Parametric Preference Orderings in Group Decision Making

Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto , Japan , 1981

PARAMETRIC PREFERENCE ORDERINGS IN GROUP DECISION MAKING T. Tanino*, H. Nakayama** and Y. Sawaragi*** ·Department of Mechanical Engineering II, Tohoku University, Sendai 980, Japan **Department of Applied Mathematics, Konan University, Kobe 658, Japan • ** Department of Computer Science, Kyoto Sangyo University, Kyoto 603, Japan

Abstract. This paper considers parametric preference orderings in group decision making. Some families of parametric group preference orderings are defined on the basis of three wellknown preference aggregation rules; the sum of individual utilities, the Rawls maximin rule and the Nash bargaining solution function. They have some desirable properties such as monotonicity along with meaningful parameters, and therefore help the group of decision makers understand the present situation objectively and proceed the decision making process smoothly. Thus considering the parametric preference orderings suggested in this paper is an effective approach to group decision making or consensus formation. Keywords. ordering.

1.

Decision theory; optimization; group decision making; preference

In this paper, we shall explain an approach for clarifying situations of group decision making problems from the above-mentioned viewpoint. Each individual's preference is assumed to be represented by a cardinal utility function and some group p reference orderings depending upon practically meaningful parameters will be defined on the basis of some wellknown aggregation rules.

INTRODUCTION

Optimization problems or decision making problems we encounter these days assume several complex aspects because they are large-scale. Due to Sage (1977), those aspects can be classified into the following three categories: 1) uncertainty, 2) a large number of decision makers, 3) a large number of incommensurable attributes. The theory of group decision making deals with the second aspect, that is, problems caused by the existence of plural decision makers and/or persons concerned whose interests and/or senses of values are different. The theme of the theory is, in a word, aggregation of plural conflicting senses of values (preferences), that is, formation of consensus. A number of methods have been developed for the purpose (Sen, 1970; Fishburn, 1973). However, it is far from easy to get consensus in a group, as has been shown by Arrow's general impossibility theorem. Namely, there is no dominant method for aggregating individual preferences into a group one in such a manner as it satisfies some rationality conditions simultaneously. Therefore, it is getting a great significance to regard group decision making as a process of attaining a final goal gradually and to study ways of proceeding it smoothly. In other words, effective (systems analytic) methods, which can clarify the situation of the present group decision making problem objectively and plainly, and can help the group proceed the decision making process, should be developed as early as possible.

The organization of the paper is as follows. In Section 2, we show points of similarity and differences between group decision making problems with individual cardinal utility functions and ordinary multiobjective optimization problems. We also discuss interpersonal comparison of utilities. In Section 3, we provide a family of group preference orderings which has the strengths of individual veto right as parameters and includes the Pareto partial ordering and the ordering by the sum of individual utilities as two extreme cases. Section 4 is concerned with the concept of justice . Another family of group preference orderings is defined on the basis of the Rawls maximin relation. Section 5 considers parametric preference orderings constructed from the Nash bargaining solution function (that is, the product of individual utilities). Finally, in Section 6, we provide an illustrative example. 2.

GROUP PREFERENCE ORDERINGS BASED ON INDIVIDUAL CARDINAL UTILITIES

As has been already stated, this paper deals with the case where individual preferences are represented by cardinal utility func-

1749

1750

T. Tanino, H. Nakayama and Y. Sawaragi

tions (utility values), that is, the case where intrapersonal comparis'on of utilities is pcssible. Suppcse that the group concerned, N, consists of n individuals (decision makers). The individuals are denoted by i=1,2, ... ,n. Namely N = {1,2, .. . . ,n } . Let X be a set of feasible (admissible) alternatives and suppcse that each individual's preference can be represented by a cardinal utility function ui (i € N) on X. In this event, the group 's preference attitude for an alternative x € X can be completely specified by the n -dimensional vector u(x)=(u (x),u (x) , ... ,un(x». Therel 2 fore the group decision making problem we should consider results in the multiobjective optimization p roblem which maximizes n objective functions u ' " .,u on the set X n l

the individual veto right as parameters, based on the additive group utility function (that is, the sum of individual utilities). Let x R y indicate that the group considers x to be at least as good as y (x, y € X) • Moreover, suppcse that the individual utility functions uis are normalized so that the weights assigned to them should be equal to one another. The following two orderings have been often used for determining the group preference: 1) Pareto partial ordering (unanimity rule) ; x R Y ~ u

s~multaneously.

Hence it is natural to apply existing results in multiobjective optimization (for instance, several interactive optimization methods or multiattribute utility theory) to the above case of group decision making. However, it is quite difficult in practice , because the trade-off analysis which is considerably practicable in multiobjective problems is indeed an ethical problem of interpersonal comparison of utilities and because setting up a reference pcint also contains another ethical problem. Therefore, as has been stated in Section 1, we focus on showing in intelligible forms how individual preferences are different from one another. One of effective methods f6r it is, as was proved by the extended contributive rule method (Nakayama et al., 1979), to generate a family of preference orderings containing meaningful parameters and to show rankings of alternatives (with the help of directed graphs in case of a finite number of alternatives) . This paper is concerned with this approach for more cases. Comparison of the utility value vectors u(x) and u(y) for two alternatives x and y respectively has the following two different cases: 1) comparison based on the relative location of u(x) and u(y), that is, based only on the vector u(x)-u(y) (or u(y)-u(x», 2) comparison based on the absolute locations of u(x) and u(y). The ordering by an additive group utility function is a representative of the former and Rawls ' maximin rule or the Nash bargaining solution is that of the latter. The group should judge which case is more appropriate for the problem . We will introduce parametric preference orderings for both cases . The parametric orderings will be obtained as partial orders so that they can clarify the preference attitude of the whole group objectively.

i

(x) -u

~

(y)

0

i for Ifi=l, ... ,n,

2) ordering by the additive group utility function (the sum of individual utilities) ; n

x R y ~ ~ [u. (x)-u. (y)] ~ O. i=l ~ ~ -

PARAHETRIC PREFERENCE ORDERINGS BASED ON THE SUM OF INDIVIDUAL UTILITIES

This section will be concerned with parametric orderings containing the strengths of

(2)

The former is generally recognized to be a reasonable preference ordering, but it selsom gives order relations among alternatives in practice. In other words, the information given by the Pareto partial ordering is very stable and credible, but the amount of the information is very small. On the other hand, the latter is often used for ranking alternatives and selecting one alternative. It is difficult, however, to normalize the ui's (that is, to compare degrees of impcrtance among the individuals) and the ordering does not pay enough attention to the equity among the individuals, the variance of the individual preferences and so on. Therefore, the ordering obtained may not be reliable enough to be used for the final decision. In order to cover up these defects of both orderings, we take into account the individual veto right. Namely we introduce the following as s '.lInpt ion : if a member of the group strongly oppcses the preference of an alternative x to an alternative y, the group might not prefer x to y (i.e. not x R y). In this event, we may clarify the group decision situation (for example, the variance of the individual preferences in the group) by changing strengths of the individual veto right as parameters. The above discussions can be formulated mathematically as follows. First, note that the equations (1) and (2) can be rewritten as x R Y

~


> 0

If p

(3)

E:

€

and x R Y

~

> 0 for e= (1, ... , 1)

3.

(1)

€

En (4)

respectively. Here <', ' > denotes the inner product in the n-dimensional Euclidean space n

E.

Let

A= (A , A , ... , A ) n 2 l

€

n

E+ (=the

Parametric Preference Orderings in Group Decision Making n-dimensional nonnegative orthant) be a parameter vector and define a group preference ordering R depending on A by

A

i

x RA y==?
°

(5) €

En.

1

i-th element n y ~

)

)

° (6)

It can be immediately understood by the formula (6) that Ai is a parameter indicating the strength of the individual i's veto right. The parametric preference orderings RA have the following properties. Proposition 1. i) RA is a partial order on X for each A

'(Monotonicity) If Ai > Ai then RA c RA'

(Le.

for ~i=l, ... ,

x RAY ~ x RA' y

the ordering (2) or (4) by the additive utility function. iv) If u (x) ~ u (y) for ~i=l, ... ,n, then i i x RA y for ~A € E:. Moreover RA approaches to the Pareto partial oedering (1) or (3) as A -+ (+00, ••• , +00) • Proof. Since the proofs of the statements are easy, they are omitted here. A defect of the preference ordering given by (5) lies in the fact that it takes into account each individual's veto right separately. Hence preference orderings in which coalitions of the individuals having opposite opinions are considered might be better. Therefore we define another family of preference orderings RA for a nonnegative parameter vector A =(Al, ... ,A ) by n -I x RA y ~

°

for VI

C

N,

(7)

is the n-dimensional vector whose

i-th element is l+ A.

1

if i

€ I and 1 if i

i I.

This formula (7), which can be rewritten as n

~

(x)-u. (y)] +A. min 1 1

Remark. If we take A = A = ... =A , the 2 n l n-dimensional parameter vector A becomes essentially unidimensional. Namely, we may define a group prefer~nce ordering RX for a nonnegative scalar A by n

L

for ~I eN,

[u.(x)-u.(y)] + >; L i=l 1 1 i=l min(O,u (x)-u (y» ~ 0. i i This ordering is essentially nothing but the ordering given in the extended contributive rule method (Nakayama et al., 1979).

4.

PARAMETRIC PREFERENCE ORDERINGS BASED ON THE RAWLS MAXIMIN RELATION

In the previous section, we have defined a family of parametric preference orderings based on the interpersonal comparison of utility differences. In this section we will consider parametric orderings depending on absolute locations of utility vectors (see Section 2). In other words, we would like to incorporate the concept of justice or equity into group preference orderings. We start from Rawls' theory of justice. Rawls considered that the group utility for an alternative should be determined by the utility value of the worst-off individual and required that the most preferred alternative should be chosen to make the worstoff best-off (Sen, 1970, Chap. 9). This rule, called Rawls' maximin rule, can be expressed mathematically as follows. We rearrange ul(x) , ... ,un(x) in numerical

v (x) ; ... ; vn(x». The extended Rawls 2 maximin relation (lexicographic maximin relation) is given by

R

x

L

[u.(x)-u.(y)] . 1 1 1 1= + A.[u.(x)-u.(y)] i€I 1 1 1

n

L

order and thus have a new n-dimensional bector V(X)=(V1(x), ••. ,Vn(X»(i.e. v1(x) <

which is an intuitive extension from RA'

e~

1

By the way, there is a close relationship between these preference orderings and the concept of partial comparability by Sen (1970) (see Tanino et al., 1980).

x R>; y ~

for \Ix, y € X). ni) If A =(0, ... ,0), then RA coincides with

Here

L ([u.

i=l

(O'U (x)-u (y») ~ 0. (9) i i The proof of the equivalence between (8) and (9) can be seen in Tanino et al. (1980).

€

E: n,

n

XRAy{==?

L [u. (x)-u. (y)]

j=l

+ Ai [u (x)-u (y)] > i i for "i=l, ... ,no

ii)

very attractive in the p ractical use:

Proposition 2. The properties given in Proposition 1 hold even if RA is replaced by

That is,

x \

1751

>

° (8)

seems to be very complicated and impracticable. However, it has another simple expression as follows fortunately and so is

Ry

~ 3k € {l, ... ,n } such that

vi (x)=v (y) for Vi=l, ... ,k-l i and vk(x) > vk(y) or

v(x) = v(y).

(10)

An intuitive extension (parameterization) of this ordering is given as follows: take p € {l, ... ,n} and define an ordering R by p

1752

T. Tanino, H. Nakayama and Y. Sawaragi

Ry

x R Y~ x p

and

for The relation R

P

~

vi (x)

vi (y)

i=l •...• p.

(11)

has the meaning that x R

P

y

if and only if x is preferred to y in the between the p worse-off utility values. It also has the following properties as can be checked easily. Proposition 3. i) R is a partial order for each p=l •...• n. p

il)

If P > P '. then Rp

lli)

RI

Rp'.

C

R (lexicographic

maximin relation)

iv) Rn coincides with the Pareto partial ordering among {v (x)

Ix

E

X}.

It is obvious from this definition that R is crucially dependent upon u. For example. if the value of U. decreases with the other u~s

Let v =(vl •...• v ) be an n-dimensional nonn negative parameter vector and define a preference relation Rv by x R" Y <=> [uk(x)-ii +v ] IT [u. (x)-u.] k i~ ~ v K ~

Since the Rawls maximin relation gives decisive power to the worst-off position (the value of VI)' it thinks much of the position

> [11. (y)-11 +v ] IT [u. (y)-u.] K le k i~k ~ ~

for ~k=l •...• n n IT [u. (x)-u.])/( IT [u. (y)-u.]) i=l ~ ~ i=l ~ ~

in the utility vector in the positive sense. While. we may think much of the order of utility values in a negative sense. That is. for an n-dimmensional nonnegative vector ~ = (~l····'~n) with ~l ~ ~2 ~ ... ~ ~n' we may define a relation R n

x R

~

Y <.=9

L {[v. (x) -v . (y)] ~

i=l

~

+ ~imin

(O.v (x)-v (y»)} ~ O. i i

(12)

This family of preference orderings is useful for grasping the variance of individual preferences in the group under the assumption that interpersonal comparison of utility levels is possible.

In this section. we will consider parametric preference orderings based on the concept of the Nash bargaining solution. The Nash bargaining solution has the conspicuous property of being invariant with respect to arbitrary positive linear transformations of individual utility functions (Sen. 1970). The Nash bargaining soluti~n depends on a distinguished alternative x. which we may call the status quo point. Denote u(x) by u. Since the region where u (x) > u (i=l •...• n) i i has the practical meaning. we assume that the set of admissible alternatives X is contained in the above region. The group preference ordering R by the Nash bargaining solution function is given by IT [u. (x)-u.] ~

~

>

n IT [u. (y)-u.].

i=l

~

(1+

k

()_

x -u k (14) R has the following properties and is u~eful for clarifying the decision situation connected with the Nash bargaining solution. ~

Proposition 4. n i) Rv is a partial order for each v E E+. il) (Monotonicity) I f Vi then Rv C Rv'·

~

vi (i=l •...• n).

If v =(O •...• 0). then R

v

coincides with

V

PARAMETRIC PREFERENCE ORDERINGS BASED ON THE NASH BARGAINING SOLUTION FUNCTION

n

v

k

() _ ) / u k y -~

the ordering defined by the Na~h bargaining solution function. i.e .• R = R.

~

by (12) has similar properties to those of the orderings (6) and (9) in the previous section .

i=l

v > max ( 1 +

lli)

It is quite obvious that the family of the parametric preference orderings R defined

5.

n

k

by

~

1

~

fixed. the value o f u for the bari gaining solution becomes smaller. Because the ratio of u (x)-u to u (y)-u tends to 1 i i i i as u + - 00 . Thus. making u small means i i thinking little of the preference of the individual i. We will define a family of parametric group preference orderings based on this fact. (j~i)

~

(13)

iv) I f u. (x) > u (y) for i=l, ... ,n, then i ~ x Rv y for any v E En. Moreover. in case + of n=2. R tends to the Pareto partial v ordering as v + (+oo •...• +oo). Proof. The proof is tedious. though not difficult; whence it is omitted here Thus the preference orderings Rv show how it exerts influence on the preference relation by the Nash bargaining solution function to think little of individual preferences appropriately. Similar types of parametric orderings to (14) might be considered. 6.

AN EXAMPLE

In this section we will show an example of the approach to group decision making via the parametric preference orderings suggested in this paper. Numerical data are taken from Inoue et al. (1980). In TABLE 1. the characters A - 0 stand for fifteen al ternatives (to be concrete. research subjects in environmental science) and the numbers 1 -16 sixteen individuals. Individual utility

Parametric Preference Orderings in Group Decision Making values are normalized between 0 and 10. We will show some group preference orderings among the alternatives A - 0 as (directed) graphs. Each point in graphs represents an alternative and a line between two points indicates that the upper point is preferred to the lower point.

tions even by the Pareto partial ordering if the assumption of comparability of individual utility levels is satisfied. A few orderings by R~

First, Fig. 1 shows three group preference orderings by RA (orderings based on the sum of individual utilities) with A.= 0, 0.5 and 1.

20 (i=1, ... ,16). We can conclude that the alternative(s) G (B and C) is (are) preferred (disliked) considerably, though not completely, unanimously in the group. Fig. 2 shows two extreme cases of orderings by R , which are based on the Rawls maximin p relation. The graph in case of p=l (namely the ordering by the leximin relation) is rather different from the result in Fig. 1 in the upper part (especially the alternative F). However, the graph in case of p=16 is more similar to Fig. 1. It also illustrates that there remain several preference rela-

TABLE 1

A

1

2 3 4 5 6

7

8 9

10 11

12 13

14 15 16

B

C

0.0 1.9 4.0 3.1 2.3 1.5 0 . 8 3.8 0.0 7.4 8.7 0.0 3.9 1.3 0 . 0 4.6 10.0 0.0 10.0 8.1 8.1 0.0 0.6 1.3 9.3 7.3 3.7 0.6 1.2 0.0 5.2 0.0 4.8 5.3 0.0 0.7 7 .1 0.0 10.0 1. 7 0.0 0.0 5.3 1.6 0.0 0 .4 1.3 0.0

(eq. (12)) are depicted in Fig. 3. The graph (b) shows, if compared with Fig. 1, more alternatives can be ranked under the above assumption. While, (c) is the result in the case where the concept of justice by Rawls is added to the ordering by the additive group utility function, and we must pay attention to the alternatives J and F. Finally, Fig. 4 shows parametric group preference orderings based on the Nash bargaining solution, where the status quo value is taken as u = -1 (i=1, ... ,16). The ordering i in case of vi= 0 (i=1, ... ,16) is surprisingly similar to the left graph in Fig. 1. This fact implies that we may rank the alternatives rather generally. The ordering in case of v.= 10 (i=1, ... ,16) shows there are 1.

some variance among the individual preferences.

Individual Utility Values

D

E

F

3.6 5.2 3.8 6.1 9.1 5.0 7.0 8.8 3.0 5.4 5 .7 1.9 2.7 8.5 9.8 2.6

2.1 4.2 8.0 6.5 7.8 6.2 5.7 7.9 5.0 5.8 3·8 8.9 2.0 7.4 9.3 4.4

6.0 6.2 4.5 4.1 2.5 2.7 6.8 4.8 5.0 4.2 5.7 8.2 7.8 4.7 4.0 3.1

G

8.9 4.6 9.4 10.0 10.0 8 .3 7.2 10.0 2.7

8.8 1.9 10.0 3.1 9.6 4.0 8.0

G

I

H

J

K

5.5 8.9 5.7 10.0 0 . 0 8 . 8 10.0 2.3 7.3 8.3 7.2 10.0 9 .6 5.4 3.7 7.6 4.9 9.9 5.1 5.7 7.9 2.5 3.7 3.1 5.8 0.0 8 .7 4.2 2.9 7.3 5.6 4.8 6.7 0.0 8.7 4.7 1.7 8.1 7.2 10.0 1.9 10.0 5 .7 7.6 1.2 4.0 3.5 2.8 2.7 5.8 2.2 5.6 10.0 3.2 4.0 0 . 2 0.9 10.0 8.8 6.0 6.7 10.0 8 .5 5.0

L

M

N

6.0 6.6 6.6 0.8 2.5 7.5 4.8 5.9 5.9 2.6 3.9 3.5 5.4 3.3 7.9 4.8 6.3 8.7 3.4 4.2 1.9 5.6 5.4 4.2 2 . 0 9.3 10.0 9.3 2.8 6.4 7.1 10.0 7.6 3.2 6.0 7.0 7.3 7.1 7.6 1.3 0.0 4.5 6.3 7.2 3.5 1.7 2.2 8.0

o 3.8 1.5 2 .3 2.4 1.6 7.1 6.8 2.7 7.7 2.5 8.6 8.2 9.1 2.3 3.5 2.0

F

I

J E

J

E K

K

M

E N G

o

H

o

o

N

J

M F H

o L

Ai

A

=

0.5

A- B- C- D- E- F- G- HI- Je Ke L. M- N- Oe

B C

A i

1753

=0

Fig. 1.

A.1. = 20 Group preference orderings by RA

o

o

L K M H I A B

H

p

16

C

P

=1

Fig. 2.

Group preference orderings by R P

1754

T. Tanino, H. Nakayama and Y. Sawaragi

G I N

J E K D M

F H 0 L

I

N

D

K

M

F

0

L

A

A

B C

B

I

A

B C

C ~. =

1.

0

1.

(a)

~ l=

0 .5

~.

D

~2 =

20,

~3=

10,

~4= ·· ·=~16= 0

(b)

Fig. 3.

30,

(c)

Group preference orderings by

G N

R~

G

J

E I D

l'

D

H

o

K

F M

preference as a weak order in most cases. As a matter of fact, however, this is a too severe requirement, particularly in an early phase of a decision making process and the weak order obtained might not be reliable enough to determine a final choice. Sensitivity analysis is an effective approach for making up for this unreliability. For instance, in case of the sum of individual utilities, it can be made by changing the weighting coeeficient vector slightly. However, the sensitivity analysis approach is not always satisfactory in clarifying the preference structure in the group in intelligible forms. Because it is rather difficult to find through the sensitivity analysis where the differences among individual preferences exist and how much the degrees of the differences are. In contrast with this, the parametric preference orderings suggested in this paper have meaningful parameters and, by observing several group preference orderings (which may be depicted as directed graphs in case of a finite number of alternatives) obtained by changing the values of the parameters, it is possible for all the group members to objectively grasp the differences among the individual preferences, the degrpes of the differences and the appropriateness of the basic aggregation rule. Therefore, the group can proceed the subsequent group decision making process more smoothly with the help of them.

L 0 H

B

A

C

B

C

Fig. 4.

7.

Group preference orderings by Rv

CONCLUSION

In this paper we have defined several families of parametric group preference orderings based on some aggregation rules in group decision making, that is, the sum of individual utilities, the lexicographic maximin rule and the Nash bargaining solution function. The families have the following common features: 1) They include practically meaningful pa rameters. 2) The ordering corresponding to each value of parameter is a partial order. 3) They have the monotonicity property with respect to the parameters. 4) They have two extreme cases and one of them coincides with the weak order by the basic aggregation rule. 5) The other extreme case coincides with the Pareto partial ordering (except for the case of the ordering from the Nash bargaining solution function with n > 2).

REFERENCES Fishburn, P. C. (1973). The Theory of Social Choice. Princeton Univ. Press, Princeton. Inoue, K., T. Tanino, H. Nakayama and Y. Sawaragi (1980). A trial towards group decisions in structuring environmental science. The Fourth International Conference on 11ultiple Criteria Decision Making, Delaware. Nakayama, H., T. Tanino, K. Natsumoto, H. l1atsuo, K. Inoue and Y. Sawaragi (1979). Methodology for group decision support with an application to assessment of residential environment. IEEE Trans. Syst., Man & Cybern., 19, 477485. Sage, A. P. (1977). Methodology for Large Scale Systems. McGraw-Hill, New York. Sen, A. K. (1970). Collective Choice and Social Welfare. Holden-Day, San Francisco. Tanino, T., H. Nakayama and Y. Sawaragi (1980). On methodology for group decision support. The Fourth International Conference on Multiple Criteria Decision Making, Delaware.

The pr ime object of the theory of group decision making has been obtaining a group

For Discussion see page 1785