J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
55
i 21 i 21
Cr 2 Cr Cr 2 Cr
k k 21
i i 21
]]]]] ]]]]]
x 5
O
q 2 q
O
q 2 q
i 21 k
k 21 k
k 21
r 2 r r 2 r
k k 21
i i 21
k 51 k 51
Cr 2 Cr Cr 2 Cr
x 2 x n 2 i 1 1
i i 21
i i 21
i i 21
]]]]] ]]]]] ]]]
]]] 5
q 5
q 5
q ,
i 21 i 21
i 21
r 2 r n 2 i 1 1
r 2 r q 2 q
i i 21
i i 21
i i 21
proving that x q ? ? ? x q . The remaining three sets of inequalities are proved in
1 1
n n
a similar fashion. h
As a consequence of Lemma 1, one can determine bounds on agent-specific prices even in the more general case of cost functions considered in the following sections.
Furthermore, Lemma 1 makes it possible to compare x and y with average and marginal cost pricing, see Section 5.
3. Upper and lower bounds on payments
In the literature, one has considered bounds on payments of agent i which depend only on q . It follows from Moulin and Shenker 1992 that if C is convex, then
i
payments x are bounded from above by the unanimity cost Cnq n and from below by
i i
the stand-alone costs C q , thus I C q x Cnq n. Furthermore, it follows from
i i
i i
de Frutos 1998 and Moulin 1996 that if C is concave, then II Cnq n y , III
i i
0 x C q . The bounds I–III cannot in general be improved if they are to depend
i i
on q only. However, the asymmetric nature of these bounds is a disadvantage from the
i
point of view of mixed serial cost sharing as will become apparent in Section 4; for example, there is no upper bound for y in II. In order to overcome this asymmetry and
obtain stronger bounds, define for any vector of demands q the ith lower sum Q and
i i
i
upper sum Q by Q 5 q 1 ? ? ? 1 q if i 2, Q 5 q
1 ? ? ? 1 q if i n 2 1, and
i 1
i 21 i 11
n n
Q 5 Q 5 0. In this section we establish upper and lower bounds on x and y which are
1 i
i i
stronger than I–III and depend only on q , Q , and Q .
i i
If demand of agent i is fixed, then x depends only on demands q , . . . , q and y
i 1
i 21 i
depends only on demands q , . . . , q . For any index i 5 1, . . . , n and fixed demand
i 11 n
vector q define the sets V q 5 q , . . . , q 0 q ? ? ? q
q and q 5 q if j i ,
u
h j
i 1
n 1
i 21 i
j j
i 5 2, . . . , n,
i
V q 5 q , . . . , q q 5 q if j i and q q ? ? ? q ,
u
h j
1 n
j j
i i 11
n
i 5 1, . . . , n 2 1,
n
and V q 5 V q 5
h
q
j
. It turns out that over these domains, j and h are
1 i
i
order-preserving functions with respect to the partial ordering of majorization see Appendix B. Thus
j C, q is Schur-concave on V q and h C, q is Schur-convex on
i i
i i
V q . For example, under mechanism j, agent n pays less the more the demands of the other agents are spread out in the sense of majorization; thus agents can manipulate
j by
56 J
.L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
averaging their demands. Moreover, manipulation of h is possible by spreading out
demands; for example, if q , q
and agents n 2 1and n demand q 2
´ and
n 22 n 21
n 21
q 1 ´, for ´ . 0 sufficiently small, then by Schur-convexity the payment of agent n 2 2
n
will increase. Consequently, neither j nor h are immune to reallocations of demand.
Now, consider some demand vector q and index i with q . 0. We define two vectors
i
related to the pair q, i which are more spread out than q, and one vector related to q, i which is less spread out than q. Firstly, for i 2 let h [ 1, . . . , i 2 1 be the number
h j
i 1
uniquely determined by i 2 h 2 1q , Q i 2 hq and define the vector q by q 5 q
i i
i
and
i
q 5 0, . . . , 0, Q 2 i 2 h 2 1q , q , . . . ,q , q , . . . ,q ,
i 5 2, . . . , n. 11
i i
i i
i 11 n
3 3
3 3
3 3
h 21 i 2h
n 2i i
Secondly, define the vector q by i 5 n 2 j 1 1 and
n 2j 11 i
q 5 q , . . . , q
, q , q
, . . . ,q , Q 2 j 2 2q
,
1 n 2j
n 2j 11 n 2j 11
n 2j 11 n 2j 11
3 3
3 3
n 2j j 21
n
j 5 2, . . . , n, and q 5 q. 12
i 1
1 1
n
¯ ¯
¯ Finally, define the vector q by q 5 q , Q n 2 1, . . . , Q n 2 1, q 5 Q n 2
1 n
1, . . . , Q n 2 1, q , and
n n
i i
i
¯q 5 Q i 2 1, . . . , Q i 2 1, q , Q n 2 i, . . . , Q n 2 i ,
i i
i
3 3
3 3
n 2i i 21
i 5 2, . . . , n 2 1.
Lemma 2. Let a denote the ordering of majorization , see Appendix B. For any pair
i i
i i
¯ q, i , i 5 1, . . . , n, one has q a q aq , q . In particular, q is a 2 maximal in
V q
i i
i
and q is a 2 maximal in V q. Furthermore, by replacing the last n 2 i or the first
i
¯ i 2 1 elements of q by those of q
, one obtains the a 2 minimal element in V q or
i i
V q, respectively.
Proof. This follows by application of the difference criterion, Appendix B, B.2. h
One can now establish the following bounds.
Proposition 1. Consider a given demand vector q and index i with q . 0. If C is convex,
i
then
i i
¯ j C, q j C, qj C, q .
14
i i
i
In particular , for i 5 2, . . . , n,
q Cr Q
Cnq 1
i 2 1
i i
i i
]] ]]]
S
]]
S
]]
DD
]] C q
x Cr 2
C n
. 15
i i
i
r n 2 i 1 1
n i 2 1
n
i
Furthermore, if C is concave, then
J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
57
i i
¯ h C, q h C, q hiC, q ,
16
i i
and for j 5 2, . . . , n,
i
Cnq 1
j 2 1 Q
n 2j 11
]]] ]]]
]] ]]
Cs 2 C
n y
S S DD
j n 2j 11
n n 2 j 1 1
n j 2 1
i
Cs C nQ 2 j 2 2q
s d
j n 2j 11
]] ]]]]]]]
2 .
17 n 2 1
nn 2 1
Proof. Inequalities 14, 16 follow from Lemma 2, Schur-concavity, and Schur- convexity. The first and the last inequality of 15 follows from convexity of C.
Furthermore, the second inequality of 14 is equivalent to the third inequality of 15. In order to prove the second inequality of 15, we proceed as follows.
i
If i 5 2, . . . , n, then payments x , . . . , x corresponding to q are determined by
1 i
21
x , . . . , x 5 0, x 5 n 2 h 1 1
C n 2 h 1 1 Q 2 i 2 h 2 1q , and
s s
dd
1 h 21
h i
i
x 5 ? ? ? 5x
h 11 i
1 1
]] ]]]
F G
5 C r 2
C n 2 h 1 1 Q 2 i 2 h 2 1q 18
s d
s s
dd
i i
i
n 2 h n 2 h 1 1
where r 5 n 2 i 1 1q 1 Q . Note that n 2 h 1 1 5 n 2 i 1 2 1 i 2 h 2 1 , n 2 i 1
i i
i 2
2 1 Q q and define g 5 i 2 hq 2 Q 2 q r 1 q . Then by 18 and convexity of
i i
i i
i i
i i
C, C n 2 i 1 2 1 Q q Q 2 i 2 h 2 1q
s s
dd
i i
i i
]]]]]]]]]]]] n 2 hx C r 2
s d
i i
n 2 i 1 2 1 Q q
i i
r Cr Cr 2 C r 2
g r 1 q q r Cr
Cr
s d
i i
i i
i i
i i
i i
i
]] ]]]]]]]]
]] ]]
5 1
g 1
g
i i
r 1 q r 1 q
r g r 1 q q
i i
i i
i i
i i
i
q Cr
i i
]] 5 n 2 h
. 19
r
i
By 19, the second inequality of 15 follows from the first one of 14. Finally, consider inequality 17. Under mechanism
h agents n and n 2 1 each pays
i
y 5 Cnq n and y 5 Cs n 2 1 2 Cs nn 2 1. Thus the vector s corre-
n n
n 21 2
1 i
i i
i i
sponding to q , see 5, satisfies s 5 nQ 2 j 2 2q and s 5 Q 1 n 2 j 1
1 n 2j 11
2
1q 5 s . Consequently,
n 2j 11 j
i i
Cs Cs
2 1
]] ]]]
y y
5y 5
2
n 2j 11 n 2j 11
n 21
n 2 1 nn 2 1
i
Cs C nQ 2 j 22q
s d
j n 2j 11
]] ]]]]]]]
5 2
. 20
n 2 1 nn 21
Then 17 follows by concavity of C, 16 and 20. h
The second inequality of 15 establishes a lower bound on x in terms of average
i
costs of r . In particular, x q CQ Q meaning that the agent with the largest
i n
n
58 J
.L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
demand never pays a price below average costs; this inequality can also be derived from Lemma 1. Note that in 17 there is no corresponding upper bound for y in terms of
average costs since y . 0 is possible even if q 5 0. One further important asymmetry
1 1
between 15 and 17 remains in the sense that in 17, there is no upper bound in terms of stand-alone costs C q . In order to establish such a bound one needs additional
i
assumptions on C and q, see Hougaard and Thorlund-Petersen 2000.
4. Mixed serial cost sharing
