Analysis of Procedures in Collective and Multi-Criterial Decision Making

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ANALYSIS OF PROCEDURES IN COLLECTIVE AND MUL TI·CRITERIAL DECISION MAKING F. T. Aleskerov and V. I. Vol 'skiy I 'I , II! III,

HI ( .,,'ill,,1 ) 11, lit I'. ' \/ '''111 ;1 . ( ·."t .) U

Abstract. Colle c tive and multi-criterial decision making procedures of three types where the resultant de c ision with original "estimates" as a totality of criteria or binary relations is repreaented as a) a criterion or binary relation; b) a choice func tion; and c ) a choice function with the original "eatimates" as a totality of choice functions are stadied. Keywords. Decision theory; graph theory; set theory; optimization; choice function; local operator.

INTRODUCTION Choice of the best variants from a given set is the main stage in decision making. In actual situations the variants c an be plans of sectoral development, design options, nominees for off ices, sites of future factories, etc. Whenever the varisnts are estimated in terms of a set of indices or these "estimates" are made by a collective of individuals the "private estimatea" have to be aggregated into "overall preference". The problems of aggregating individual "opinions" (represented, as a rule, as binary relations) are referr9d to as problems of collective choice and the problem of aggregating criterial estimates as problema of multi-criterial choice. Three types of procedures may be regarded as aggregation procedures. The first type maps a totality of criterial estimates {If,(X.j into an "integral" estimate ffi (X.) (such are widely known procedures of criterion convolution) or a totality of binary relations {G-i.} into a resultant relation ~~ (such procedures are treated in the theory of collective choice where the relations (;, are associated with preference relations of individuals and the relation &~ , with the overall collective "preference"). In procedures of the second type the totality of relations {Gd or of a criterion totality {!fi(X)} are mapped into the resultant choice function C ~ (X) , which makes it possible to indicate "preferable" variants on each subset X of the entire set A of admissible variants. This mapping is performed, for instance, by the Pareto choice rule with criterial estimates i(Ji(X),i.= 1., ..., n. I lJC3-J

1363

Using choice fun c tions for theoretical description of decision making set is justified by the following important fact. }'irst, numerous actual decision making procedures can be described in terms of the properties of choice functions in a unified way, these properties making possible a plain meaningful interpretation; second, the analysis of these properties makes it possible to develop new types of procedures which can be employed in actual problems. Finally, procedures of the third type maps totality of choice functions {C.(·n into a resulting function C*(·). Such procedures have not been studied until very recently (Grether snd Plott, 1982). Let us study all the three types with the following initial notation and concepts: A ={ x •.... , Am 1, m) 2 , - is a finite set of all admissible variants; I ={ 1• ... , 11.), 11. > j , is a set of indices (individuals in the problem of c ollective choice or criteria in the problem of multi-criterial choice. Every subs cript t from I is associated with some binary relation G, = ({XI ~n
A

The totality of relations {G, 1~ or criteria ['i, (x)r or functions (C,(o)l~ is referred to as the profile. Relations G, are represented herea f ter by weak order relations. Ea ch auch

F. T. Aleskerov and V. I. Vol'skiy

1364

relation can be associated with a numerical function f. (::x:) so that (X,~)E G, i f and only i f f, (x) > f, (~) • The class of weak order relations is denoted as WO • The analysis can proceed in any of the following three ways: a) in terms of the properties of the procedures as mappings of the profile into the relation or criterion or choice function. Such mappings are referred to as operators. This approach makes it possible to investigate classes of procedures without fixing their specific representation; b) indicating the specific procedure such as a Pareto rule for design of the function C*(o), or a rule for design of the criterion f * (x) as a linear sum of particular criteria; and c) indicating the properties of the generated function C*Co) Let us have a closer look at the three techniques of analyzing the multi-criterial and collective choice. PROCEDURES OP THE FIRST TYPE Let us identify the class of locel operators which map the profiles 1. G, 1 into the resultant relation G* • Let V (x,~ i £G, )) denote a set of subscripts i. from I for which (X, ~ ) E G, , i . e. V(X,~;{Gd)=tt.ell(:x:,~)EGd (in the criterial case V(:x:,~; {'fd)={lEl/fJx:»fJ~)}). Definition 1. An operator F is referred to as local i f for any pair (x,~) and for any two profiles {. G, } and [G: that satisfy the condition V(:x:,~; {G-, = V (x,~; {G:}) it follows that (X,~)E G*~(X,~)EG-'''where G"= F({ G, 1) and G'" = F' ({ G-: 1) •

1 D=

By the definition of locality the decision (X,~)E G* on inclusion of the pair (x,~) into the relation G* is made independently of other pairs (u, t) Let us introduce a unified way to describe local operators. Let W ~ I be a certain subset of the set of indices. Associate each ordered pair (x''i) € A x A with a list, which is a certain totality (set) of groups Q (x U)={w(X,~) w(J<,'J) 1. '4

"

... ,

S(x''J) j

and define the rule for design of the relation G* in the following way.

(x''i)e G*~ V (x,'J; {G.)€ Q (x,~)

(1)

The totality of liat {Q (X,~)lvx,~ and the rule (1) be referred to as listlike representation of the operator F.

1 (Q (::X:''j)} = Q

Let us now consider list-like conatraints imposed on local operators F which perform the mapping WO -- Q, (l = i, 2,3 ) where a, is a set of a_cyclic, Qz ,of transitive, and Q3 , of negative transitive (:x:G~~ 'iG'l.~xGz)binary relations. By operators F which perform the mapping WO -- Q.. we mean the operators F which map the profiles {Gj !, G € WO J \fd ,into the relation G*€ a·l . The intersection n Q 2 n Q3 can be shown to coincide with the set of weak orders, or n 2 n Q3 = WO and the intersection n Qz ' to coincide with the set of strict partial orders, or irreflexive transitive relations. These classes Q, n Qz ,and WO are most extensively used in the theory of collective choice in decision making.

at

a, a at

at,

Let us introduce three more constraints on the lists.

Wit eo Q (XI. ,X,)

We Q (::x:,,/)

Then

The conditione 1 0 , ••• , 40 are referred to are referred to as conditions of poaitive unanimity, negative unanimity, monotonicity, neutrality to variants and neutrality to subscripts, respactively. By Condition 1~ the pair (X,'i) for which all i's from I accepted should be included into o resultant relation G* • By Condition the pair (x,~) for which none of i's from I accepted has not to be included into G-• • Monotonicity Condition 2 0 can be easily interpreted in tha following way: i f for the profile (G.! the pair (:x,~) is included into the relation c,* and i f the profile 1. G-~ 1 is such that for all relations Gc for which the pair (x''i) is accepted thia pair is also ac cepted for the corresponding relations c,: and if for some relation G. for which (x,~) •J is not accepted, G accepts this pair, i then (x,~) is by all means included in G*' • By Condition )0 the operator F is independent of the names of pairs (i.e. is independent of the specific variants 0 in the pair (x,'i) ) while Condition 4 "prescribes" independence of the operator F of re-enumerstion of the voters or criteria.

~~. V(x''i;lG-d)=l~(x,~)EG-·; 1~. V(x''i; {Gd) • ; 20

0

(:X''i) '

(1)

'* (x,'i)' G·

•

and 1(Q(x,~f»=[Hw(X'~)>l

Let us introduce some constraints on the local operatora in terms of their listlike representation: V::x:, ~ E A

,. et>

0

Let ? be a bijection of I into 1. Denote HWd = UJEW, Hi) ; 4

Q (x,,;/) :. Q

for any k (2~ k,. rn) and for any CA (L=I, ... ,k-1) W. E Q (x., X,+I) and

n~=,

W l '"

rp

i t follows that

1365

Col l ect i ve and Mu l t i- crite ri al Decis i on Maki ng

(2) for aD,y W., Wz and W such that Wf fi Q (Xf,X Z ) ' W z € Q (x z , x~) ,and W, n Wl. !;; W & W. uWz it followa that [;j e Q (X f , X,,) and such thst () for aD,y w., W z , Wz Q (x" X 3 ) W f , Q (x" xl.) it follows that and W, n W z !< W

w 1-

Wf-

Q (x"

X,,)

•

€ WO Vi.. Then is included in the class 3. 0. 3 iff the 1. 2. Qz. and operator P sstisfies the conditions 1. (1); 2. (2) and 3. (), respectively.

Theorem 1.

Let

G-i

G" = F' ({G, \)

at

;

Now let us take up some corollaries of Theorem 1 for operators which satisfy the conditions 1~ , 1~ , 2°, 3 0 , and 4°. When the conditions 1~ , 1~, 20 , and 3 0 are satisfied, direct use of the list-like representation of local operators leads to formulae for design of the relation G* on the knowledge of &, from the profile. In this case G*" = U w,€ Q n i4iWs &, ,or aD,y pai; (x, ~) is included into the relation G if it is unanimously accepted by all individuala in at least one group Ws. Let us consider three particular cases of such an operator

(4) (5) (6 )

n lEW G, , W, ' G = niEI G-i. ; G*" -- G('" t' . E 1 (7 *" :;

..

~

I;

Lemma. The rule (4) is a combination of conditions (1) and (2) and the rule (6), of conditions (1)~ (2 and ~3) while Conditions 1~ , 1_, 2 and 3 hold. The rule (5) is in effect the rule (4) while Condition 4 0 holds and therefore, the rule (5) is a combination of conditions (1) and (2) while Conditions 1~ , 1~ , 2° 40 hold.

6,

Now let us formulate two Corrolaries of Theorem 1. Corrolary 1. Let G-. E WO Vi., and F satisfy the Conditions 1~ , 1~ , 20 and 3 0 • Then the rela tion c,.* :; F' ([ Gi } ) is included in the class 1. Q, n Qz. and 2. n Qz n Q 3 = WO lff P is a type 1. (4) and 2. (6) operstor, respectively.

a,

Corollary 1 is another formulation of the famous Arrow paradox (Arrow, 1963, Sen, 1970) which is studied in the theory of collective choice. In its proof Conditions )0 is not required to hold for it is held by virtue of tr lIlBitivity of the relations (1* from the classes Q. n Q.z and WO • Corollary 2. Let G-i. € WO, Vi., and P satisfy the Conditions 1~ , 19 , 2 0 , )0 o and 4 • Then the relation G* = P ([Gi 1) is included in the class 1. with

a,

rn ~ h. and 2. (5) operator.

Q,

n Uz.

iff F is a type

Proposition 2 obtained above in Corollary 1 can be formulated in terms of multicriterial choice, viz. that any local procedure of designing criterion convolution (or mapping Ui (:C) 1 - f"(x) that sa0 tisfies conditions 1~ , 1~ , 2 , and )0 results in the presence of a single subscript L" € I such that f*(x) = f,* (x) for all x EA Let ua now have alook at a nonlocal procedure of designing the relation Gfrom the profile (Gd (G, € WO, Yi) Associating every variant :J:E A with a set of m numbers tcard (V (X, ~ ; {G;\ >}Y" or the number of relations Gi (or criteria i'. ) in which variant x is "bet ter" than ~ • By the definition of the set V(x,~; fG,}) it is true that V(X,x; fGcl)=~

The results of this pairwise comparison can be conveniently represented as a quadratic m" m matrix]) = Md (x,~)U where d(x,~)=ca1.d(V(x,~;{GJ) with d (x,x )=0 • The matrix D can be regarded as a Table of the result of some n-circle tournament if the variants JeE A are viewed as players while d(x,~) is regarded as the number of games in which player x wins player ~ • Let us refer to the matrix D as a tournament matrix. A generalized numerical estimate of the variant xe A in the matrix.D will S ,5/ 5 ' be the number M (X)=VLd (x,~), ~EA se l ~, 00 1, which depends on the number S as a parameter. Let us design the relation following way (X,~)E

G"

~

MS (x»

G'"

in the

MS ('t)

It is obvious that with any fixed values S the relation G* belongs to the class WO This procedure of designing the relation (1* obviously satisfiea conditions 1 ~ , 1~ , )0, and 4 0 and does not satisfy the locality condition and, unlike local operators, the presence of the subscript ~ for which G* = Gc" cannot be established in advance. S =4 ,ther;, M'(x,'i)='-~cJ(X,~) and the relation G coincideJ with an ordering which is provided by the generally applied rule of determining the "victors" in circular competitions according to the sum of pointa.

If

If

S

=00

,

then

r.

1366

T. Ales k e r ov and V. I . Vo l' s ki v

ann the relation G* coincides with the ordering which is obtained by using the Kramer procedure (Studied in Vol'skiy, 1982). This term is borrowed from the dynamic theory of voting (Kramer, 1977) where a similar procedure is employed. PROCEDURES OF THi SECOND TYPE This Section will be devoted to procedures of mapping the profiles tG,1 into a collective choice function C* (.) The results of Sect. 2 can be interpreted as axiomatic definition of multi-criterial choice rules. Indeed, let us consider G" = n ieI Cr, and define the choice function which associates each subset X of the set of admissible variants with a set of "preferred" ones so tha t C (X) .. = [~EX \ 13 nX: X& ..~} 'VX • I f initiall:' the variant estimates are specified in the "criterial" form (recall that it is assumed that G, € WO) Vi ), 1. e. in the form tf,(X)} then this rule can be represented in the form C*(X)=[~EX\

13xeX: fL(X»fL('i), V'i.dl • The set C* (X) is referred to a s a set of Slater optimal variants and the rule, as the Slater rule (Hurwitcz, 1958). The Pareto rule can be defined in a aimilar way but in this case the definition of locality and the procedure of designing a list-like representation must be extended. Now let us take up a ment", procedure for t ion C" (.) from the study the properties

nonlocal, "tournadesigning the funcprofile {G-i } and of this function.

The choice function C" (0) is obtained in the following way. For every X £ A the choice is made on the submatrix D,x of the tournament matrix.D associated with a given X (the matrix ~ was defined in Sect. 2). The generalized numetical index has in this case the form

and depends, unlike the tournament procedure considered above on the set X s; A aubmitted for choice. Consequently, a single-parametric family of procedures generates a single parametric family of choice functions

c; (X)

=

{~EX I M~ (~)

=

=mdxM;(x)1, SE[t,c><> X&X

Theorem 2. With any fixed it follows that

SE

[l,

<><»

C; (X) ~ Cp<7!'(X) , VX q"A . Therefore, choice functions from a family

can be employed to identify part of the Pareto set, which is often required in multi-criterial choice (Yemel'yanov et al., 1973). The extreme cases of the family {C$ (.) SE [~, <><>11 are the following choice functions: with s=l C~(·)~ Csu ".,(·), function of choice by the sum of points; with 5=<><> (·)~Ckr(·)' the Kramer choice function. In the theory of choice the choice functions sre described in terms of characteristic conditions which describe the properties of choice functions with certain distortions of the sets submitted for choice. The most widely used characteristic conditions are (Aizerman and 1Ialishevski, 1981 and the References there) ;

C:

Heritage (H) VX',XsA :X'!:X=>C(X'):2C(X)(1X'; , " ( . X )1-P=:-CV')= lv' Constance (Co)VX,X!:A :XsX.Xnc :C(X')=c(X)nX';

Concordance (C) VX',X"sA =>C(x')nc(X·)~ C(X'UX"); Independence of rejecting the outcast variants (0) VX', X~A: C(X)£X'£ X =- c (X') = C(X); Choice maintenance VX ' , X£A: X'S C(X) =C(X') =X' . Many choice functions are described in terms of characteristic conditions, for instance Pareto choice functions C~r(') and only functions which satisfy conditions H,C and 0 simultaneously; scalar optimization choice functions and they alone satisfy condition Co The func tions Csu ," (. ) and Ckr (. ) do not, however, satisfy these conditions. It is therefore necessary to find the characteristic conditions for description of such choice functions. Below new characteristic conditions are introduced in which the concept of choice function superposition is employed. These characteristic conditions are intended to be used with functions which identify part of the Pareto set in problems of multi-criterial choice. The new characteristic conditions are obtained by using the relation of the function being studied, (C(·» and another choice function (E (. assuming that qx) se (X) ,'VX sA • The function C(o) is referred to hereafter as the embracing choice function and it is assumed that E(.) belorl6s to some known class of choice functions such HnCno which is generated by Pareto choice functions.

»

Definition 2. The function C(·) will be sai~ to satisfy, for the embracing function C(o) ,the conditions of: Independence from rejection of variants

Co llectiv e and

~lulti - cr i te r ial

which do not belong to the embracing func) if VX, tion C(·) (condition IR C(X)=C (X \ X') if X'nC(X) = 'P; X'sA representability as superposition (condition R5 ) i f ceX)=C(C(X)), VX~A; inverse representability as superposition (condition IRS) i f C (X)=C (C (X»), VX ; superpositiQn co~utability (condition se) i f C(C(X»=C(C(X»,

VX,

amplified superposition commutability (condition ASC) if C()<)=C(C(x»= = C(C(X»), YXsA .

13 67

Decis i on }lak i n g

C or 0 satisfies IR and, by virtue of Theorem 5, conditions RS, SC, and ASe. In multi-criterial choice a procedure is often used whereby the choice is made according to "proximity" of the variants to some "ideal" point which is determined at each Xs; A • Such models are studied in (Aizerman and Malishevski, 1981) and Yu and Leitmonn, 1974). The procedures of designing a function C (.) of this kind are also nonlocal. The functions CC·) implemented by these procedures do not meet conditions H, C or 0 but can be described in terms of the characteristic conditions introduced in Definition 2. PROC~DURES

OF THE THIRD TYPE

The characteristic conditions listed in Definition 2 may be interpreted in simple meaningful terms. Thus IR _ in the case of the embracing function C(·) being represented as a Pareto set CPg, C·) specified that the variants which are not members in the Pareto set do not influence the choice. For the functions C(·) which are intended for identification of part of the Pareto set this condition looks natural.

Now let us take up operators which map the functional profile [C, (.) 1 into the collective choice function C*(·) This statement of the problem is of special interest in the theory of collective choice because it enables representing the initial "estimates" by individuals in a very general forrr of choice functions, not necessarily confined to pairwise comparisons of variants as was the case of the first and second types.

The Theorems below establish the relations between the characteristic conditions depending on the class in~which the embracing choice function C (.) is a member.

denote a set of individuals who include variant x into choice C, (X) from the set X ,or V ( :x., X; (. Cd' = { i. El I :x: f Cl (X)} .

Theorem 3. Let C(.) satisfy condition Then C(·) satisfies c ondition RS with respect to t(·) iff the function C(·) satisfies condition IR with respect to C(·)

o.

Theorem 4. Let C(·) satisfy condition CM • Then C (.) satisfies condition IRS.

n)

Definition 3. The operator F will be referred to as local functional if for any and (. C; (.») and two profiles ~ CL (. )} an,y :x: and X that meet the condition

V(x,X; {Cd'>})

= V(x,X;

{((.)})

it is true that X€C"(x)<=> X€ C,..'(X) where e*(')=F([C,(.»)) and e*'(·) =F ({C~ (.))) .

Corrolary. Let C(·) satisfy conditions CM and 0 • Then the function C(·) satisfies condition IRS with respect to E(.) Hf C (.) meet condition IR with respect to C(·)

Thus a decision to include variant :x: into the choice C"(X) from the set X is dependent on whether variant :x: is by indi viincluded into choice from X duals and this decision is independent of variants ~ from the set X\ {x 1

Theorem 5. Let the function C(.} meet conditions H ,C and 0 • Then for an,y function C(.) such that C(X)£C(X) conditions IR, RS, se and ASe are equivalent.

Let us introduce a unified way to describe local functional operators as their list-like representation. The list is defined for each pair (x,X) os a totality of subsets of the set 1={1"t1.1 ,i.e.

The region H (lenO in the choice function space was shown above to be generated only by Pareto functions. Then from Theorem 5 it follows that for an,y function C (.) which specifies part of the Pareto set the Theorem conditions are equivalent. Such functions are, in particular, functions of choice by a tournament matrix, which are generated by essentially nonlocal procedures. If C(-) is represented as Ckt-C') ,then this function which does not satisfy H,

Q ( X) f W (x,X) w(x,X)} where x, = t / , ... , s(x,X) c..p:,X) S I Vi and the rule f or desig-

, ' ning the function

e*(·)

has the form

Let us impose constraints on local functional operators. These conditioBs are identical with conditions 4 imposed

,0 -

F . 1. Al eskerov and V. I . Vol ' skiy

13 68

on local operators that were discussed in Sect. 2 and therefore are denoted by the same numbers and have the same names. Let us now enumerate the characteristic conditions: Q (x ,X) ; '~.lEQ(X)<); "ix,X : X€X!;A ; 1~ W ~ W £ 1 = WEQ(X,X); Z~WE Q (x,X) ~ IJw )0. Q(X,X)=Q ; 40 • Let Z be a bijection from I on I. Denote 1 (w L ) = UjeW, 1(J) and

. rp,

)(x,XI 1 1 (0 (X,X»)= f HW (X,X)), ···,1 ('VJS(x.X»)J· t

Then

1 (Q (x,X») = Q (x,X)'

These conditions have the same sense that Conditions 10 - 40 have for local operators of the first type.

a

Definition 4. The class of choice functions will be referred to as closed for a local functional operator F if for any profile (CL (.)) such that Cl (.) E Q, Vi., i t is true that p({ed- H)=C"(-)EQ. The largest (in set inclusion) class of operators F for each of which the class Q is closed will be referred to as complete class of operator closeness for Q and will be denoted as

J\Q In terms of list-like representation a genersl form of operstors csn be established from complete classes of operator closeness f1.H ' Ac ' and Aa for clasces of choice functions H , C ,and 0 of the preceding Section. Let us consider for brevity only operators from the ~­ tral class Ae , or those which satisfy the conditions 1~, 1~ , 20 , and 30 and operators from the symmetricslly central sc class f1. = 1; nL~ nzon3°n4° • The operators from the central class define the function C*(o) on the knowledge of a function from the profile in the following way

fe,{,»);

(1)

C*(X) =U W . EQ n iew C,(X) . J

d

Let us consider particular cases of operator (1) (2)

C""(X)=nl€W,C,(X);

()

C*(X)=U i €€, C,(X),

(4)

C"(X)=Ci,,(X);

(5)

CIt(X)=U Jrl s= (~)

s

n.leW · C-(X) l d

The operator (5) is fy condition 4 0 and member C/I(X) i f and indi vidus Is include from the set X by CL (0 ) The () The (2)

c,!;.;I; wherelw. \=k,

VJ = "

d ,S;

easily seen to satisX is seen to be a only i f at least k :x:. in the choice CL (X) their choice functions

operstor (5) with k = I (the operstor with c, = I ) will be denoted ss "V' • operator (5) with k=n.. (the operator with W, = r ) is denoted as "U· •

Denote operator

classes defined by for-

mulae (1) - (5) as AV~An,f1.v,A' and f1.un~ • Consequently, AC=/l.lJn

N e = Aun~

and

Theorem 6. The complete class of operator closeness AH is bound to contain the entire central class of operators, or !lH=>AIJ~ The intersection of the central class and complete classes of operator closeness a) Ac ; b)

11. 0

;

c) AWl, d)

A.. no

e) ACllo

;

f)!lH~no; and g) A~coincides with the ope-

rator classes a)f1.n ; b) AV c) An; d) 1I. v ; e) J\.' f) 11.' ; and g) At , respectively. IlliFERENC};S Aizerman M.A., Malishevski A.V. (1981). General Theory of Best Variants Choice: Some Aspects.- I};EE Trans. Automat. Control, No.26, p.1030-1040. Arrow K.J. (196). Social Choice and Individual Values. New Haven and London: Yale Univ. Press, 2-nd ed. Grether D. M., Plot t C. R. (1982). Nonbinary Social Choice. An Impossibility Theorem. Rev. of Econ. Stid., Vo149, No.2, p.14)-149. Hurwitcz L. (1958). Programming in linear topological spaces.- In. : Arrow K.J., Hurwitcz L., Uzava H. Studies in Linear and Non-Linear Programming. Stanford Univ. Press, Stanford, CaliL Kramer G. H. (1977). A Dynamical Model of Political Equilibriurn.- J. Econ. Theo!:Ji.., Vo1.16, No.2, p.)10-334. Sen A.K. (1970). Collective Choice snd Social Welfare.- San Francisco; Holden-Day. Volskiy V.I. (1982). Use of the Kramer Method to Identify Part of the Pareto Set in Multi-Criterial Optimization.Avtomatika i telemekhanika (in Russian) Vo1.4), No.12, pp.111-119. Yemel'yanov S.V., Borisov V.I., Malevi ch A.A., Cherkashin A.M. (1973). Models and Methods of Vector Optimization. In : Tekhnicheska a kibernetika Ito i nauki i tekniki Control ~n­ gineering Results in Science and Technology) (in Russian) Moscow: VINITI, Vol.5, pp. 386-448. Yu. P.L., Leitmann G. (1974). Compromise Solutions, Domination Structures and Salykvadze's Solution.- J. O~tim. Theory Appl., No.1), p.362-) S.