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
As mentioned in Section 1, the mechanism of mixed serial cost sharing will be defined on the class of cost functions which equal a sum of a convex and a concave cost
1 ↑
↓
function, denoted D 5 D 1 D . In general, we are interested in a cost sharing
1 ↑
↓
mechanism on D which is completely determined by its values on D
and D .
1 ↑
↓
Therefore, a decomposition rule is defined as a mapping G : D
→ D 3 D
where C 5 R 1 S for
GC 5 R, S and normalized by the requirement that the right-derivative of R at Q 5 0 equals zero. Furthermore, consider a pair of additive mechanisms
r on R
1
and s on S. Then a corresponding decomposition mechanism y
is defined on D by
G ; r,s
y C, q 5
rR, q 1 sS, q and GC 5 R, S. Clearly, the particular choice of
G ; r,s
decomposition rule G is crucial for the allocation of costs via the mechanism y
. Note
G ; r,s
also that only additive mechanisms r and s are considered.
Mixed serial cost sharing z relates to the particular decomposition rule, in short
denoted G , studied in Thon and Thorlund-Petersen 1986 and defined as follows.
CS 1
Every function C [ D has a right-derivative C9, possibly with C90 5 `, on [0,`[. For
2 21
´ ± 0 define the quotient D CQ 5 ´ C9Q 1
´ 2 C9Q for Q 1 ´ 0. Then the s
d
´ ↑
↓
CS-decomposition C 5 R 1 S, is determined by functions R [ D , S [ D such that
2 2
i lim sup D RQ D SQ 5 0; ii R90 5 0.
21
´ ´
´ →
If C 5 R 1 S is twice continuously differentiable on ]0, `], then 21i is equivalent to ;Q . 0: R0QS0Q 5 0.
22 If C 5 R 1 S is piecewise affine with respect to a finite subdivision of [0, `[, then 21i
means that at any Q . 0, at least one of the functions R and S is affine on an open neighbourhood of Q. It follows from Thon and Thorlund-Petersen 1986 that among all
possible decompositions the CS-decomposition maximizes the concave component.
As a simple example, consider the following cost function. Suppose that a good is sold at a price of 1 dollar per unit and furthermore that a single bundle of 10 units is
offered at a price of 80 cents per unit. Then the relevant cost function C is determined by CQ 5 Q for Q , 8, CQ 5 8 if 8 Q , 10, and CQ 5 Q 2 2 if 10 Q with
unique decomposition as a sum of a convex and concave cost function given by CQ 5 RQ 1 SQ 5 max 0, Q 2 10 1 min Q, 8 . However, if a 10 cent excise tax
h j
h j
is added, then total costs equal CQ 1 0.1Q with CS-decomposition max 0, Q 2 10 1 h
j min 1.1Q, 0.1Q 1 8 , which is not the only possible decomposition as for example
h j
CQ 1 0.1Q 5 max 0, 1.1Q 2 11 1 min 1.1Q, 0.1Q 1 8, 9 . h
j h
j
J .L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
59
1
Now, define mixed serial cost sharing as the mechanism z 5 y
on D , thus for
G ;
j ,h
CS
1
any C [ D with CS-decomposition C 5 R 1 S, one has
zC, q 5 jR, q 1hS, q. 23
Therefore, if C is convex, then zC, q 5 jC, q 5 x and if C is concave, then
zC, q 5hC, q 5 y. Note that even though payment scheme y may be negative for non-concave C, the vector of payments under
z, denoted z 5 zC, q, is always nonnegative. Consequently,
z defines a mechanism in the sense of 3. In order to motivate the choice of the CS-decomposition consider some general
decomposition mechanism y
which satisfies the following properties.
G ; r,s ↑
↓
a Extension Mechanism y
is identical to j on D and to h on D .
G ; r,s
b Independence of irrelevant cost levels Let q be given. If two cost functions C ,
1 1
C [ D coincide on the interval [a, b] , [0, `[ and a nq , nq b, then
y C ,
2 1
n G ; r,s
1
q 5 y
C , q.
G ; r,s 2
1
c Fixed-costs additivity For any cost function C [ D and fixed-costs function
1
F [ D , y
C 1 F, q 5 y
C, q 1 y
F, q.
G ; r,s G ; r,s
G ; r,s
Property a reflects the basic idea behind mixed serial cost sharing as a mechanism which is an extension of both
j and h. Hence consider for some G the mechanism y .
G ; j ,h
Since the CS-decomposition prescribes that if the function C is concave, then it is decomposed according to C 5 0 1 C, mixed serial costs sharing clearly satisfies a. On
the other hand, replacing the CS-decomposition by what would seem to be its natural counterpart, i.e. a decomposition which maximizes the convex component, leads to a
mechanism which violates property a; consider for example the concave function CQ 5 min 2Q, Q 1 1 5 max 0, Q 2 1 1 min 2Q, 2 . Clearly, b holds for both
j h
j h
j h
j and
h and reflects a basic property of the serial method as only cost levels between nq
1
and nq matter. Finally, property c appears to be a natural requirement of any cost
n
sharing mechanism on D . Note that the original serial scheme in Moulin and Shenker 1992 is additive and hence fixed-costs additive. Moreover, properties a and c jointly
imply that if for example the cost function equals fixed costs plus a linear function, then even an agent having zero demand pays an equal share of fixed costs; this can be
interpreted as an equal split of a common entrance fee.
1
It turns out that mixed serial cost sharing is the only decomposition mechanism on D which satisfies all three properties:
Theorem 1. Let
G and r, s be a given decomposition rule and a pair of additive mechanisms as above. Then the decomposition mechanism
y satisfies properties
G ; r,s
a –c if and only if it is identical to mixed serial cost sharing, i.e. G 5 G
and r 5 j,
CS
s 5h.
Proof. It is easily verified that z satisfies a–c. Thus we prove the converse. It suffices
to consider a piecewise affine cost function C which is determined by some subdivision
a, b
0 , Q , Q , . . . of [0, `[. If C equals the convex angle function C Q 5 max 0,
h
1 2
aQ 2 b , a, b . 0, then C has no other decomposition than C 5 C 1 0. Consequently, j
a, b a, b
by property a, rC
, q 5 jC
, q holds for any demand vector q. Every piecewise
60 J
.L. Hougaard, L. Thorlund-Petersen Mathematical Social Sciences 41 2001 51 –68
affine convex cost function equals a finite sum of convex angle functions on any bounded interval and by assumption, mechanism
r is additive over its domain; therefore r 5 j. In a similar fashion, consideration of concave angle functions C
Q 5 min aQ,
h
a, b
b shows that s 5h. j
1
Now, consider a cost function C [ D decomposed by
GC 5 R, S and a demand vector q confined by the interval a, b in the sense that a nq , nq b for some a,
f g
1 n
∨
b . 0. Firstly, if C is convex on a, b , then there exists a convex cost function C and a f
g
∨ ∨
∨
fixed-costs function F such that C 1 F
coincides with C on a, b . Hence, from
f g
r 5 j, s 5h, b; c; a and q . 0, it follows that
1 ∨
∨ ∨
∨
jR, q 1hS, q 5 y C 1 F , q 5
y C , q 1
y F , q
G ; j ,h G ; j ,h
G ; j ,h ∨
∨ ∨
∨
5 y
C , q 1 hF , q 5 jR 1 S 2 F , q 1hF , q
G ; j ,h
5 jR 1 S, q.
Thus as jR, q 1hS, q 5 jR 1 S, q for any q being confined by a, b , S must be
f g
affine on this interval.
∧
Secondly, if C is concave on a, b , then there exists concave cost function C and a f
g
∧ ∧
∧
fixed-costs function F such that C 1 F coincides with C on a, b . Then again using f
g
∧ ∧
∧
a–c, one obtains hR 1 S 1 F , q 5 y
R 1 S, q 1 hF , q 5 jR, q 1hS 1 F ,
G ; j ,h
q, and it follows that R must be affine on a, b . f
g Consequently, by Thon and Thorlund-Petersen 1986 the piecewise affine function C
is decomposed according to the CS-decomposition, meaning that G 5 G .
h
CS
It follows from 23 that mixed serial cost sharing z inherits positive cross-
monotonicity from j and h on their respective domain. Furthermore, for C 5 R 1 S,
inserting C 5 R in 15 and C 5 S in 17 one obtains directly by additivity in the sense of 23 upper and lower bounds for the mechanism
z. For example, the payment of agent 1 satisfies Cnq n z Rnq n 1 SQ n 2 1 2 S nQ 2 n 2 1q nn 2 1.
s d
1 1
1 1
For given C and strictly positive q, consider corresponding agent-specific prices z q , i 5 1, . . . , n. Such prices satisfy the following condition.
i i
Proposition 2. Suppose that q . 0. Let i , i , ? ? ? , i be an odd number of
1 1
2 2m 11
agents in N. Then the vector of prices z q , . . . , z q satisfies
1 1
n n
z q 1 z q 1 ? ? ? 1 z q
z q 1 z q 1 ? ? ? 1 z q
24
i i
i i
i i
i i
i i
i i
2 2
4 4
2m 2m
1 1
3 3
2m 11 2m 11
Proof. This follows by Lemma 1 since the vector z q , . . . , z q equals the sum of
1 1
n n
two nonnegative vectors x q , . . . , x q and y q , . . . , y q , each of which are
1 1
n n
1 1
n n
increasing and decreasing, respectively, in index i, and therefore satisfy 24. h
In particular, Proposition 2 implies that for all 1 h , i , k n one has z q z
i i
h
q 1 z q . Thus for i 5 2, . . . , n 2 1, z q 2 max z q .
h j
h k
k i
i j ±i
j j
Finally, a few comments on the stand-alone test are appropriate. This test requires that no coalition of agents can be asked to pay more than the cost of serving the coalition
alone. Since mixed serial cost sharing applies the decreasing rule on the class of concave
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
