American Journal of Oper ations Research, 2011, 1, 51-56
doi:10.4236/ajor.2011.12008 Published Online June 2011 (http://www.SciRP.org/journal/ajor/)
Copyright © 2011 SciRes. AJOR
A Museum Cost Sharing Problem
Yuntong Wang
Department of Economics, University of Windsor, Windsor, Canada
E-mail: yuntong@uwindsor.ca
Received March 29, 2011; revised April 22, 2011; accepted May 11, 2011
Abstract
Ginsburgh and Zang [1] consider a revenue sharing problem for the museum pass program, in which several
museums jointly offer museum passes that allow visitors an unlimited access to participating museums in a
certain period of time. We consider a cost sharing problem that can be regarded as the dual problem of the
above revenue sharing problem. We assume that all museums are public goods and have various (e.g., ser-
vice) costs. These costs must be shared by museum visitors. We propose a cost sharing method and provide
an axiomatic characterization of the method. We then define a game for the problem and show that the cost
sharing method is the Shapley value of the game. We also provide a comparative statics analysis for both the
Shapley value of the museum pass game and the Shapley value for the cost sharing game when the number
of museums and/or the number of visitors change.
Keywords: Cost Sharing, Shapley Value
1. Introduction
Ginsburgh and Zang [1] consider a revenue sharing pro-
blem for museums participating in a museum pass pro-
gram. In the museum pass program, museum passes pro-
vide visitors an unlimited access to participating mu-
seums in a certain period of time. The total revenue from
sales of museum passes must be shared among the par-
ticipating museums. Since some museums may have
more visitors than others, museums with more visitors
should share more revenue. In general, each visitor may
visit more than one museum. Thus, one cannot simply
distribute the total revenue to museums in proportion to
the number of visitors each museum has had. Ginsburgh
and Zang define a cooperative game in which museums
team up in offering the access passes. They apply the
Shapley [2] value to allocate the joint income from the
sale of passes among the museums.
We consider a cost sharing problem that can be re-
garded as the dual problem of the above revenue sharing
problem. We assume that all museums are public goods
and have various costs. These costs must be shared by
museum users. Thus, museum users (customers) pay
different fees for the services provided by different mu-
seums. This is in contrast to the museum pass program in
which users pay the same fee for a pass. In the museum
pass program, some users may subsidize for some other
users. Here, we require that different users pay different
fees depending on their different uses of the services
provided by different museums.
We propose a cost sharing method for the cost sh aring
problem. The method simply allocates each museum’s
cost equally among the visitors who have visited the
museum. We first provide an axiomatic characterization
of the method by the axioms of Additivity, Anonymity
and No Blind Cost. Additivity and Anonymity are the
standard axioms in the cost sharing literature ([2,3]). No
Blind Cost says that if the museums visited by a visitor
all have zero costs, then this visitor shouldn't pay any
cost. We then define a game for the cost sharing problem.
The game can be considered as the dual game of the
museum pass game. We show that the cost sharing
method is the Shapley value of the game.
We also provide a comparative statics analysis for
both the Shapley value of the museum pass game and the
Shapley value of the cost sharing game when the number
of museums and/or the number of visitors change. For
the museum pass game, we demonstrate by an example
that adding one more museum will decrease all muse-
ums’ revenue if there is no increase in the number of
visitors. But if the additional museum brings additional
visitors to the museum itself as well as some other mu-
seums, then it is possible that some museums are better
off and some are worse off. For the cost sharing game,
we demonstrate by using a similar example that adding
one more museum will increase all visitors’ cost if there
52 Y. T. WANG
is no increase in the number of visitors. But if the addi-
tional museum brings additional visitors to the museum
itself as well as some other museums, then it is possible
that some visitors are better off and some are worse off.
c1
c2
In general, our model can be used to deal with cost
sharing problems where a finite number of users use a
finite number of production functions (or service facili-
ties). There are many cost sharing problems involving
two groups of agents in which each agent from the first
group obtains services from some agents in the second
group. And the total service costs incurred to the second
group must be shared among the agents in the first group
(see Moulin and Laigret [4]).
c3
2. The Model and Proposed Method
Let
1, ,
M
m denote the set of customers (e.g., mu-
seum users), and let
1, ,N
1,,
n
c
jM
n
denote the set of service
providers (e.g., museums), where are two positive
integers. Let be the cost vector, where
, is the cost incurred by service provider .
nm,
0iN
Cc
i
ci
For each customer , let
K
j denote the set
of service providers whose services were utilized by
customer . Denote such a mapping from
j K
M
to
. For each service provider , let
2NiN

1
K
i
be the
set of customers who use the service of provider . De-
note i
1
K
such an inverse of . Assume that for each
service provider i, K
N
Ki
1. Denote the
set of all possible mappings from
M
to . A prob-
lem is a list 2N

,,,
M
NKC where and
K
n
RC
.
A solution is a vector
,,
m
m1
x
xxR
such that.
j
i
jM iN
x
c


A method is a mapping that assigns to each problem
,,,
M
NKC a solution
,,,
x
MNKC . Except Sec-
tion 5, through out the paper, we fix the sets
M
and
. Thus, we simply call
N

,
K
C a problem and

,
x
KC a solution.
In this paper, we propose and study the following
method which allocates each service provider's cost
equally to the customers who have utilized that service:
 

1
,,
i
jiKj
c1,,
x
KCj m
Ki

(1)
where

1
K
i
is the number of elements in the set

1
K
i
.
Example 1. In Figure 1, there are three customers
1,2,3, and three service providers 1,2,3. Customer 1 uses
all three service providers, customer 2 uses only service
providers 1 and 2 and customer 3 uses service providers
2 and 3. Suppose that service providers’ costs are given
by the vector .

123
,,Cccc
Figure 1. The left side denotes the three customers. The right
side denotes the three service providers along with their costs,
respectively.
Then the cost sharing method (1) gives the following
solution:
112
111
,
232
3
x
cc c
21
11
,
23
2
x
cc
32
11
,
32
3
x
cc
3. Axiomatic Characterization of the Method
In this section, we provide an axiomatic characterization
of the method (1). For this purpose, we require the fol-
lowing axioms.
Additivity: Fix
K
. For any

11 1
1,, n
n
Cc cR
 and
22 2
1,, n
n
Cc cR
, we have

12 1
,,
jj
x
KCCx KC
2
,
j
x
KC for all jM
.
Additivity is a classical axiom in the cooperative game
theory [2] and in the cost-sharing literature [3]. In the
present context of museum cost sharing problem, we can
provide the following interpretation. If museums’ costs
are split into two parts, for example, the capital cost and
the operating cost, and the cost allocation of each part of
these costs is computed, then the sum of these two cost
allocations would be equal to the cost allocation obtained
by applying the method to the unsplit total costs.
A permutation
of
1, ,
M
M is a one-to-one
mapping from
M
to
M
, i.e., :
M
M
and for all
,ij M
,

ij
if and only ij.
Anonymity: For any permutation
of
M
and any
n
CR
,


,,
x
KCx KC

where
,,
K
CK

C

,

K
iK i

, and

j
x
x
m
for
x
R
.
In words, Anonymity requires that the costs allocated
to the customers do not depend on their names.
No Blind Cost: For any , if for all jM
iKj,
0
i
c
, then
,0
j
xKC
.
Copyright © 2011 SciRes. AJOR
Y. T. WANG
53
In words, for any customer, if the costs of all museums
he or she has visited are all zeros, then the customer
shouldn’t pay any cost. In other words, there are no
cross-subsidizations.
Theorem 1 The method defined in (1) is the only
method that satisfies the axioms of Additivity, Anonymity,
and No Blind Cost.
Proof: It is easy to check that the method (1) satisfies
the axioms of Additivity, Anonymity, and No Blind
Cost.
Now we show that the method (1) is the only method
that satisfies the three axioms. Suppose that a cost sharing
method
satisfies the three axioms. Fix an arbitrary , K
and a cost vector . For any , let
where 1 is the th component of
the -dimensional vector . Note that
1,n
Cc c
, ,0
i
C

1, ,in
0, ,0,1,0
i
C
ni
1
K
i
i

1
\K i
is the
set of customers that are served by museum which has
one unit cost in No Blind Cost and Anonymity, we
have for all , and
i
C. By
0

,
i
jKC
jM
1
,1
i/
j
K
CK
i
for all .
1
jK
i
Now for any giv en , it can be uniquely written
n
RC
i
c
as . By Additivity,1 we have

1,,
ni
iN
Cc cC








11
\
,0
,,
i
ii
jiNKj iKjiKj
ii
jij
iN iN
cc
KC
K
iK
KcCcKC


 

 




 

i
for all .
jM
This completes the proof of the theorem. Q.E.D.
4. The Game and the Shapley Value
We now show that the method (1) defined above is, in
fact, the Shapley value of the following game
c
that
is associated to the problem

,
K
C:


,
i
iKS
cScSM

,
(2)
Where
 
jS
K
SK

S.
j, i.e., the set of service
providers whose services have been utilized by the cus-
tomers in
Recall that the Shapley value of a game
c
is de-
fined by


 

:
1! !\,1,,.
(3)
!
jSMjS
SmS
ccScSjjm
m





Proposition 1. The Shapley value of the cost game
c
defined in (2) coincides with the method (1).
Proof. For any iN
, consider the unit vector in
n
R
0, ,0,1,0, ,0,C
i
where 1 is the -th component of the vector . The
corresponding cost game defined by (2) is given by
ii
C
 
1.
0,
iifK Si
cSS M
otherwise


(4)
Clearly, all agents in

1
\
M
Ki M
are dummy
agents in the game and all agents in
i
c
1
K
i
c are
symmetric. Thus the Shapley value of the game is
given by
i

 
1
1
1,
0,
ijK i
Ki
c
otherwise
(5)
for all jM
.
Since the cost vectors,
1, ,
i
Ci n
n
R
, form a ba-
sis of , for any
n
RC
C
, it can be uniquely written
as i
i
iN
c
C
. By the definition of the cost game,
for SM
 , we have

 


i
ki
k
kKSkKS iN
ii
ii
k
iN kKSiN
cScc C
cCcc

 


 





 





S
where
k
C is the th component of the vector .
Because the Shapley value satisfies Additivity (and thus,
Linearity, see footnote 1), we have
kC







1
\
1
0
ii
jij
iNiNKj iKj
i
iKj
c
ccc
K
i
c
Ki


 

For all jM
. This completes the proof of the
proposition. Q.E.D.
5. Comparative Statics Analysis
Now we consider the effect of adding one more museum
on the solution of the problem. This is important because
when several museums consider the museum pass pro-
gram, they need to decide which museums shall be in-
cluded themselves. If each museum makes its decision
about joining the program independently, it becomes a
strategic game for all potential museums in a given geo-
graphic region. Of course, the revenue sharing method
they choose would eventually affect the equilibrium
1It can be easily shown that Additivity plus Positivity (non-negative
cost shares) implies Linearity, i.e.,
is linear with respect to the cost
vector . See Aczel [5].
C
Copyright © 2011 SciRes. AJOR
54 Y. T. WANG
number of museums joining the program. A complete
analysis of this strateg ic game is beyond th e scope of this
paper. Nevertheless, it is helpful to consider the effect of
adding one more museum (and likely generating more
visitors) on the revenue (cost) allocation.
For completeness, we first provide a comparative stat-
ics analysis of the Shapley value of the museum pass
game. We demonstrate by an example that adding one
more museum will decrease all museums’ revenue if
there is no increase in the number of visitors. But if the
additional museum brings additional visitors to the mu-
seum itself as well as some other museums, then it is
possible that some museums are better off and some are
worse off.
Next, we provide a parallel analysis for the Shapley
value of the cost sharing game. We also demonstrate by
using the same example as for the museum pass game
that adding one more museum will increase all visitors'
cost if there is no increase in the number of visitors. But
if the additional museum brings additional visitors to the
museum itself as well as some other museums, then it is
possible that some visitors are better off and some are
worse off.
Now consider the following example:
Assume that initially there are three museums and
three visitors (Figure 2(a)). Suppose that visitor 1 visits
museums 1, 2, and 3, visitor 2 visits museums 1 and 2,
and visitor 3 visitor museum 3 only. The Shapley value
solution of the revenue sharing problem is the following:
  
1111 1
1,2,3
32323
 
 
1
Suppose that one more museum, museum 4, joins the
museum pass program as shown in Figure 2(b). If mu-
seum 4 brings no additional visitors, then the solution is
the following:
 
11 11
1,2
42 42

 ,
 
11 11
3,4.
42 42


Obviously, all museums except the newly added mu-
seum are worse off than before.
But as shown in Figure 2(c), if the new museum 4
brings one additional visitor 4, who also visits other mu-
seums, e.g., museum 1, then museum 1 is better off while
museums 2 and 3 are worse off as shown below (com-
pared to the case without the m useum 4 and the visitor 4):
 
111 11
1,2
422 42

,
 
11 111
3,4
42422

 
(a) (b) (c)
Figure 2.In this revenue sharing problem, the left side de-
notes visitors and the right side museums. The Shapley value
solution assigns to each museum a proportional share of the
total revenue. The total revenue is omitted.
In general, adding one more museum without bringing
in more visitors will make all existing museums worse
off. But if more visitors are generated by the additional
museum and these additional visitors also visit other
museums in addition to the newly added museum, then
some museums may be better off and some may be
worse off, depending on how these visitors are distrib-
uted to the museums.
In parallel, we have similar results for the cost sharing
problem. We use the same example as above but now
assume that museums 1,,4 have costs and
, respectively. 123
,,ccc
4
c
Again, initially there are three museums and three visi-
tors as shown in Figure 3(a). Also suppose that visitor 1
visits museums 1, 2, and 3, visitor 2 visits museums 1 and
2, and visitor 3 visitor museum 3 only. The Shapley value
of the cost sharing problem is the following:

12
111
1,
222
ccc

3

12
11
2,
22
cc


3
1
3.
2c
Suppose that one more museum, museum 4, joins with
cost (
Figure 3(b)). If museum 4 brings no additional
visitors, then the new cost sharing problem has the fol-
lowing Shapley value:
4
c

1234
1111
1,
2222
cccc


12
11
2,
22
cc


34
11
3.
22
cc

Apparently all visitors are either worse off or no better
off than before.
But as shown in Figure 3(c), if the new museum 4
brings one additional visitor 4, who also visits museum 1,
.
Copyright © 2011 SciRes. AJOR
Y. T. WANG
55
then visitor 2 is better off while visitor 3 is worse off, but
it is unclear whether visitor 1 is better off or worse off as
shown below (compared to the case without the museum
4 and the visitor 4)2:

123
1111
1
3223
cccc
 
4
,

12
11
2,
32
cc


34
11
3,
23
cc


14
11
4.
33
cc

In general, adding one more museum without bringing
in more visitors will make all visitors worse off. But if
more visitors are generated by the additional museum
and these additional visitors also visit other museums in
addition to the newly added museum, then some visitors
may be better off and some may be worse off, depending
on the new distribution of visitors to museums.
6. Discussion
It is straightforward to show that the game (2) is convex,
and consequently its Shapley value is contained in its
core [6].
Our model contributes to the recent literature on cost
sharing problems involving cost functions that are gener-
ated from multiple technologies. In the traditional cost
sharing models, it is usually assumed that all agents share
a commonly owned technology given by a single cost
function (or production function). And the cost function is
independent of how the agents form coalitions. But in the
cost sharing problem discussed in this paper, the coali-
tions of visitors determine the costs of the museums they
have visited. In a recent paper, Trudeau [7] considers a
cost sharing problem with multiple technologies, where
(a) (b) (c)
Figure 3.This is a cost sharing version of the comparative
statics problem.
gains come from the presence of agents rather than from
the returns to scale. As pointed out in Trudeau [7], this
literature on cost sharing problem with multiple tech-
nologies is closely related to network formation problems
in which gains come from the presence of agents.
As we have pointed out in Section 5, the Shapley
value (in fact, any revenue sharing method) of the mu-
seum pass game induces a strategic game for the muse-
ums with regard to whether or not they should join the
museum pass program in the first place. For example, if
the number of visitors has increased and more revenue
has been received, the Shapley value allocation just dis-
tributes the additional revenue equally to museums that
are visited by these visitors. However, if one more mu-
seum joins the museum pass program, a new game must
be defined and the Shapley value must be recalculated.
Depending on how the additional visitors are distributed
among the museums, it is possible that some museums
may be better off and some worse off. But this, in effect,
will determine which museums actually join the program
(or are allowed to join by existing museum members if
they vote on accepting new members). Indeed, each
revenue sharing method would generate a strategic game
for all potential museums when they contemplate par-
ticipating the museum pass program. An important ques-
tion is: which revenue sharing method would generate an
efficient coalition of museums?
For the cost sharing problem, if one more museum is
added, its cost is equally allocated to those visitors who
have visited the museum. If one more visitor has visited
a number of museums, all existing visitors will benefit.
In general, when one more museum is added it usually
generates additional visitors to museums at the same
time. In this case, as demonstrated by the example in
Section 5, it is possible that some visitors are better off
and some are worse off. When the number of visitors is
large, choosing which museums to visit is an independ-
ent decision. Thus, unlike the museums in the museum
pass program, visitors usually do not act strategically.
Finally, we point out that the cost sharing model pro-
posed in this paper is more suitable in dealing with cost
sharing problems involving two groups of agents in
which each agent from the first group obtains services
from some agents in the second group. And the total ser-
vice costs incurred to the second group must be shared
among the agents in the first group. When the number of
agents (users and service providers) is not so numerous,
our cost sharing method provides a straightforward cal-
culation for the allocation of the costs. As ag ents are not
so many, choosing which service providers once again
becomes a strategic decision for the users, which, in turn,
depends on the cost sharing method.
2 For this example, it depends on whether or not 11
111
>
233
ccc4
.
C1
C2
C3
C4
C1 C1
C2 C2
C3 C3
C4
Copyright © 2011 SciRes. AJOR
Y. T. WANG
Copyright © 2011 SciRes. AJOR
56
7. References
[1] V. Ginsburgh and I. Zang, “The Museum Pass Game an d
Its Value,” Games and Economic Behavior, Vol. 43, No.
2, 2003. pp. 322-325.
doi:10.1016/S0899-8256(03)00013-7
[2] L. S. Shapley, “A Value for n-Person Games,” In: H. W.
Kuhn and A. W. Tucker, Eds., Contributions to the The-
ory of Games II, Annals of Mathematics Studies,
Princeton University Press, Princeton, Vol. 28, 1953, pp.
307-317.
[3] H. Moulin, “Axiomatic Cost and Surplus Sharing,” In: K.
J. Arrow, A. K. Sen and K. Suzumura, Eds., Handbook of
Social Choice and Welfare, North-Holland, Amsterdam,
Ch.6, 2002, pp. 289-357
[4] H. Moulin and F. Laigret, “Equal-Need Sharing of a
Network under Connectivity Constraints,” Games and
Economic Behavior, Vol. 72, No. 1, 2011, pp. 314-320.
doi:10.1016/j.geb.2010.08.002
[5] J. Aczel, “Function Equations and Their Applications,”
Academic Press, New York, 1966.
[6] L. S. Shapley, “Cores of Convex Games,” International
Journal of Game Theory, Vol. 1, No. 1, 1971, pp. 11-26.
doi:10.1007/BF01753431
[7] C. Trudeau, “Cost Sharing with Multiple Technologies,”
Working Paper, University of Montreal, Montreal, 2007.