj 2 [0,1] andPn
j¼1wj ¼ 1. Here bj is thejth largest of the ai’s, the argument values [13]. Let k be an indexfunction so that k(j) is the index of the jth largest of the argumentvalues, hence bj = ak(j). We refer to the collection of the wj as theOWA weighting vector. Yager [14] suggested using an aggregationattitude function g:[0,1] ? [0,1] where g(0) = 0, g(1) = 1 and g ismonotonic, g(x) P g(y) if x > y to obtain the OWA weights. Usingthis function we obtain the weights for an n dimension OWAaggregation as
wj ¼ gjn
� �� g
j� 1n
� �for j = 1 to n. Some notable examples of the function g are Linear:g(y) = yj, Power: g(y) = yr with r P0 and step type function:g(y) = 0 for y < b and g(y) = 1 for y P b.
Based on his work on importance weighted OWA aggregators[14] Yager suggests an approach for obtaining a probabilisticallyweighted OWA aggregator. Assume we have n pairs of arguments
s surrogates for stochastic dominance in decision making, Knowl.
6 R.R. YagerQ1 , N. Alajlan / Knowledge-Based Systems xxx (2014) xxx–xxx
KNOSYS 2817 No. of Pages 7, Model 5G
2 May 2014
Q1
(ai, pi) where pi is the probability associated with the occurrence ofthe argument value ai. Then the probability weight OWA operatoris
OWAg ½ðai;piÞ� ¼Xn
j¼1
wjakðjÞ:
Here again k(j) is the index of the j largest argument however herewj = g(Tj) � g(Tj�1) where Tj ¼
PjK¼1pkðjÞ. Here Tj is the sum of prob-
abilities of jth largest arguments.We look at some special cases of g. If g is linear, g(y) = g(y), then
gðTjÞ ¼ Tj ¼Pj
k¼1pkðkÞ and wj = g(Tj) � g(Tj�1) = pk(j). From this itfollows
OWAðai; piÞ ¼Xn
j¼1
pkðjÞakðjÞ ¼Xn
j¼1
pjaj
It is the usual expected value.If we use the step type with b = 0.5, then wj = 1 for j such that
Tj P 0.5 and Tj�1 < 0.5. In this case OWA(ai,pi) is the usual medianvalue.
If we let b ? 0 in the step type then OWA(ai,Pi) is the maximumai having a non-zero probability, it is essentially the Max. If we letb ? 1 then this step type is the minimal ai having a non-zeroprobability.
Let us now see if generally the OWAg[(ai,Pi)] satisfies therequired conditions for it to a PWM. For simplicity in the following,and without any loss of generality, we shall assume that the ai havebeen indexed in ascending order, ai+1 > ai. First we see that OWAg((-
ai, piÞÞ ¼Pn
j¼1wjakðjÞ where wj = g(Tj) � g(Tj�1) and Tj ¼Pj
k¼1pkðkÞ.Here k(j) is the index of jth largest argument. We note that herek(j) = n + 1 � j because of the indexing. It is easy to see thatwj 2 [0,1] since Tj P Tj�1 and g is monotonic. Furthermore sincePn
j¼1wj ¼Pn
j¼1ðgðTjÞ � gðTj�1ÞÞ ¼ gðTnÞ � g(T0). Since Tn = 1 andT0 = 0 then g(Tn) = 1 and g(T0) = 0 we have
Pnj¼1wj ¼ 1. Thus the
wj are a classic set of weights and hence Minj½aj� 6Pn
j¼1wjakðjÞ 6
Maxj½aj�.Consider now the second condition, assume pk = 1 and all other
pi = 0. Here then with j = n + 1 � i, let us denote r = n + 1 � k. Wesee Tj = 0 for j < r and Tj = 1 for j P r. From this we get g(Tj) = 0 forj < r and g(Tj) = 1 for j P r. Thus we have wj = 0 for j – r andwr = 1. Thus we see in this case OWAg(ai, Pi) = ak(r) = ak.
Consider now the third condition, assume pk = 0 and assumek(r) = k. We see that here wr = g(Tr) � g(Tr� 1) but Tr ¼
Prj¼1pkðjÞ
and Tr�1 ¼Pr�1
i¼1 pkðjÞ since Pk(r) = Pk = 0 then Tr = Tr�1 and wr = 0.From this we see that zero weight is assigned to aK, and hence ak
has no effect in the aggregation.Now we consider the fourth condition. Assume B={(ai,qi)} is
another alternative such A > SDB thusPk
i¼1qi ‘‘>’’Pk
i¼1pi for all k.Here RðAÞ ¼
Pnj¼1wjakðjÞ with wj = g(Tj) � g(Tj�1) where Tj ¼Pj
Please cite this article in press as: R.R. Yager, N. Alajlan, Probability weighted mBased Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
Similarly we get RðBÞ ¼ �gðS0Þakð1Þ þPn�1
j¼1 gðSjÞðakðjÞ � akðjþ1ÞÞþgðSnÞakðnÞ. Again since S0 = 0 and S1 = 1 we have RðBÞ ¼
Pn�1j¼1 gðSjÞ
ðakðjÞ � aðkþ1ÞÞ þ akðnÞ
We see that the only difference between R(A) and R(B) are thatthe positive terms (ak(j) � ak�1)) which in R(A) are multiplied byg(Tj) and those in R(B) are multiplied by g(Sj). Let us now compare
Tj and Sj. We see Tj ¼Pj
k¼1PkðkÞ ¼Pj
k¼1pnþ1�k ¼Pnþ1�j
i¼n pi ¼Pn
i¼nþ1�j
pi. We recall thatPn
i¼1pi ¼ 1 ¼Pn�j
i¼1pi þPn
i¼nþ1�jPi ¼Pn�j
i¼1pi þ Tj
hence Tj ¼ 1�Pn�j
i¼1pi. Similarly we get that Sj ¼ 1�Pn�j
i¼1qi. As aresult of the stochastic dominance relationship between A and B
we havePn�j
i¼1qi PPn�j
i¼1pi for all j and at least one satisfies >. Thuswe have Tj P Sj for all j and for at least one j we have Tj > Sj. Becauseof the monotonicity of g we have for all j, g(Tj) P g(Sj) and we haveR(A) P R(B). Furthermore if g is strictly monotonic, g(y) > g(x) ifx > y then in this case we have R(A) > R(B). Thus the probabilityweighted OWA operator is a PWM if g is strictly monotonic other-wise, if g is just monotonic, then it is a weak PWM.
Another general class of means are the power means defined in[15] as
Mða1; . . . ;anÞ¼1n
Xn
i¼1
xri
!1=r
for r 2 ½�1;1�; r–0 and all ai P 0
Among the special members of this family are the following
e
r
ans as surrogates fo
Form
r stochastic dominance in de
Name
1
1n
Pni¼1aj� �
Average
-1
nPn
i¼11ai
Harmonic mean
2
1n
Pni¼1a2
i
� �1=2
Root mean square
0
(Pai)1/n Geometric mean �1 Mini(ai) Min 1 Maxi(ai) Max
Using this class we now suggest a class of PWM. If A = {(ai,pi) fori = 1 to n} we define RðAÞ ¼
Pni¼1pia
ri
� �1=r . We now show that thishas the four required properties of a PWM.
(1) Let aMax = Maxi[ai] and aMin = Mini[ai]. We see
RðAÞ 6Xn
i¼1
piarMax
!1=r
6 arMax
� �1=r6 aMax
RðAÞPXn
i¼1
piarMin
!1=r
6 arMin
� �1=r P aMin
(2) If pk = 1 and all other pi = 0, i – k then RðAÞ ¼ ark
� �1=r ¼ ak
(3) If pk = 0 then RðAÞ ¼Pn
i ¼ 1i–k
piari
0B@1CA
1=r
(4) Assume B = {(ai,qi), i = 1 to n}. For simplicity in the followingwe will assume the indexing of the ai is such that ai < ai+1.Assume that A > SDB j thus for all k we have
Pkj¼1pj 6Pk
j¼1qj and for at least one k we havePk
j¼1pj <Pk
j¼1qj.
We shall first consider the case where 0 < r <1. Let us definedi = (ai)r. We note in this case with r > 0 then the ordering of di isthe same as the ordering of the ai, di < di+1. Consider nowRðAþÞ ¼
Pnj¼1pjdj and RðBþÞ ¼
Pnj¼1qjdj each of these is a simple
case of expected value. Furthermore we here have already shownin this case that if A > SDB then R(A+) > R(B+). Additionally we haveR(A) = (R(A+)1/r and R(B) = (R(A+)1/r and since R(A+) > R(B+) then
R.R. YagerQ1 , N. Alajlan / Knowledge-Based Systems xxx (2014) xxx–xxx 7
KNOSYS 2817 No. of Pages 7, Model 5G
2 May 2014
Q1
R(A) > R(B). We note in the case where r ? / then all we can showis that R(A) P R(B). In a similar manner we can show that if0 > r > �1 then R(A) > R(B) and if r ? � / then all we can showis that R(A) P R(B).
A related family PWM can be based on the quasi-arithmeticmean [15]. Here we let A = {(ai,qi), i = 1 to n}, where the ai notrestricted to be non-negative, ai 2 [�1,1). Again we shall assumethat the ai are indexed increasing order, ai+1 P ai. In addition let g:[�1 ,1] ? [�1,1] be a mapping of the argument values. Herewe define R(A) = g�1(R pig(ai)) where g�1 is the inverse of g. Wecan show that if g is a strictly monotone increasing function, forx > y that g(x) > g(y) or strictly monotone decreasing function, forx > y that g(x) < g(y), that R(A) is a PWM. If we replace g(x) > g(y)or g(x) < g(y) by g(x) P g(y) or g(x) 6 g(y) then R is a weak PWM.
6. Aggregating PWM
We now provide an interesting general result about the PWM.Assume R1, . . . , Rm are a collection of PWM and let w1, . . . , wm
be a set of weights, wi 2 [0,1] andPm
i¼1wi ¼ 1. Then R defined sothat for any uncertain alternative A is RðAÞ ¼
Pmi¼1wiRiðAÞ is also a
PWM. Let us see that R(A) satisfies the four required conditions.Again here we let A = {(aj,pj),j = 1 to n} with the aj ordered inascending order, ai < ai+1.
(1) We see for each Ri we have a1 6 Ri(A) 6 an. From this we seeRðAÞP
Pmi¼1wiai P a1 and RðAÞ 6
Pmi¼1wian 6 an.
(2) If pk = 1 then each Ri(A) = ak and hence R(A) = ak.(3) If pk = 0 and if each of the Ri is indifferent to the value of ak
then R is also indifferent to this value.(4) Finally assume A and B are two alternative so that A > SDB.
Then since for each i we have Ri(A) P Ri(B) and henceRðAÞ ¼
Pmi¼1wiRiðAÞP
Pmi¼1wiðRiAÞÞP RðBÞ. Furthermore if
each Ri is such that Ri(A) > Ri(B) then R(A) > R(B).
We also observe that if R1, . . . , Rm are a collection of PWM thenR(A) = Maxj[Rj(A)] is a PWM where A = {(a1,pj)}.
(1) Since each Mini[ai] 6 Rj(A) 6Maxi[ai] then Minj[ai] 6Maxj
[Rj(A) 6Maxi[ai].(2) If pK = 1 then for each Rj(A) = aK and we easily see that
Maxj[Rj(A)] = aK.(3) If pK = 0 then each of the Rj(A) is independent of ak and hence
Maxj[R(A)] is independent of the value of ak.(4) Finally assume A and B are two alternative so that A > SDB
then for each j, Rj(A) P R(B) and it follows that Maxj[-R(A)] P Maxj[R(B)]. Here R(A) = Maxj[Rj(A)] is a weak PWM.
We can easily show that R(A) = Minj[Rj(A)] is also a weak PWN.More generally we can show that this holds for mean operator
[15]. We recall that a mean operator M is an bounded andmonotonic aggregation operator M(x1, . . . , xq) = x so that
(1) Mini[xi] 6M(x1, . . . , xq) 6Maxi[xi].(2) M(x1, . . . , xq) P M(y1, . . . , yn) if xi P yi for all i.
It is clear a mean is idempotent, M(xj , . . . ,x) = x.
Please cite this article in press as: R.R. Yager, N. Alajlan, Probability weighted mBased Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.04.024
Assume Rj j = 1 to q are a collection of PWM and letR(A) = M(R1(A), . . . , Rq(A)) then R(A) is always a weak PWM. Formany cases of M it is a PWM.
7. Conclusion
We discussed the role of stochastic dominance in comparinguncertain payoff alternatives. However, we noted the fact that thisis a very strong condition and in most cases a stochastic dominancerelationship does not exist between alternatives. This required usto consider the use of surrogates for stochastic dominance to com-pare alternatives. Here we provided a class of surrogates that wecalled Probability Weighted Means (PWM). These surrogates arenumeric values associated with an uncertain alternative and assuch comparisons can be based on these values. A PWM is consis-tent with stochastic dominance in the sense that if alternative Astochastically dominated alternative B then its PWM value is lar-ger. We looked at a number of different examples of probabilityweighted means.
Acknowledgement
Ronald Yager’s was supported by an ARO Multidisciplinary Uni-versity Research Initiative (MURI) Grant (Number W911NF-09-1-0392) and by the ONR grant for ‘‘Modeling Human Behavior withFuzzy and Soft Computing Methods‘‘, award Number N00014-13-1-0626. The authors would like to acknowledge the support fromthe Distinguished Scientist Fellowship Program at King SaudUniversity
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