A counterexample on adding utilities

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Economics Letters 1 (1978) 11-l 3 0 North-Holland Publishing Company

A COUNTEREXAMPLE

ON ADDING

UTILITIES

G. YUVAL Computer Science Department, Ifebrew University. Jerusalem. Israel

Received

April 1978

Given utility scales for two individuals their sum can be used to indicate the Pareto optima. conjecture is that any social utility that is consistent with the Pareto criterion can be constructed as the sum of some pair of individual utility representations. This is disproven by counterexample.

A

Given the preference orders of two individuals, X and Y, there are many ways to translate them into utility values for the various alternatives. If we add X’S and Y’s utility scales, we set a third scale, from which an order of combined preferences is implied. Because of the non-uniqueness, given the preference orders, of the two utility scales, we can get many different orders of combined preferences by this method. The only obvious limitation on the combined order is the Pareto rule that, if both X and Y prefer alternative A to alternative B, the combined order must do so too. The question now arises whether the Pareto rule is the only limitation on the combined order. Note that, in the two cases where X always agrees, or always disagrees, with Y, this is indeed the case. In this paper we shall show that the Pareto rule is not the only limitation. We shall give three orders of preference, one for X, one for Y, and a third, which obeys the Pareto rule, but which cannot be obtained by assigning utility scales to X and Y and adding them up. Let there be eight alternatives, number 1-8, where X’s preference (in increasing order) is 25618347, and Y’s preference is 41236785. Fig. 1 shows the eight alternatives, plotted so that X’s preferences give the X axis, and Y’s preferences give the Y axis. The preference order 12345678 obeys the Pareto rule, as may be seen by inspection. But we shall now show that it cannot be obtained by assigning utilities to the two orders and adding. The basic argument is seen in fig. 1. Consider the four rectangles given by the pairs of corners 12, 34,45,67. In each case, the combined preference between the

12

G. Yuval /A counterexample

on adding utilities

5 ----7,

~._-_---__.-.~ !

1 I

L____

.

.

..___

__7

~_----_“‘___~

I ‘__---________,

-x Fig. 1.

two points is that given by the rectangle’s short side. Thus, fig. 1 is badly drawn, and should be redrawn with each rectangle’s short side getting long enough to dominate its long side. But because each short side is a subinterval of some other long side, this cannot be done. To turn the last paragraph’s hand-waving into a proof, let X1 be X’s utility value for alternative 1, and so on. Suppose preference order 12345678 is obtained by adding two utility scales. Then the order within the pairs 12,34, 56, 78 implies Xz+Ya>Xr+Yr, Xq+Y4>Xa+Ya, xlj+Y6>xs+Y5, Xs+Ys>X7+Y7, which implies xr - xa < Y, - Yr ) Ys-Yq
imply

xg-xs
G. Yuval /A counterexample on adding utilities

because, in each case, the smaller interval is a subinterval Combining the last eight inequalities, we set x*-x~
-x5


-x2

of the larger one.

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