Inequality comparisons of payment schemes

J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68 61 functions, then it must necessarily violate this principle because individual rationality is violated by the decreasing serial rule. Here, we take the view that in a number of cost sharing problems individual rationality is irrelevant as agents may be forced to cooperate due to regulatory economic or organizational constraints. Sufficient conditions for the decreasing serial rule to pass the stand-alone test are given in Hougaard and Thorlund- Petersen 2000.

5. Inequality comparisons of payment schemes

In this section, mixed serial cost sharing z is related, in terms of economic inequality, to the well-known schemes of average and marginal cost pricing as well as to the original serial scheme in Moulin and Shenker 1992. Moreover, in the same spirit, we compare the use of the CS-decomposition with any alternative decomposition. Define the mechanism of average cost pricing a on D by the vector of payments 1 a 5 CQ Q q. Secondly, for C [ D and q, define the vector of payments m by s d 1 ] m 5 q C9Q 1 CQ 2 QC9Q , i 5 1, . . . , n. 25 s d i i n AC ↓ Let D denote the functions in D for which average costs CQ Q are decreasing in Q . 0. If 1 AC ↓ C [ D D , 26 then the vector m is nonnegative. Hence define the mechanism of marginal cost pricing m on the domain 26 by mC,q 5 m. As an example of 26, consider the concave– convex function CQ 5 Q if Q , 1, CQ 5 1 if 1 Q , 2, and CQ 5 1 2Q if Q 2. Our first result shows that under schemes x and y, payment vectors are more less spread out than the payment vector of average cost pricing if the cost function is convex concave. Furthermore, marginal cost pricing yields payments which are more spread out than under increasing and decreasing serial cost sharing, respectively. Lemma 3. Let a denote the ordering of majorization, see Appendix B. Consider a given demand vector q and cost function C [ D . If C is convex, then a a x a m and a a y. If C is concave, then y a m a a and x a a. Proof. Suppose that C is convex or concave and that q . 0. Then by Lemma 1, x q 1 i i and y q are increasing or decreasing in i . Hence the four majorizations involving x, a i i or y, a follow from application of the quotient criterion, Appendix B, B.3. If C is convex and i 5 1, . . . , n 2 1, then by 6, 7, m 2 m 5 q 2 q C9Q i 11 i i 11 i Cr 2 Cr n 2 i 5 x 2 x , hence x a m follows by the difference criterion, i 11 i i 11 i Appendix B, B.2. If C is concave and j 5 1, . . . , n 2 1, then m 2 m 5 n 2j 11 n 2j q 2 q C9Q Cs 2 Cs n 2 j 5 y 2 y thus again by 6, n 2j 11 n 2j j j 11 n 2j 11 n 2j 62 J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68 7, and the difference criterion, it follows that y a m . Finally, m a a follows from the quotient criterion. h Lemma 3 shows that no definite conclusion can be drawn concerning majorization between the payment vectors of mixed serial cost sharing and of average cost pricing. With respect to marginal cost pricing, there is such a relation. Moreover, in order to compare mixed serial cost sharing z with the original serial mechanism determined by x 1 on C [ D one needs the following inequalities. Lemma 4. Consider the payment schemes x and y of 8 and 9. For a given demand q and index k [ N, we have ↑ C [ D implies i x 1 ? ? ? 1 x 1 y 1 ? ? ? 1 y CQ, ii x a y, 27 1 k k 11 n ↓ C [ D implies i x 1 ? ? ? 1 x 1 y 1 ? ? ? 1 y CQ, ii y a x. 28 1 k k 11 n Proof. We prove 28, as 27 can be proved in a similar manner. Thus suppose throughout in this proof that C is concave. Then if k 5 1, . . . , n, it follows from A.2, A.4, A.5 of Appendix A that Cr Cr Cr k 1 2 ]]] ]]] ]]]] x 1 ? ? ? 1 x 5 1 n 2 k 1 1 ? ? ? S 1 k n 2 k 1 1 nn 2 1 n 2 1n 2 2 Cr k 21 ]]]]]]] 1 D n 2 k 1 2n 2 k 1 1 Cr k ]]] 1 n 2 k s 2 s n n 2i 11 n 2 k 1 1 r r 1 2 21 ]]] ]]]] 3 C s 2 s 1 1 ? ? ? S S n n 2i 11 nn 2 1 n 2 1n 2 2 r i 21 ]]]]]] 1 DD n 2 i 1 2n 2 i 1 1 q 1 ? ? ? 1 q 1 n 2 kk 2 1 1 k 21 ]]] ]]]] ]]]]] S S D D 5 Cr 1 C n k n 2 k 1 1 n k 2 1 k n ] ]]]] 5 Cr S k n kn 2 k 1 1 q 1 ? ? ? 1 q n 2 kk 2 1 1 k 21 ]]]] ]]]]] S D 1 C n D k 2 1 kn 2 k 1 1 k n nn 2 k ] ]]]] ]]]] C r 1 q 1 ? ? ? 1 q s d S D k 1 k 21 n kn 2 k 1 1 kn 2 k 1 1 k n ] ] 5 C S q 1 ? ? ? 1 q D . 29 s d 1 k n k Furthermore, by A.3 it follows for k 5 1, . . . , n 2 1 that J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68 63 Cs n 2k ]]] y 1 ? ? ? 1 y 5 k 11 n k 1 1 Cs Cs Cs 1 2 n 2k 21 ]]] ]]]] ]]]] 1 k 1 1 ? ? ? 1 S D nn 2 1 n 2 1n 2 2 k 1 2k 1 1 Cs s n 2k 1 21 ]]] ]]] 1 k s 2 sn 2 j 1 1C s 2 s S S n n n 2j 11 k 1 1 nn 2 1 s n s s 2 n 2k 21 ]]]]] ]]]] 1 1 ? ? ? 1 DD n 2 1n 2 2 s k 1 2k 1 1 n q 1 ? ? ? 1 q 1 kn 2 k 2 1 n k 12 ]] ]]]] ]]]]] S S D D 5 Cs 1 C n n 2k k 1 1 n n 2 k 2 1 q 1 ? ? ? 1 q n 2 k n kn 2 k 2 1 n k 12 ]] ]]]] ]]]] ]]]]] S D 5 Cs 1 C n S D n 2k n n 2 k 2 1 n 2 kk 1 1 n 2 kk 1 1 n 2 k n nk ]] ]]]] ]]]] C s 1 q 1 ? ? ? 1 q s d S D n 2k n k 12 n n 2 kk 1 1 n 2 kk 1 1 n 2 k n ]] ]] 5 C q 1 ? ? ? 1 q . 30 s d S D n k 11 n n 2 k Adding 29 and 30 and using concavity yields 28i . Thus the pair of vectors x, y satisfies B.1 of Appendix B thus 28ii follows. h Now the following result can be established. 1 Proposition 3. For demand q and cost function C [ D , the payment vector x is more spread out than z, zC, q a x. Furthermore, if average costs are nonincreasing, 26, then payments under marginal cost pricing are more spread out than under mixed serial cost sharing, zC, q a mC, q. Proof. Since x a x, this follows from Lemmas 3 and 4. h Consider the CS-decomposition, C 5 R 1 S as used in mixed serial cost sharing. It follows from Lemma 4 that on the class of concave cost functions, choosing the payment scheme y rather than x amounts to selecting the minimum with respect to a of the set x, y . Similarly, choosing payment scheme x over y on the class of convex functions is h j equivalent to selecting the a-minimal payment scheme. In other words, choosing payment schemes x for increasing marginal costs and y for decreasing marginal costs is tantamount to choosing the most egalitarian payment scheme under the given circum- stances. On the other hand, it turns out that the vector of payments z based on the CS-decomposition is maximal with respect to a in the convex set of n-vectors hjC , 1 ↑ ↓ q 1 hC , q uC [ D , C [ D j. 2 1 2 64 J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68 ↑ ↓ Theorem 2. Consider functions R , R [ D , S , S [ D such that the cost function 1 1 C 5 R 1 S has CS-decomposition C 5 R 1 S. Then for any vector of demands q, 1 1 jR , q 1hS , q a zC, q, 31 1 1 Proof. The function G 5 R 2 R is increasing convex as can rather easily be proved 1 from Thon and Thorlund-Petersen 1986. In particular, if C is twice continuously 99 99 99 99 differentiable, then one has 0 5 R0 R 2 R0 1 S 2 S0 5 R0R 2 R0 1 R0S , thus s d 1 1 1 1 9 9 G0 0. As G9 5 R 0 2 R90 5 R 0 0, the function G is increasing convex and 1 1 Lemma 4 applies. One must prove that for any k 5 1, . . . , n k k O j R, q 1h S, q O j R , q 1h S , q . 32 s d s d i i i 1 i 1 i 51 i 51 Since G 5 R 2 R 5 S 2 S , 32 is equivalent to 1 1 k k k O h S, q 2h S , q O j R , q 2 j R, q 5 O j G, q, s d s d i i 1 i 1 i i i 51 i 51 i 51 which follows from 27ii . h At first sight the result of Theorem 2 is rather striking; as the CS-decomposition picks the maximal concave component, this method intuitively maximizes the role of h which one might expect to lead to more equal payment vectors. On the other hand, it follows 1 from Proposition 3 that the use of payment scheme x on the entire domain D leads to a more unequal payment vector than what results from z. It remains an open question whether there exists a counterpart to Theorem 2 in terms of an alternative decomposition rule leading to a unique payment vector of minimal inequality. An obvious candidate would be a decomposition which maximizes the convex component. However, a cost sharing rule based on such a decomposition, will violate the condition of Extension defined in Section 4; consider, as in Section 4, the concave cost function CQ 5 min 2Q, Q 1 1 which has CS-decomposition C 5 0 1 C as well as the decomposition h j CQ 5 max 0, Q 2 1 1 min 2Q, 2 . h j h j

6. Final remarks