Journal of Multi-Criteria Decision Analysis, Vol. 2, 167-170 (1993)

Group Decision as a Twice Multicriteria Optimum Problem

PAOLO D’ALESSANDRO DIS, University of Rome, ’La Sapienza‘, V. Eudossiana IS, 1-00 184 Rome, Italy

ABSTRACT This paper studies the problem of aggregation of preferences, preserving the multicriteria nature of the individual decision process. The resulting group decision problem is in effect a twice multicriteria optimum problem. We give a structural theory for the efficient set of this optimum problem. Moreover, on the basis of this theory we study the relationships between optimality, consensus and evaluation. We confirm the pathological nature of aggregation of preferences, but in a new setting and with a different meaning.

KEY WORDS Group decisions Multicriteria optimization Pareto optimality

1. INTRODUCTION

The purpose of this paper is to study the group decision problem preserving the multicriteria nature of the individual decision problems of members of the group. The further multicriteria dimension of the problem is of course given by the presence of a multiplicity of decision makers. The other key feature of our approach is the central role played by Pareto optimality and hence by the study of the efficient region of the problem. We confirm the pathological nature of aggregation of preferences discovered by Arrow (1963), but in a new setting and with a different meaning. We shall omit some straightforward verifications and we shall comment briefly on our results. The interested reader can find more details in a former version of this paper (d’Alessandro, 1992).

2. THE MATHEMATICAL MODEL

Consider the following three finite sets: the set X of alternatives (with m elements), the set D of decision makers (with n elements) and finally the set A of criteria (also called attributes, with p elements). Each DM (decision maker) ranks by means of a finite scale S (any finite linearly ordered set with enough elements will do: to fix the ideas, we stipulate that the scale is the set

1057-92 14/93/030 167-04$07 .OO 0 1993 by John Wiley & Sons, Ltd.

Received 19 June 1992 Accepted 27 April 1993

168 Twice Multicriteria Optimum Problem

(1, . . ., 10)) each alternative against each criterion. Thus this evaluation process defines a function f on X x A x D to S such that f (x ,a ,d) is the score that the decision maker d assigns to the alternative x with respect to the criterion a. Of course, f ( x , -, a ) can be regarded as a matrix (denoted by f ( x ) for short) and the whole function f either as a three-dimensional matrix or as a matrix-valued function on X . To solve the decision problem, we have to maximize this matrix- valued function over X . In the next section we shall study the structure of the efficient set E, i.e. the set of solutions.

3 . STRUCTURE OF THE EFFICIENT SET

First of all we note that the efficient set E cannot be void. This is a trivial consequence of Zorn’s lemma and the fact that all sets are finite. Next consider for each decision maker d the set e(d) of his own efficient solutions, obtained by maximizing f( * , * ,d) (viewed as a vector-valued function) over X . By the same argument each e(d) is non-void. Let U=U{e(d): dED). Then we can state the following.

Theorem 1 The set U is contained in the set E. It can be properly contained unless there are only two alternatives.

Pro0 f The proof of the first part of the theorem is rather obvious and can therefore be safely omitted. Next suppose that an alternative, e.g. xl (without restriction of generality), is dominated for all decision makers. If there is only one other alternative x 2 , then for each decision maker the corresponding column off (x2) dominates that of f ( x l ) . Hence the alternative x1 is dominated too. For the general case m > 2 there are (rn - 1 ) x d columns in the matrices corresponding to the remaining alternatives, and d of them are constrained to make x1 dominated for a11 decision makers. Clearly they can be placed in such a way that, out of the remaining (m - 2) x d columns, there is one in each of the matrices f ( x 2 ) , . . . , f (x,) which is dominated by the corresponding

As is clear from the proof, the presence of alternatives in R = E - U reveals conflicting evaluations of the decision makers. On the other hand, a non-void set set I= n(e(d): d € D ) reveals consensus on the alternatives in this set. Introducing the set S = U - I , we obtain a total of five sets (E, U , I , R and S) which together with the family (e(d): d € D ) give a first structural characterization of the problem. The sets E and I/ are always non-void, and if I is void, then S is necessarily non-void. Thus we are left with the following six combinations classifying any possible problem (a plus denotes non-void and a minus void):

column of xl. Thus x1 can be optimal even though it belongs to no e(d).

Group decision problem table

R S I 1 + + + 2 + + - 3 + - + 4 - + + 5 - + - 6 - - +

P. d’Alessandro 169

Clearly combination number 2 corresponds to decision situations with the most conflict and combination number 6 to the most concordant-but are all these combinations actually possible? The next theorem ensures that that is so (excluding, of course, the case of two alternatives).

Theorem 2 All six possibilities stated in the group decision problem table can occur provided that there are more than two alternatives.

Our proof will be constructive: we shall give an example for each of the combinations of the table. Before getting to this, however, we will introduce some further characterizations of the decisional situation. The consensus function c on E is defined by letting c(x) be the number of e(d)s in which x appears. The discord function s on E is given by s(x) = n - c(x) . Notice that R = ( x : c(x)=O], I= (x : c(x)= n) and S=(x : O<c(x)<nJ. Clearly if I is void and the maximum of c on S is below the majority (the less the worse), we have a very litigious situation, the most litigious if, in addition, R is non-void.

To sum up, including scalarization (to fix ideas and for simplicity we shall consider even weights) and variance, the decisional situation is characterized by the following parameters: the collection of sets E , U, I , R and S , in particular the pattern for R , Sand I ; the family of sets [e(d): dEDj; the values of c (and s) on each element of the set S and its maximum on this set; the absolute evaluation ~ ( x ) (sum of all scores got by x); and finally the variance u,(x) for each element x in E.

We now give the examples, leaving, for reasons of space, all the necessary verifications to the reader. Let n = 4 and p = 3, whereas m will vary according to the example. Consider the following matrices along with the corresponding absolute values and variances:

8 9 7 6 9 8 5 2 2 9 2 3 M3=l 9 3 2

7 7 9 7 8 6 3 6 1 8 1 1

2 2 8 1 5 4 3 7 10 10 10 10

2 3 1 0 4 2 2 1 8 9 3 3 3

2 9 2 2 1 1 9 1 10 10 10 10 M7= 10 10 10 10, Mg= 1 1 9 1 , M9= 1 1 1 1

7 7 7 7 10 10 10 10 1 1 1 1

Ml=6 8 8 9, M2=7 7 4 8,

M4=1 3 9 1 , M 5 = l 2 4 10, M6= 9 3 3 3

v (XI ) = 91, V (X3 ) = 42,

~ ( ~ 7 ) = 8 3 , a,(x7)= 3.0945, v (xg ) = 64, u , ( x ~ ) = 4.3461 v ( x ~ ) = 48,

U, ( X I ) = 1.0374, u , ( x ~ ) = 3.0686,

V ( X 2 ) = 73, v ( ~ 4 ) = 46,

U, ( ~ 2 ) = 2.0999 U, ( X 4 ) = 3.13 13

v(x5) = 49, U, (X5 ) = 2.7826, V ( X 6 ) = 76, @n(X6)= 3.3499

U, ( X 9 ) = 4.2426

We shall denote the examples by the letter E followed by a number, which refers to the corresponding row of the table.

E2: f ( x i ) = M i , f (x2)=M2, f (x3)=M3, f@4)=M4, f(xs)=M5

El: f ( x l ) = M ~ , f ( x 2 ) = M ~ , f(x3)=M39 f(x4)=M4; f(x5)=M59 f(x6)=M6

E4: f (Xl )=M3, f(X2)=%, f(X3)=M5, f(Xq)=Mci

E5: f ( x l ) = M 3 , f(X2)=M4, f(X3)=M5

4. CONTRADICTIONS OF PREFERENCE AGGREGATION

It is obviously possible that the family (e(d): d € D ) be disjoint. Thus in this extreme case c is uniformly equal to 1 on S= CJ. This means that any alternative that can be proposed as winner by a member of the group has as an overwhelming majority (the rest of the group) against. An even more striking pathology regards elements of R . In fact, even though they may well be the most rational candidates for the choice of a final winner (as is seen by the examples, they have typically uniformly high scores), they cannot be proposed or voted for by any member of the group by the very definition of R itself. The situation is much like there were a further decision maker, the group as a whole, who is independent of the individuals and whose preferences can clash with those of the individuals.

In either case any final choice of the group will be contradictory. We may assist (e.g. example E2) to a stall due to fierce litigations over worst-valued choices, whereas the best alternatives in terms of both absolute value and variance are not even taken into consideration. This, incidentally, yields evidence against the existence of a social wisdom.

Given the structure of the decision problem, these contradictions are intrinsic to its nature and therefore we cannot expect to devise really satisfactory solutions. Thus an important point is to characterize such structure and compare it with other possible structures in the light of our results. The group decision structure is characterized by the power held by individuals, the possibility of blind defence of personal interest, the risk of fierce and paralysing litigations and of poor decision quality and finally the dilution of responsibility (an excellent excuse when the consequences of such poor decision quality emerge). Other structures may be promoted by the presence of a high community spirit, envisaging the presence of a superior authority representing public interest, to which the group simply consults. He/she will carry the whole responsibility of the decision and, being unaffected by the interests of individual group members, can consider the optimal solutions that are withstood by them, guaranteeing a much higher decision quality.

In any case, awareness of the present theory (which requires a minimum mathematical background to be grasped) and of the illustrated characterization of a decision problem would clearly be very helpful in assisting a decision and improving its quality. The author is working on a group decision support system based on this theory, whose description will be the topic of a forthcoming paper.

REFERENCES

d’Alessandro, P., ‘Group decision as a twice multicriteria optimum problem’, Research Report 48/92,

Arrow, K . J . , Social Choice and Individual Values, 2nd edn, New York: Wiley, 1963. Department of Electrical Engineering, University of L’Aquila, February 1992.