Multicriterial interval choice models

INFORMA TION SCIENCES 80,254 1 ( 1994) 2s

Multicriterial Interval Choice Models

FUAD ALESKEROV

Institute of Control Sciences, 65, Profsqyuznaya St., 117806 Moscow, Russia

ABSTRACT

The multicriterial choice models are considered based on interval estimations of options. Several new rules of choice are introduced, and the properties of such rules are studied.

1. INTRODUCTION

In multicriterial choice, determination of accurate criteria-based option estimates presents difficulties such as in analysis of development forecasts. Nevertheless, interval option estimation is quite possible within the framework of the Lute model for the one-criteria1 problem. This approach needs choice rules generalizing the well-known Pareto and Sleuter multicriterial choice rules.

This paper is devoted precisely to this problem. Section 2 discusses the one-criteria1 problem. Section 3 generalizes the multicriterial rules to the case of interval option estimates. Section 4 then generalizes the rules to the case where options are chosen so that they are optimal with respect to a family of criteria from the criteria set under consideration, rather than to the entire set. The characteristics introduced in rules in each section are formulated in terms of the choice functions they generate.

2. ONE-CRITERIAL CHOICE

Let A={x,,..., x,,J be finite set of options, m 2 2 and let scalar criterion cp(.> be defined on the elements of A. For the sake of definite- ness, assume further that the criterion pc.1 is maximized.

The choice rule is defined, VXcA, X#@, as

C(X)={y~XIl3x~X:cp(x)>cp(Y)}, (1)

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26 F. ALESKEROV

with C(X) for the set of chosen options. According to (l), the options with maximal cp(.> are chosen.

The rule (1) can be rewritten equivalently as

C(X)={y~XltlX~X:cp(y)~;(~)}. (1’)

Let us note that the choice rule is defined not only for the entire set A, but also for any of its nonempty subsets.

Within the framework of this approach, the rule (1) defines choice function C(.>, and choice procedures may be studied for rationality in terms of deformations of sets X from A.

As was shown by Arrow [S], the constancy condition K: VX, X’: X’ c_X, C(X) nX’ #J3 * C(X’> = C(X) nX’ is necessary and sufficient for representability of an arbitrary choice function such as (0, that is, for any function C(e) there exists a criterion cp(.> such that, based on it, choice of maximal options on any X coincides with C(X) if and only if C(X) satisfies the condition K.

Lute [B] suggested a generalization of the notion of criteria to the case where an interval of values [q(x)- E(X), q(x) + E(X)] involving the unknown so(x) corresponds to each option x from A, rather than the exact value of p(x). In doing so, he assumed that E = const > 0. The choice rule (1) can be written here, VXcA, X#O, as

EXAMPLE 1. Consider the case of A = {x, y, z}, where option estimates are shown in Figure 1. The choice C(A) by (2), then, includes the options y and z, and not x, because q(y) - E(Y) > ye + E(X). The rule (2) can be readily seen to be a direct generalization of the extremizational choice rule (1): if E(X) is equal to 0 for all x, (2) boils down to (1).

The situation where precise measurement by criterion is impossible provided by archaeology, where radiocarbon dating, for example, always involves error (e.g., 400 + 50 yr). The choice models of the kind of (2) were investigated by Mirkin [91, Fishburn [6, 71, and Aleskerov [2].

V(Y)-F:(Y) P(Y)+R(Y)

r x 1 r Y I >

I z 1

Fig. 1

MULTICRITERIAL INTERVAL CHOICE MODELS 27

Let us introduce some characteristic conditions for choice functions that will be required in the following text.

DEFINITION 1. The function C(o) will be said to satisfy the following conditions:

1. Heritage (H) if VX, X’: X’ GX - C(X’> 3, C(X) nX’. 2. Concordance (0 if VX’, X”: C(X’> n C(X”) c C(X’ UX” ). 3. Outcast (0) if VX, X’: X’ cX\C(X) * C(X\X’> = C(X).

These conditions were used by Sen [121. The necessary and sufficient representability condition for the choice

function simultaneously satisfying H, C, and 0 as (2) was given by Fishburn [7] in the following form: denote R(X) =X\C(X); then, VX, and X, from A, C(X,)nR(X,)#pl~C(X,)nR(X,)=a.

3. MULTICRITERIAL INTERVAL CHOICE

Prior to presenting the results concerning multicriterial interval choice, we present facts known to the theory of usual (with exact estimates) multicriterial choice theory. The vector criterion cp(.> = (cp,(*>, . . . , cp,(*)), II > 1, is assumed to be defined on A. Denote the criteria index set as N={l,..., n). Here, the generalized rule (1’) can be written, VXcA, as

C(X) ={yEXlVxEX & ViENq$(Y) >-cp,(x)), (2’)

that is, the option y is chosen if its estimates by all criteria from N are not less than those of any other option.

Rule (1) can be generalized, VXcA, as

C(X)={yEXl~3xEX:vi~Ncp,(x)~~;(y)}, (3)

that is, the option y is regarded as chosen if there exists no other option x “superior” by all criteria.

Another possible generalization of (21, VXG~, is

c(x)=(yExI ~3xEX:(viENcp,(x)~cp;(y)

& Ji,=N: cp,,(x> > cp,,(Y)))Y (4)

that is, the option y is regarded as chosen if there exists no other option x with all estimates, but possibly one, equal to those of y and one estimate better than that of y.

F. ALESKEROV 28

!x , 6

I * I I

Fig. 2

fP,(')

Rule (3) is usually referred to as the Sleuter rule or as the weak Pareto rule, and (4) is referred to as the Pareto rule.

EXAMPLE 2. Figure 2 shows positions of estimates of options from A={+.., x9) in the criteria1 space (cp,, qoz). The choice by (2’) on A is empty, that is, C(A) =fl, and one can easily see that it can be nonempty only if there is an option in A with maximal estimates by both criteria: C,,(A) = ix,, . . . , x6} by the Sleuter rule and C,,,(A)=&, xj, XJ by the Pareto rule.

As follows directly from the definitions (2)-(41, C,(.) c Cs,(.> c CPar(.), where C,(e) is the function generated by (29, but the choice C,(e), as a rule, is empty.

THEOREM 1 (Aizeman andAl&erou (II). The condition HnCf10 is necessary and suficient for the choice function Cc.1 to be representable as (3) and (4).

Because representability of C(e) in different forms (3) and (4) depends on the same characteristic condition H n C n 0, the following important corollary holds.

COROLLARY 1. For the choice function C,,(.), on the n-criteria vector Q there exists a k-criteria vector I) such that the Pareto choice on I) coincides with C,,(.>. Inversely, for the choice function C,,,(q) on the k-criteria vector IJJ there exists an n-criteria vector Q such that the choice C,,,(e) on $ coincides with C,,(e) on Q.

This result was proved directly by Aleskerov, Litvakov, and Zavalishin 141.


NOW we pass to the “interval” models. Let, for each x, an n-criteria vector cp and n error functions q(x), i = 1,. . . , n, be defined, that is, let n intervals [ cp,(x> - EJx), ~Jx> + E,(x)] = [gi(xl, hi(x)1 be assigned to each X, where g,(x) = qi(x) - q(x), hi(x) = 4pi(x) + q(x). Similar to (2’), another rule can be defined, VXGA, as

C(X) ={yEXlVxEX 8~ ViENg,(y) >h,(x)}, (2”)

that is, the option y is included into choice if its lower estimates by all criteria are not smaller than the upper estimates of any other option X.

Similar to (3), the following rule is introduced:

C(X)={yEXI ~3xEX:viENgj(x)>h,(Y)}, (3’1

that is, the upper estimates of chosen option cannot be less by all criteria than the lower ones of any other option X.

Rule (4) is transformed as

& %:gJx) >JQ~~(Y))). (4’)

The criteria1 interval estimates allow one to introduce some more rules, one of which is considered below:

C(X) =(yExI 73 xEX:(viENh,(x)>lzhi(y)

& Xxi,,(x) >hioY)). (5)

Rule (5) dwells on the following fact: since here the exact value of q;(+> is unknown, it is only natural to assume that if the maximal estimate of x with allowance for error does not exceed the maximal estimate of y by all criteria and if, by one criterion at least, the maximal estimate of x is less than the minimal estimate of y, the option x is “worse” than option y.

Rules (3’) through (5) were introduced by Aleskerov (1983).

EXAMPLE 3. Figure 3 depicts the interval estimates of options from A in the criteria1 space (cpl, (~~1. The choice by (2”) on A may be seen to be empty. The choice by (3’) involves all options but x5, that by (4’) has

30 F. ALESKEROV

. . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I 8 . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , .,....,................. I . . . . . . . . . . . . . .._........

i f I i

Fig. 3

options xi, x2, and x3, and, finally, rule (5) provides a choice consisting of x2 and xg.

Denote the choice functions generated by G!“), (3’1, (4’), and (5) by C,(*>,. . . , C,(a), respectively. It follows directly from the definitions of these rules that C,(.) c C,(e) G C,(a) G C,(e).

Let us study the conditions to be satisfied by C(s) to be representable as (3’), (4’), and (5). The case of (2”) is disregarded, because the choice C,(a), usually, is empty.

THEOREM 2. The condition H n C fl0 is necessary and sufficient for the choice function to be representable as (3’1, (4’1, or (5).

The proof of this and other theorems can be found in the Appendix. Corollary 2 follows from Theorem 2.

COROLLARY 2. For any function C,(a) representable as (3’) on n,-criteria vector Q and n3 functions ~~(‘1, there exists an n,-criteria vector @ and n4 functions S,(e) such that choice C,(v) by (4’) on JI and Sic*> coincides with C,(e). Inversely, for any function C,(e) on n,-criteria vector Q and functions S,(e), there exist n3 criteria Q andfunctions q(e) such that C,(a) = C,(a), where C,(o) is the choice by (3’) on Q and q(e).

Similar relations take place for other combinations (3’)-(5) and (4’1-6). Obtain from Theorems 1 and 2 the following corollary.


COROLLARY 3. For any function C,(s) [or C,(.), C,(*)l on n-criterkd vector Q and error functions e;(e), there exist k criteria Q, and the Pareto choice on JI coincides with C,(e) [or C,(*>, C,(->I. Znver.sdy, for any Pareto choice function on the k-critical vector +, there exist an n-criteriql vector Q and n error fimctions q(a), such that the function C&*) [or C&.1, C&.)1 coincides with the Pareto choice function.

Similar assertions exist for the Sleuter rule. By virtue of the equivalence of different models, in the following text,

consideration will be given only to rule (3’), which will be called, for the sake of simplicity, the Pareto rule.

4. MULTICRITERIAL CHOICE: A GENERALIZATION

Let Z be a subset of the set N of all criteria, i.e., ZCN. For I, (3’) may be rewritten as

C(X)={YEXI 73xEX:ViEZ gi(x)>h,(y)]. (6)

This rule may be called the partial Pareto rule. Assume that several subsets Z 1,. . . , Zk are defined on N. This brings up the question of determining the optimal option for this case.

Consider the following two generalizations of (6) to this case. The first generalization can be constructed as

C(X)={~EXI~Z~, j=l,...,k

s.t.(73xEX:ViEZj g,(r) >h,(y))}, (7)

that is, the option y belongs to a choice from X if it is Pareto-optimal at least on one of the criteria families from {Z,}:.

Another generalization is constructed as

C(X) ={y~XltfZ~, j=l,...,k

s.t.( 73xEX:ViEZj g,(x) >h,(y))}, (8)

that is, the option is included into choice if it is Pareto-optimal in each criteria family from {Zj};“.

32 F. ALESKEROV

TABLE I

Criteria

Options h,(.) g,(.) h,(.) g*(.) h,(.) gd.1

X 4 5 4 4 3 4 Y 1 2 5 5 5 7 z 3 3 3 3 1 2

EXAMPLE 4. Consider criteria1 estimates of the options from A = {x, y, z} by the interval criteria [g,(s), hi(*)], i = 1, 2, 3 (Table 1). The options x and y are Pareto-optimal on N. If the set {I,) makes up the criteria1 family I, ={l, 21, Z2 ={l, 3}, and {Z, ={2, 31, the choice on A by (8) has only the option y that is Pareto-optimal on any two-criteria subset.

The choice by (7) on the same set (4) includes the same options. Attention is now drawn to the fact that if in the second interval

criterion [g,(e), h,(*)] the estimates of x and y are interchanged, X, y, and z become Pareto-optimal options on N, and the choice by (7) on {Ii}, where I, = {l}, Z2 = {2}, and Z, = (3) are x and y, is empty by (8).

As follows directly from the definitions of (7) and (81,

where C,(e) and C,(e) are the functions generated by (7) and (8) on the same set (Ii}.

Let us explore the following two questions: What are the conditions for the (S&generated choice to be nonempty under arbitrary estimates of options by the criteria, and what are the conditions satisfied by the choice functions generated by (7) and (8)?

Notice that for (7), investigation of nonemptiness is senseless because this choice is always nonempty.

The first question is answered below.

DEFINITION 2. Let a criteria family /= {Z& be defined. Call a subset 3’ 0 a cyclic family if

The element-minimal cyclic family Jr0 will be called minimal cyclic family, and the number of its elements will be denoted by p,, that is, p0 = card(Y).


LEMMA 1. Zf I Al >, pO, there ahvuys exist estimates of options [h,(e), gi(.)], i=l , . . . , N, such that C,(A) = 8.

LEMMA 2. Zf JAI <pO, then VX C,(X)#fl.

The result about the conditions for nonempty choice by (8) follows from these lemmas.

THEOREM 3. At arbitrary estimation of options by criteria and when I AI > p, holds, th e choice generated by (8) on some set {I,): is nonempty if and only if the criteria families I;, j = 1,. . , , k, satisfi the condition

;I Ij+fl, (9) i=l

while under I Al < pu the choice generated by (8) is always nonempty.

Now let us investigate the properties of choice functions generated by (7) and (8).

THEOREM 4. The choice function C(e) is generated by (7) for any set {Z,} if and only if it satisfies the characteristic conditions H n 0 and it satisfies the condition H f? C n 0 only, where the set {Ii} has a single set I.

The function C(e) is generated by (8) for any set (Zj} if and only if it satisfies the characteristic conditions H n C and it satisfies the condition H n C n 0 only where the set {Z,} contains the single set N.

As far as is known to the author, the following generalization of the multicriterial choice model has never been considered in the multicriterial choice theory. First, introduce the notion of dominating set for the option y in A by [gi(*>, hi(*)I:

Q(Y) = {x-k(x) %(Y)}.

Obviously, the dominating set of y in X by [g,(*), hi(.)] is XnD,(y), and one can readily see that the Pareto rule can be rewritten as

c(x) = (yt~l n [xnq(r>] =O). isN

34

or as

F. ALESKEROV

Its generalization can be now written as

that is, y can be included in the choice from X even if there are p options at most having estimates better than estimates of y by the criteria from N.

We refer to this rule as p-Pareto rule, and to the options chosen by it as p-optimal Pareto options.

In what cases can this rule be used in spite of the fact that it chooses options such that there are other options in X with even better estimates? Assume that option estimation by criteria is so inaccurate that even the interval estimates do not guarantee a confident choice of true Pareto-optimal elements. It is only natural to introduce, then, a threshold p ensuring that the choice of p-optimal Pareto options misses no true Pareto elements. Such situations are possible if, for example, accurate estimation of options by criteria is rather costly. Understandably, the mean number of options chosen by this rule exceeds that by (3’). This rule, however, can be used at the first stage to reduce the total number of options; the remaining options would be estimated by costlier procedures.

This rule may be generalized as

C(X) = {y EX(Card( n [XnD,(y)]) a,i}, iGI

that is, for a set of Z criteria, p’-optimal options are chosen. Denote the choice function generated by (10) as C(X, I, p’). Attention is drawn to the fact that if p’= m - 1, with m for the number

of options in A, C(X, I, m - 1) =X for all X. The definition of (10) can be extended in the same manner to the case of p’ < 0. Here, we assume that C(X, I, p’) =@ for all X. Therefore, in the following text, consideration always will be given to nontrivial cases of 0 <p <m - 1.


Now we write the most general form of the rule under study. Let a set {Z,}: be defined of criteria sets Z, GN. The rule

C(X) = (J 0 C(x, 'j, P’,) (11) j=l iEI

I

is called the generalized multicriterial interval choice rule. This rule chooses the options that are p-optimal Pareto options on one

criteria set Zj at least. Now we study the conditions satisfied by the choice function generated

by (11).

THEOREM 5. The choice function is generated by (11) iff it satisfies the heritage condition H; it satisfies H fl0 ifffor anyj, card(Z,) = 1, and for k = 1 and p’l= 0, Cc.1 is generated by (11) ifs it satisfies the condition H n C n 0.

The graphics obtained by numerical modeling of the average value of chosen options depending on the cardinality of the set A when card(A) = 10, 20, 30, 40, and 50 are represented in Figures 4-6. The number of criteria is 5, and the options estimates are assumed to be integer numbers uniformly distributed over the interval [O, 1001. In Figure 4 the bold line refers to Sleuter’s rule, dashed line refers to the rule C’(A) = ll I E9 C(A, I, O), where 9 is all possible sets of criteria of the cardinality 4, i.e., card(Z) = 4, the chain-dotted line corresponds to the rule C”(A) =

30 .

40 20 JO 40 =' cd(A)

Fig. 4

Fig. 5


n , t_Y C(A, I, l), where 3 consists of the same sets of criteria. The graphics on the Figure 4 are obtained under assumption of the pointwise estimates of options, i.e., Vx, Vi, g,(x) = hi(x).

The graphics in the Figure 5 are obtained for the same rules, but under assumption of the interval estimates of options and the constraint Vx, Vi, g,(x) -h;(x) G 10. The graphics in Figure 6 meet the constraint Vx, Vi, g;(x) -h;(x) < 20.

Let us consider these graphics. In the case of pointwise estimates (Figure 4) when lA(=30, one can obtain IC,,(A)I = 23, IC’(A>I =5, and K”‘(A)1 = 10, i.e., the number of chosen options for the rule C’(e) is approximately 5 times less than for Sleuter’s rule. If the estimates are given in interval form, then the number of chosen options naturally decreases. Thus for the case shown in Figure 5 when (A( = 30, we have K’,,(A)] = 26, IC’(A>I = 8, and IC”(A>I = 14. Correspondingly, when the range of interval increases, then for IAl =30 one can see ICs,(A>I -27, ICY’(A)\ ~11, and IC”(A)( = 19.

5. CONCLUSION

Note that these rules have been obtained by Aleskerov [3] within the framework of the axiomatic approach for the general case where option “estimates” are represented as weak-order binary relations, which, for the criteria1 case, correspond to exact option estimation by criteria.

It might be well to note that the idea of choosing Pareto-optimal elements by criteria that are Pareto-optimal on a family of criteria sets from N, rather than on the entire criteria set N, was suggested by Nogin (see Podinovskiy and Nogin [ll]). As follows from Aleskerov [3], these rules are only possible within the framework of a certain axiomatic system.

As one can see from Figure 4, these choice rules are also of interest because they allow one to avoid the well-known demerit of Pareto-like procedures in multicriterial problems where the number of options in the chosen set is too large. With the appropriate choice of parameters p’ and criteria families {Z,), procedures like (8) or (11) would reduce the number of chosen options.

APPENDIX

Proof of Theorem 2. (a> First we demonstrate that one can construct, by the criteria1 estimates of options [sic.), hi(.)], binary relations enabling the choice of nondominant options coinciding with the initial functions defined

38 F. ALESKEROV

by rules (3’)~(5). Using the option criteria1 estimates, construct the following three binary relations &, &, and &:

X&Y w (Vi EN g,(x) >&(y)),

xP4Y e (ViEN gi(‘) ahi & 3io:gi,(x) >hi,(Y)),

Consider the three choice functions i = 3, 4, 5:

It follows directly from the definitions that the functions C,(*>-C,(e) coincide with their corresponding functions Cp,C*>-Cp,<*>.

(b) Demonstrate that the relations &-ps are irreflexive and transitive. Assume that x&x, that is, Vi EN cp,(x> - E,(X) > cp,(x> + E&X), which is

possible at E&X) < 0. Irreflexivity of the relationships fi4 and & can be shown similarly. Demonstrate transitivity of &. Let x&y and y&z, that is, let ViEN

pi(X) - Et(X) > pipi + Ei(y) and Cpi(y> - Ei(y) > vi(z) + Ei(t). By summing these inequalities, obtain that &) - E,(X) > cp,(z> + ~~(2) + 2~,(y), that is, cp,(x> - EJX) > y+(z) + ei(z). Therefore, x&z.

Transitivity of pq and & is proved similarly. (c> According to the generalized Sen theorem (Aizerman and Aleskerov

[ll>, the functions Cp,<.>-Cp,<*> satisfy H n C n 0. (d) Since C,(e), i = 3, 4, 5, are choice functions on the irreflexive

relation pi, Ore’s [lo] theorem can be used, leading, in the terms of this paper, to the following result: for any function Cp$*>, there exists on the irreflexive relation pi a criteria vector cp such that the Pareto choice on it coincides with the initial function C,,C*>.

(e> Consider now the criteria vector cp and choice by the Pareto rule on it.

For pointwise criteria1 estimates, the Pareto rule is, obviously, a special case of rules (4’) and (5).

(f> As follows from Corollary 1, there exists, for the choice function defined by the Pareto rule on some criteria vector cp, a criteria vector + where Sleuter choice coincides with that by the Pareto rule on cp, and since for exact option estimates, (3’) coincides with the Sleuter rule, Theorem 2 and Corollaries 2 and 3 are proved. n

Proof of Lemma 1. Let there be p, sets Zr, . . . , Z,,, in the minimal cyclic set. Since n;:, Zj =fl, a total of 2 p. - 1 relations of these elements and


their negations is possible. Consider the index set (1, 2,. .., p,> and construct its cyclic permutations ( p,, 1, 2,. . . , p. - l>, ( p, - 1, pO, 1, 2,. . . , p, -

21,. . . ,a 3,. . . , PO - 1, PO, 1). Now,consider sets I, nZ,n +.. n&“, ZpOnZln ... nZ,o-,n~po_,,...,Z2n

zj n **. n Zp, n I,, generating p, combinations of the sets I,, . . . , Zp, involving precisely one complement of each of these sets. Assume now for each criterion ~EZ~+~ nzk+2n a.- n&n ... nZ,_, nfk that &,+,)> cpi(x,+,) > .‘. > ‘pi(X,_,)> Cp(X~).

Consider the sets Z = Z; nZ; n ... nZ&, where Z; is either Zj or Z,, the number of-sets involved “negatively” m i being more than 1. For any profile i E I, construct now the criteria1 estimates cpi(Xj> > qi(x,l, if Zj and Z, belong to I’ and j <k. Let v be the maximal number of sets involved positively in i, that is, in the form Zr and not I,. Assume that q$x,) > (P(x~), where Ik E I’, and that 4, Zk E Z for any cpi(Xj>= cpi(X,).

Now we look at the fact that cpi(xj> > cpi(xj+ r) for Vi E Z, and, therefore, C,(A) =A\(xj+,}, where C,(A) is the set of Pareto elements on the criteria set Zj. Considering in (8) the special case of exact estimates (Vx hi(x) =gJx>>, obtain then that

Lemma 1 is proved. m

Proof of Lemma 2. Assume on the contrary that 3X: C,(X) =8. With- out loss of generality, assume that n , E9D,IIJ C,(X) = {xl and C,s(X) = {y). Then cpi(y>> q,(x) for tli~l~. On the other hand, there exists I’ wO, I’ f Z, such that Vi E I’, q+(x) > cpJy), because, otherwise, C&x> 3y holds.

Therefore, I’ n Z, = 0, which contradicts the minimality assumption for 7. Lemma 2 is proved. n

Proof of Theorem 3. Let I Al 2 p,. Then it follows from Lemma 1 that criteria1 estimates are possible such that C,(A) =O, whence (9) is a necessary condition.

Let now n f= I Zj #fl, that is, there exists at least one criterion i belonging to all criteria groups Zj, j = 1,. . ., k. Then, the i-maximal elements belong to the Pareto elements on any group Zj, whence the choice C,(X) is nonempty for all X.

The second part of Theorem 3 follows immediately from Lemma 2. Theorem 3 is proved. n

Proof of Theorem 4. The second parts of each of the assertions in Theorem 4 follow from Theorem 2.

40 F. ALESKEROV

We prove the first part of the first assertion. Introduce a choice function C,(a), i = 1,. , . , k, such that

C,(X) ={yEXl -13xEX:ViEzj g,(x) >/2,(y)}.

The choice function C(a) generated by (7) is representable as C(.) = t-l I”= i C,(e). Now we see that the choice function C,(e) is a Pareto choice function, i.e., it satisfies the conditions H f’ C n 0 according to Theorem 2, and that the union of such functions belong to the domain H n 0 by virtue of Theorem 2 presented in Chapter 5 of the book by Aizerman and Aleskerov [ 11.

According to Theorem 6 of Chapter 6 of [l], any function C(s) belonging to the domain H fl0 is representable as the union of functions from Hn Cn 0 and, in particular, as that of functions C,<.> by virtue of Theorem 2.

Now, we prove the first part of the second assertion. Obviously, the function C(s) generated by (8) is representable as C(*> = n /“=, C,(.> where, as mentioned earlier, C,(o) E H n C n 0 and, according to Theorem 5 in Chapter 6 of [l], the intersection of functions from H n C n 0 belongs generally to H r? C.

The inverse statement follows from Theorem 5 in Chapter 5 of Aizer- man and Aleskerov [l] and Theorem 2 of this paper. Therefore, Theorem 4 is proved completely. n

Proof of Theorem 5. First, we prove the following lemma.

LEMMA A.l. Rule (10) generates the choicefunction Cc., I, p’>, belonging generally to H n 0, and only in the case of p’ = 0 does the choice function Cc., I, 0) belongto HnCnO.

Proof of Lemma A.]. The case of p’ = 0 is an evident consequence of Theorem 4.

(a) Let now 0 <p <m - 1. Let, for some X, x E C(X) and x EX’ cX. Then, p’ > card( fl 1 E ,[ Xn Q(x)]> a card( n i E ,[ X’ n DJx)]) and, thus, x E C’(X), that is, the condition H is satisfied.

(b) Let x E C(X) and y, z GG C(X), which means that

p’>card( jcl [Xnoi(x)]). p’<card( ,?, [XnD,(Y)]),

p’<card( n [XnD;(Z)]). iEI

Consider the set X’ =X\{z} and demonstrate that x E C(X’>. Assume that x E C(X’>, that is, p’ < card( II i t JX’ nD,(x)l>, implying that there


exists Z’ @ n iE ,[XnDj(z)l and z’ E n iS ,[X’ nDJX)I. Therefore, there exists a criterion i, EZ such that z’ EX~D;,$X>, and then exclusion of z from X cannot change the dominating set Di,$x>, that is, z’ G fl , t ,[X’ n D;(x)l.

On the other hand, if ye C(X) and y EC(X’), we obtain that card( n it ,[X’ n Dj(x)l) <p’, but card( rl it JXn D,(x)l) >p’.

SO, ZE ni,,[XnDi(y)l><p’, but card(n,,,[XnD,(z)l)~p’ and ZE C(X) hold by virtue of y E C(X), which conflicts with the assumption.

Thus, we have proved that x E C(X\IzJ) and y @ C(X\{zl) for any X and x, y, z such that x E C(X), y, z @ C(X).

According to Lemma 2 of Aizerman and Aleskerov ([ll, p. 1261) this condition is equivalent to the condition 0.

Finally, we show that the function Cc., I, p’) does not satisfy the concordance condition C. Consider the criteria option estimates in Table 1 and the choice function C(*, (2, 31, 1). Obviously, z is included in the choice from sets (x, zl and {y, z), but z @ C(A, {2, 31, 1). The lemma is proved. 0

Assertion of Theorem 5 follows immediately from Theorems 1 and 2 of this paper and Theorem 4 in Chapter 5 of the book by Aizerman and Aleskerov [ 11. n

REFERENCES

1. M. Aizerman and F. Aleskerov, Choice of Options (Foundations of the Theory). Nauka, Moscow, 1990 (in Russian).

2. F. Aleskerov, Interval choice and its decomposition, Automat. Remote Control 6 (1980).

3. F. Aleskerov, Relational-functional voting operators, Social Science Working Paper 818, California Institute of Technology, 1992.

4. F. Aleskerov, B. Litvakov, and N. Zavalishin, On the decomposition of choice functions on the system of interval choices, Automat. Remote Control 7, (1981).

5. K. J. Arrow, Rational choice functions and orderings, Economica (NJ.) 26 (1959). 6. P. C. Fishburn, Interval Orders and Intenlal Graphs, A Study of Partially Ordered Sets,

Wiley, New York, 1985. 7. P. C. Fishburn, Semiorders and choice functions, Economettica 43 (1975). 8. R. D. Lute, Semi-orders and a theory of utility discrimination, Economettica 24

(1956). 9. B. Mirkin, The Problem of Group Choice, Nauka, Moscow, 1974 (in Russian).

10. 0. Ore, Theory of graphs, Amer. Math. Sot. Colloq. Publ. 38 (1962). 11. V. Podinovskiy and V. Nogin, Pareto-Optimal Solutions in Multicriterial Problems,

Nauka, Moscow, 1982 (in Russian). 12. A. K. Sen, Collective Choice and Social Welfare, Holden-Day, San Francisco, CA,

1970.

Received 15 January 1993; relised 15 February 1994

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Multicriterial interval choice models

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