Utilities for distributive justice

GEOFFREY ROSS

U T I L I T I E S F O R D I S T R I B U T I V E J U S T I C E *

The Meshing Problem and a Solution to it

ABSTRACT. This paper falls within the field of Distributive Justice and (as the title indi- cates) addresses itself specifically to the 'meshing problem'. Briefly stated, the 'meshing problem' is the difficulty encountered when one tries to aggregate the two parameters of beneficence and equity in a way that results in determining which of two or more alternative utility distributions is most just. A solution to this problem, in the form of a formal welfare measure, is presented in the paper. This formula incorporates the notions of equity and beneficence (which are defined earlier by the author) and weighs them against each other to compute a numerical value which represents the 'degree of justice' a given distribution possesses. This value can in turn be used comparatively to select which utility scheme, of those being considered, is best.

Three fundamental adequacy requirements, which any acceptable welfare measuring method must satisfy, are presented and subsequently demonstrated to be formally de- ducible as theorems of the author's system. A practical application of the method is then considered as well as a comparison of it with Nicholas Rescher's method (found in his book, Distributive Justice.) The conclusion reached is that Rescher's system is unacceptable, since it computes counter-intuitive results. Objections to the author's welfare measure are considered and answered. Finally, a suggestion for expanding the system to cover cases it was not originally designed to handle (i.e. situations where two alternative utility distributions vary with regard to the number of individuals they contain) is made. The conclusion reached at the close of the paper is that an acceptable solution to the 'meshing problem' has been established.

One of the most problematical and impor tan t difficulties in the field o f

distributive justice is the 'meshing problem' . To determine which of

several utility distributions is the most just, many philosophers claim

that one needs to employ two criteria: beneficence and equity. Simply

stated, the 'meshing problem' is the difficulty encountered when at tempt-

ing to aggregate these two parameters in such a way that one may deter-

mine which of two or more alternative utility schemes is the most just.

Intuitively, we want a procedure which will pick the distribution (out o f

several) which comes as close as possible to maximizing both total utility

and equity. Clearcut cases where one scheme is more equitable and beneficent than another present no problem. The difficulty arises when these

two principles conflict (e.g. when scheme A is more beneficent but less

equitable than scheme B). W h a t is needed is a method which balances the

equity vis-a-vis the beneficence o f a given utility scheme and computes its

Theory and Decision 4 (1974) 239-258. All Rights Reserved Copyright �9 1974 by D. Reidel Publishing Company, Dordrecht-Holland

240 G E O F F R E Y ROSS

total welfare (i.e. gives us a welfare measure of the distribution). The welfare measure of a distribution then, is the numerical value that is obtained when the amount of beneficence of a given utility scheme is measured or weighed against the scheme's equity. A satisfactory welfare measure will allow us to compare alternative utility allotments and to decide which one (or ones if two or more are equally just) is preferable. Before setting out a welfare measure, it is necessary to first arrive at acceptable definitions of beneficence and equity. When this is accomplished, it can then be determined if the proposed welfare measure incorporates these notions in a satisfactory way. Below are presented what I feel are acceptable definitions of beneficence and equity as applied to utility distributions.

I. THE B E N E F I C E N C E OF A D I S T R I B U T I O N

There doesn't seem to be anything controversial here, since it seems natural to use the total sum of utility as a measure of the beneficence of a distribution. Employing this concept, a procedure for computing the beneficence of a distribution can be easily formulated:

Given individuals (1, 2, ..., N) and the utility distribution (X1, X2, ..., XN), where X~ equals the amount of utility that an individual receives, the beneficence of a distribution is equal to the total sum of utility, allotted to all the individuals of that distribution.

Formalizing the above we have: The beneficence of a distribution (X1, X2,..., Xn)=

N

(X1 + X2 + ' " + XN) = ~ X~. f = l

II . THE E Q U I T Y OF A D I S T R I B U T I O N

Arriving at an acceptable definition of equity is a bit more difficult and problematic. Before setting out to do this, it should be noted that any adequate definition of equity must take account of certain criteria that the concept of an equitable distribution includes. When defining equity, it must be remembered that we are considering it as one of two parameters that will be used to determine how just a given distribution is. Therefore, we are considering equity sirnpliciter O.e. no secondary considerations like whether or not a certain individual deserves or needs more than an-

U T I L I T I E S FOR D I S T R I B U T I V E J U S T I C E 241

other are relevant) in order to discover how equitable a particular utility scheme is. Considering equity in this way, there are three criteria which should be included in any adequate definition of it.

(1) Ideally, everyone should receive an equal share (the mean) of the distribution; if this is not possible, the closer a utility scheme approximates this, the better it is.

(2) The amount of disparity between any two individuals in the utility scheme should be as small as possible.

(3) Distributions which result in individuals receiving large amounts above or below the mean, should be avoided whenever possible.

Keeping these adequacy requirements in mind, consider an equity measure which defines equity in terms of the sum of the differences from the mean. This equity measure computes the mean of a distribution and then totals the difference between each individual in the distribution and the mean. The utility scheme (comparing two at a time) with the lowest numerical value is the most equitable. Here is a formal definition of this concept of equity:

The equity of a distribution (X1, X2,..., XN)= N

E [xj - e] j = l

where

To illustrate the problems that the above definition of equity encoun- ters, consider the following two utility distributions:

EXAMPLE ~ 1

Share Scheme I Scheme H

(a) 12 11 (b) 10 11 (c) 10 9 (d) 8 9

When we apply the above equity measure to these two schemes, we find they both have a value of four. Therefore, according to the putative

242 GEOFFREY ROSS

equity definition, they are equally preferable. This conclusion seems clearly false, however, when we examine the two distributions more closely. In Scheme II, no individual is more than one point from the mean (10) while in Scheme I both (a) and (d) are two points away from it (also 10). Al- so, no one in Scheme II is more than two points away from any other individual; but in Scheme I, (a) and (d) are separated by four points. Finally, Scheme I has both a bigger winner and loser than II does. Apply- ing the three afore-mentioned adequacy criteria, Scheme II is clearly preferable to I; Scheme II avoids the big winners and losers of I, has less disparity between any two individuals and places no one further than one point away from the mean. To further exemplify that the above definition is unsatisfactory, consider the following two utility apportionments.

EXAMPLE # 2


(a) 13 11 (b) lo 11 (c) 10 11 (d) 10 11 (e) 10 9 (f) 10 9 (g) 10 9 (h) 5 9

Using the equity measure under discussion, Scheme I and II are equally equitable, since both receive a value of eight. Basically the same arguments that led us to reject this conclusion, with regard to the previous example, and opt for II over I are applicable and even stronger here. In this example, the individuals in Scheme II are in the same situation that they were in Example #1, but the state of affairs in Scheme I has worsened. Here (a) is a bigger winner than his counterpart in Example # 1 (3.25 points vs. 2 more than the average). Also the disparity between (a) and (h) in Scheme I of Example # 2 is much worse than it was for (a) and (d) earlier (8 points vs. 4), while Scheme II, in both examples, has no one more than two points from anyone else. Finally, in Example #2, (a) and (h) in Scheme I are further from the mean than (a) and (d) in Example # 1 Scheme I (3.25 and 4.75 vs 2) and in addition, no one is more than one

U T I L I T I E S F O R D I S T R I B U T I V E J U S T I C E 243

point from the mean in Scheme II in either example. Clearly, this equity measure must be rejected.

It is evident from the previous discussion that the notion of equity is not an easy one to capture. My suggestion is to consider dividing up the goods you have, starting with the individual who is getting the most (or at least no less than anyone else). Other things being equal, X1 deserves the mean of the distribution (X1, ~T~r2, . . . , X/q).

/q

Thus the difference between X1 and X/ (the mean of X1, X2,..., XN), i=1

is the measure of the equity of the scheme with respect to -t"1. It then seems reasonable to look at X2 vis-~t-vis the total remaining utility (i.e.

N

X2, X3,..., X~). All things being equal, X2 deserves Ri ; the difference be- i = 2

tween this average and X2 is the measure of the equity of the distribution relative to X2. Proceeding on in an analogous manner, we reach XN who

N deserves what he gets since XN is identical with _~i.

i = N

Finally, it seems reasonable to take the sum of these values as the equity measure of the apportionment as a whole. If one compares two distributions with the same number of individuals, the one with the lowest total is the most equitable.

Below is a formal presentation of this definition of equity. Let it be the case that - for all i and for all j, if i is less than j,

or equal to Xj (i.e., in distribution to Nindi- so arranged that X1/> X2,... i> XN- 1 >/-IN). of equity of distribution (X1, X2,..., X/q)Df

+ ( X 2 N - 1 /

then Xi is greater viduals, they are Then the degree

N

N

+ XN N l j =i=* ~ Xi N

N N ~ Xj = E Xi__ ~=I j=i ,=, ,= ~ r T i + 1"

.Ji- �9 ~ ~ - I -

N Zx,) . / ~ |

NvS i-1- 1

2 4 4 G E O F F R E Y ROSS

What we actually have here is a measure of the inequity of a distribution since the higher the value of the measure, the less equitable is the distribution. Thus in comparing two or more utility schemes, the one with the lowest value is the most equitable. Reserving the signs of these expres- sions will transform them into measures of equity.

Given that beneficence and equity have been defined, some criteria of adequacy, for any putative method of measuring the welfare of utility schemes, are needed. Three basic adequacy requirements are needed, although others might be added.

(1)

(2)

(3)

Given that two distributions are equally benefic, the one with the higher equity is better. Given that two distributions are equally equitable, the one with the higher beneficence is better. Given two schemes A and B, if A is a Pareto Improvement on B (i.e. no individual has less in A than in B, and at least one person has more in A than in B), then A is better than B.

Let us now consider a welfare measure which I will argue is acceptable and satisfies the three conditions of adequacy. Later, I will compare this method with another well-known one and demonstrate why mine is the better of the two. The welfare of a given distribution, which I will call the WM (Welfare Measure), is computed in the following way: Given that the distributions to be compared contain a fixed number of members, Let it be the case that for all i and for all j , if i is less than j , then X~ is greater than or equal to Xj (i.e. in a distribution of N individuals, they are so arranged that X 1 >>.)(2... >i X N_ 1 >1 XN). Then the W M of a distri-

bution ( X1, X2,. . . , XN)=

N bl N

x, + x, + x, /=1 i=2 i=N

N

(Where X~ is the mean of Xs, Xj + 1, ..., XN) i = j

N

N ~ Xi ~=j

= Z N - j + 1 j = l


This procedure compares distributions two at a time and the one with the higher WM is the best. If two distributions have identical WM's then they are equally preferable. To show that our method satisfies the above adequacy conditions, I will present them in order (1-3) below and prove them as theorems of the system.

THEOREM 1. I f two distributions (X1, X2, ..., XN) and (Yx, Y2,..., YN) to N individuals have equal means X = Y, then the WM of the X-distribution will be greater than the WM of the Y-distribution if[ the X-distribution is more equitable than the Y-distribution.

Proof. Arrange the indices of both distributions such that:

(1) i< j iff Xi>~Xi, (2) i < j iff Y,>~ Yj.

By the definition of equity given earlier, the equity measure of the X- distribution is

) Z x, = y, Z y' x, . i=1 j = i N - i + l /=1 j = / N - [ i + 1 i=1

Analogously, the equity measure of the Y-distribution is

E YJ Y, = E re E ~ . /=1 ~ N - i + I /=1 . = . ~ N - i + I /=1

Then the X-distribution is more equitable than the Y-distribution iff

" ~ Xj N N ~ N Z g x / > g 5 - 2 Y , . i = l j = / N - i + l /=1 / = l j = i N - i + l /=1

Since the means of the distributions are equal, we have

N N N X / . y ~ ~v N • = Y , i.e. Y , ~ = " ~ X / = ~ Y~ /=1 ,=1 ,=1 N- N . . . . /=1 i = l i = l

Hence the X-distribution is more equitable than the Y-distribution iff

N N Xj ~ 2v

~ = l j = / N - - i + I / = l j = / N - - i + I " But

i=1 j = / N - i + 1

246 G E O F F R E Y R O S S

is just the WM of a distribution, since the WM = the sum of i from 1 to N of

Xj N N X1 E= =Y~ Y, -=~N " - i+ 1 ~=l j = ~ N - i + I

Hence the X-distribution is more equitable than the Y-distribution /ff

WM of the X-distribution is greater than the WM of the Y-distribution. Q.E.D.

THEOREM 2. I f two distributions (X1, X2,..., XN) and (I"1, Y2,..., IN) are equally equitable, the WM of the X-distribution is greater than the WM of the Y-distribution iff the X-distribution is more beneficent than the Y- distribution.

Proof. If the X-distribution and the Y-distribution are equally equitable, then the equity measure of the X-distribution is equal to the equity measure of the Y-distribution, i.e.

N U X j N N N N

E E E x , = E E rj Er , i f l j = i N - i + l t=l ~=i j f ~ N - i + l ~=l

(A) But we have just seen that the WM of the X-distribution is

U N Xj Z Z i= j = ~ N - i + l

Also the WM of the Y-distribution is

N N Yi r y,

i = i i = ~ N - i + l

Substituting these equalities in (A) gives:

_ N X WM of the X-distribution ~t= 1 ~ =

WM of the Y-distribution - ~ = 1 Yr

The WM of the X-distribution minus the WM of the Y-distribution is

N N

x,- y~ Y,. (s) i = 1 i=1

Hence: The WM of the X-distribution is greater than the WM of the Y- distribution iffthe left hand side of the Equation (B) is positive. But the


left hand side of the equation is positive iffthe right hand side of the equation is positive in value, i.e. iff

N N

Z x , > Z i = l i = I

But

Xf = N and Yf = N . i = 1 f = l i

The WM of the X-distribution is greater than the WM of the Y-distribu-

t ion / f f

N > N i

and if/" u N g = ~ . i = l i = l

Hence, if the X-distribution and the Y-distribution are equally equitable, the one with the higher mean will have the higher WM. Q.E.D.

THEOREM 3. Pareto Optimality. I f (X~, )(2, ..., XN) and (Y~, Y2, ..., YN) are two distributions (ordered as specified above) such that for all i, Xi >1 Yi and such that there exists a k such that X k > Y~, then the W M of the X- distribution is greater than the W M of the Y-distribution.

Proof. Let al = X1 - Y1,

a2 = X2 - - Y2,

i.e. in general a~=X~- Yr

Then ak = Xk-- Yk>0. While for all other i, a~ = X i - Yi >>-O. Thus we have

X1 = Y1 + a l , x 2 = + a 2 ,

X u = YN + aN.

248 GEOFFREY ROSS

The WM for X-distribution is equal by definition to

But

N X j

= i=1 ~ i=i ~ \N--" i ~ 1 (replacing X i by Yj + a j)

N N N a j

= E rj + E E i=1 j=i N - - i + 1 ~ = l j = ~ N - - i + l

E i = l / = i N - - i + 1

is equal by definition to the WM of the Y-distribution. Hence, the WM of the X-distribution is the WM

distribution N N

+ ~ = aj . ~=1 ~N i + 1

of the Y-

Since ak > 0 and for all i [a, >t 0],

N lv a j

i=1 , N i + 1 > 0 .

.'. The WM of the X-distribution is greater than the WM of the Y-distribution. Q.E.D.

Thus, it can be concluded that the proposed welfare measure satisfies the three criteria of adequacy. It is hoped that, since the WM works in the cases covered by the three adequacy requirements, it will also provide an adequate criterion to compute the best distribution between two utility schemes that vary relative to each other, both with respect to equity and beneficence. The motivation behind this is that the WM results in the highest totals when each individual receives his exact share (i.e. in the cases we are considering - the mean). Also it penalizes individuals who receive a share above the average by giving them only the mean value of the distribution. Further, the WM assigns a low value to individuals receiving a share far below the mean since they fall near the end of the eom-


putation procedure and receive low values (usually their own value or lower). In short, if distributions contain individual values which vary greatly from the mean (both high above and far below), they will receive lower WMs than those which apportion shares closer to the mean. Thus, the WM of an inequitable utility scheme will in general be low relative to a more equitable one. Also, distributions which have a larger total utility than others will in turn have higher averages and therefore will score higher even if they are somewhat less equitable (but not too much less) than other schemes which are less benefic. These properties of the WM, I feel, are commensurate with our intuitions concerning distributive justice in general.

Let's consider a practical application of the method. Below are six schemes.

EXAMPLE #3.

Share

(a) 20 (b) 10 (c) 10 (d) 0

Share Scheme I V

(a) 40 (b) 0 (c) 0 (d) 0

Scheme I Scheme H Scheme 111

10 20 10 20 10 0 10 0

Scheme V Scheme V!

22 25 6 10 6 5 6 0

The order of preference that my welfare measure gives them is II, V, I, VI, III, and IV. This order, I will argue, agrees with our intuitions regarding distributive justice. Analyzing the six schemes, we find that II is the most just since everyone receives an equal share, and that V comes in second since everyone in this distribution is allotted something while at least one individual in the remaining four schemes receives nothing. This seems to fit our intuitions about fair play and a desire to prevent extreme deprivation whenever possible (which I will describe in more detail in a later example). Obviously, the worst distribution is IV and second from

250 GEOFFREY ROSS

the bottom is III (where two people receive zero). The only remaining consideration is between I and VI to determine third and fourth place in the preference ranking. Since (d) is allotted nothing in both schemes, the ana- lysis turns to the distribution of utility between (a), (b), and (c). Clearly, Scheme I is to be preferred to VI for the following reasons: (1) (c) is better offand closer to the mean in I than in VI; (2) (c) is also in a more equitable position with regard to (b) and (a) in I than in VI; (3) (a) is in a more equitable position [closer to the mean and closer in value to (b) and (c)] in I than in VI. Thus, the WM orders the six utility allotments in accordance with our intuitions about them.

Another method which has been proposed by Nicholas Rescher 1 for computing the welfare of a given utility distribution is worth noting here. Rescher's method computes what he calls the 'effective average' of a utility scheme. The higher 'effective average' of any two distributions represents the better one. The definition of the EA (effective average) of a distribution is as follows:

EA = Average - �89 (the standard deviation) or

N X~ N

Using Rescher's method we find certain discrepancies with mine. Con- sider the following two schemes:

EXAMPLE # 4


(a) 23 20 (b) 6 10 (c) 6 10 (d) 5 0

Reseller rates II higher than I (6.47 vs 6.25) while my approach places I above II (26.2 vs 21.7). The WM method seems to be the clearcut winner since both have the same amount of beneficence (40 units) but Scheme I is more equitable than II (13.8 vs 18.3). Thus, by dint of Theorem 1 (or

U T I L I T I E S FOR D I S T R I B U T I V E JU STICE 251

the first adequacy criterion), Scheme I is preferable. This result also seems to agree with our intuitions for the following reasons: (1) Clearly, (a) is better off in I than in II. (2) Considering (b), (c) and (d), we find that in distribution I they are in an approximately equitable situation, while in II (d) has nothing whereas (b) and (c) have 10. (3) Again, looking at (b), (c) and (d), both the Rescher method and the WM prefer the distributions (b), (c) and (d) receive in I over their allotment in II (Rescher, 5.7 vs 4.33 - WM, 16.2 vs 11.7). (4) While it seems unfair for (a) to receive more in Scheme I then he has in II (since he is already in an enviably strong position in II), the positive gain of raising (d) from 0 to 5 outweighs the negative effect of (a) gaining 3 points at (b) and (c)'s expense. 2

Let's look at another pair of distributions:

EXAMPLE # 5


(a) 2 3 (b) 2 3 (c) 2 1.24

Admittedly, the problems involved in deciding which of the above two schemes is best are a bit more difficult and subtle then the ones encountered in the earlier examples. However, a close look at the two distributions will, I feel, determine which of them is preferable.

Although in distribution II (a) and (b) gain a point, it is at (c)'s loss. Mr. (c) not only loses approximately 40% of what he had, but he moves from a totally equitable situation in Scheme I to a distribution in II where he has less than one-half of what the other two individuals [(a) and (b)] receive. In Scheme II (c) is in a state of great inequity with respect to (a) and (b) (some might say that he is in a state of utility deprivation) while in I he was on even footing with them. Our sense of fair play seems to persuade us that (a) and (b) ought not to gain in a way that is so inequitable relative to (c). Thus, Scheme I seems dearly preferable to II. Rescher, however, prefers II to I (2.1 vs 2) while my method places I ahead of II (6 vs 5.77) and corresponds with our reasoning and intuitions.

252 GEOFFREY ROSS

III. POSSIBLE OBJECTIONS TO THE WM SYSTEM

An objection that could conceivably be raised against the WM system might be based on a point put forth by P. M. Brown in 'Distribution and Values' 8 in which he claims that 'cancellativity' is a necessary adequacy requirement for any putative welfare measuring procedure. This criterion is expressed in terms of the 'principle of cancellativity' (PC) which he defines as follows:

Given any three distributions A, A', and B, where neither A nor A ' has any participants in common with B, the union of A with B is better than the union of A' with B if and only if A is better than A'. 4

With the PC in hand, Professor Brown goes on to direct his attention to- ward those welfare measuring systems which "seek to maximize the average happiness (welfare, etc.) according to some assignment of numbers to levels of happiness. ''5 In order to demonstrate that methods of this type are non-cancellative and as a result compute unacceptable preference orderings, he constructs the following example:

EXAMPLE # 6


(a) 9 9.1 (b) 9 9.1 (c) 9.1

Applying the PC, he adds a new member (d) (who is given 10) to the two distributions generating the two schemes below:

EXAMPLE # 7


(a) 9 9.1 (b) 9 9.1 (c) 9.1 (d) 10 10

Employing a technique which measures welfare by simply computing the mean of distributions and then preference ordering them in descending numerical order starting with the apportionment(s) with the highest aver-


age as best and ending with the lowest as worst, Brown finds that such a procedure is both non-cancellative and unsatisfactory for the following reasons:

(1)

(2)

In Example ~ 6 , Scheme II is preferred to I (9.1 vs 9.0) and this clearly coincides with our intuitions. In Example # 7 , where a new member has been added to both the allotments of ~ 6, Scheme I is now preferred to II (9.333 vs 9.325) even though II is obviously superior to I (since II here is Pareto Improvement on I).

Although the two cases presented by Brown above definitely show that the welfare measuring procedure which he considers is unacceptable, they pose no threat to the WM system. The WM method rates Scheme II higher than I in both examples (27.3 vs 18 and 36.6 vs 27.3 respectively). However, even though the distributions which he puts forth aren't counter-examples to my system, the WM itself is non-cancellative. Examples will be presented later that will establish this fact but which will also demonstrate the unsoundness of Brown's claims that the PC is necessary adequacy requirement for welfare measuring systems.

It should first be noted that Brown does not present any arguments demonstrating the soundness of incorporating the PC as an adequacy requirement for welfare measuring systems nor does he give us a set of convincing reasons why it should be accepted. The PC is defended only by dint of finding problems with Rescher's system (which I discussed earlier) and the mean computing method we have just looked at, and then pointing out that these two procedures are non-cancellative, as well. In short, aside from showing that some systems which don't satisfy the PC also compute unacceptable preference orderings, the validity of the cancellativity criterion has not been proven. Although, at first blush, the PC seems to be satisfactory, I will argue that this is because one initially views it only in terms of a quantitative change in the distribution(s) it is applied to. I will show below that when one scrutinizes this principle more thoroughly by applying it to certain utility apportionments, it brings about qualitative changes in these distributions which will ultimately lead to the rejection of the PC as a necessary adequacy requirement. There are no a priori reasons for believing that the fairness or equity of a utility allotment will not be altered by adding more individuals to it and assigning some amount of utility to each of them. Further, it might also be the case that adding a new distribution B to two apportionments A and A' alters the

254 G E O F F R E Y ROSS

original preference ranking of the new apportionments accordingly. Below are presented two examples which show that (1) Brown's PC is not a satisfactory adequacy requirement for the systems we are considering in this paper and (2) the WM procedure by, in part, being non-cancellative, computes the preference orderings which best fit our intuitions.

EXAMPLE #8


(a) 10 14 (b) 10 9

In this example the WM ranks Scheme II slightly higher than I (20.5 vs 20). One can argue that in this situation (or society) of only two people, (b) has only to make the small sacrifice of giving up one point to increase (a)'s utility substantially (4 points). In this state of affairs, (b)'s small loss is balanced out by (a)'s proportionately higher gains. This reasoning seems non-controversial and, I would claim, agrees with our intuitions.

Let us now test the PC by adding a distribution B (four individuals each receiving 10) to both schemes in Example # 8. This produces the two apportionments below:

EXAMPLE #9


(a) 10 14 (b) 10 9 (c) 10 10 (d) 10 10 (e) 10 10 (f) 10 10

Here I have added distribution B to the two schemes represented in Example # 8 and according to the PC the relative preference ordering of Scheme I vis4t-vis II should be the same in Example # 9 as it was in # 8. This is not the case, however, since the situation has changed with the addition of B for the following reasons:

(1) One can no longer argue (as we did in Example # 8) that (b) ought to give up one point to enhance (a)'s utility. In Example #9, (b) is one of


five people who could give up some of their share to increase (a)'s happiness but he alone is being asked to make the sacrifice. He can effectively argue against this by pointing out that before he was the only one who couM help out (a), now there are four others who can share the burden and singling him out is unfair.

(2) In Scheme II of Example # 9 (b)'s equity relative to the other members of the distribution is much worse than it was in the same scheme of # 8. Before, there was only one person with a higher share than (b)'s, but in the last example, there are five. Earlier, (b) was only lower than (a), he now finds himself below all the new people as well.

Clearly, Scheme II of Example # 9 is more inequitable than the same apportionment in # 8 (for the reasons expressed in (1) and (2) above). One conclusion that distinctly comes out as a result of the two foregoing examples is that the fairness and consequently the general welfare of a particular utility apportionment can change with the addition of a new distribution to it. Further, depending on how two or more schemes are configured, their original preference ranking can be changed by adding some utility allotment B to them. Thus, Brown's PC must be rejected as a necessary adequacy requirement for welfare measuring systems, since it leads to counter-intuitive and unacceptable results.

The final question to consider is whether or not the WM system takes account of the equity change that occurs when the new distribution is added to the two utility allotments in Example # 8. If it correctly does this, the WM should alter the preference ordering of the schemes in Example # 8 by rating Scheme I higher than II in # 9. In fact, it does compute these results by assigning 60 to Scheme I and 58.3 to II. This demonstrates both that the WM method is non-cancellative and that it generates results that coincide with our intuitions.

A second objection that might be leveled at the WM system is that it leads to the type of problem illustrated by the following set of distributions:

EXAMPLE # 10


(a) 20 20 (b) 20 20 (c) 20 2o (d) 12

256 GEOFFREY ROSS

Looking at the two utility allotments above, one would determine that Scheme I is better than II for these reasons:

(1) In Scheme I, all the members are in a perfectly equitable situation while this is not the ease in II, since the new member (d) who appears in it finds himself with a share substantially lower than all the other individuals in the apportionment.

(2) It seems intuitively clear that a quantitative increase in a distribution's membership should not be made at the expense of lowering it qualitatively. Expanding the size of a given population is to be avoided if this entails bringing individuals into it who will find themselves in a deprived state relative to the members of the initial distribution. In shbrt, 'the more the merrier' is not a satisfactory general societal rule and re- quires the qualification that it is necessary to consider both the amount of beneficence and the degree of equity that the expanded distribution will contain.

Applying this reasoning, the critic of the WM will then draw attention to the fact that although Scheme I is preferable to II in Example # 10, my method rates II higher than I (63.3 vs 60) and therefore the WM should be rejected.

My rebuttal to this is that the foregoing case does not constitute a counter-example to the WM because, as is clearly stated on page 244, the WM is to be applied only to distributions which contain a fixed number of individuals. Since Scheme II in Example # 10 has more members than I, the WM can't be employed to determine which is better.

Although the WM was not formulated to preference order utility schemes which don't have a fixed number of members, I would suggest that the following method can be used to handle these types of cases:

To preference order two or more distributions which vary with regard to the number of individuals they contain:

(1) Compute the WM of each utility allotment. (2) Divide the WM of each scheme by the number of in-

dividuals in it in order to get the 'per capita welfare' of each apportionment.

(3) Arrange the results of (2) from left to right in descending numerical order. This would be the preference ranking of the distributions being compared.


Formally then, the suggestion is that if one wants to compare two distributions (X1, X2,..., Xu) and (Y1, Y2,..., YN), where M # N (X1, X2, �9 .., Xu) is preferable to (Y1, Y2,..., YN) just in case

WM of (X1, X2,..., XM) M

i.e., just in case M N

M E x , N Z v, i~= i=,i ~-" MiS~+1 iN- j+1 j = l > "

M N

WM of ( Y1, Y2, ..., YN) >

N

It should be noted that the suggestion here being advanced to handle the case of distributions with different numbers of members is a natural generalization of the original WM system. For when M = N, the above formula reduces to the formalization initially presented to compute the welfare for utility schemes with a fixed number of individuals.

Applying this procedure to Example # 10 results in Scheme I being preferred to II (20 vs 15.8) which is in accord with our intuitions regarding them. It is my belief that this method is a plausible way to determine the preference ordering of this type of utility allotments. One would, of course, have to explore it very thoroughly before incorporating it without reservations into the WM system as an amendment to handle special cases of this kind.

In conclusion, I believe I have shown that the WM is a viable method for measuring the welfare of utility distributions for the following reasons:

(1) Using the above definitions of beneficence and equity (reasons for accepting these were given), it was proven that my system satisfies three fundamental adequacy requirements set down for any acceptable welfare measuring procedure.

(2) It was shown that the preference ordering obtained by applying the WM to alternative utility apportionments corresponded to our initial intuitions about them.

(3) Further, when these distributions were analyzed more closely, it was concluded that both our initial intuitions and the results generated by the WM were the most reasonable ones.

258 GEOFFREY ROSS

(4) I demonstrated, by means of clear-cut counter-examples to it, that a well-known welfare measuring system (the one presented by Nicholas Rescher in Distributive Justice) was inadequate.

(5) Two possible objections to the WM were considered and determined to be without substance since they were answerable from the standpoint of the WM system.

(6) It was established that the WM didn't leave itself open to the counter-examples which Resche~'s system and Brown's PC ran into. Rather, the WM opted for the distributions in these cases which were shown to be the intuitively superior ones.

(7) Finally, the WM method was found to be adaptable to situations which it was not originally designed to handle (viz. the preference ordering of utility schemes which do not contain the same number of individuals), further increasing the credibility of it.

While it is true that one might want to show that the WM method ful- fills certain other adequacy requirements, and/or satisfies certain desired axioms, it was demonstrated that it satisfies the plausible requirements for any putative welfare measuring system that were set out in this paper. Thus, within the parameters considered here, a solution for the 'meshing problem' and hence a method for measuring the welfare of utility distributions has been established.

Dept. of Philosophy, Stanford University

N O T E S

* I would like to gratefully acknowledge the assistance of Michael Tooley whose positive suggestions and critical comments were invaluable in the writting of this paper. 1 Rescher, Nicholas, Distributive Justice, Bobbs-Merrill Co., New York, 1966, p. 35. 2 I would argue here that our intuitions would lead us to conclude that it is usually better to improve the state of affairs of the lowest member of a given distribution even if this means raising the highest member(s). 3 Brown, P. M., 'Distribution and Values', Journal of Philosophy 66 (1969) 197-213. 4 Ibid., p. 200. 5 Ibid., p. 201-2.

Utilities for distributive justice

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