We consider natural axioms for allocating the income of museum pass programs. Two allocation rules are characterized and are shown to coincide with the Shapley value and the equal division solution of the associated TU-game introduced by Ginsburgh and Zang (2003).Museum pass program; fair treatment; Shapley value; equal division solution

Béal, Sylvain

Solal, Philippe

English

Research Papers in Economics

Allocation rules for museum pass programs
Sylvain Béaly, Philippe Solalz
December 9, 2009
Abstract
We consider natural axioms for allocating the income of museum pass programs.
Two allocation rules are characterized and are shown to coincide with the Shap-
ley value and the equal division solution of the associated TU-game introduced by
Ginsburgh and Zang [1].
Keywords: Museum pass program, fair treatment, Shapley value, equal division
solution.
JEL Classiﬁcation number: C71.
1 Introduction
The museums of many big cities oﬀer a joint entry pass which gives pass buyers unlimited
access to several museums during a limited period. As an example the Paris museum pass
involves 60 museums and monuments in and around Paris. Three options are available
at diﬀerent prices for 2, 4 or 6 days. For the Louvre museum, 430000 visits in 2006 have
been attributed to the Paris museum pass. This note examines the problem of sharing
the income of a museum pass program among the participating museums.
In section 2, we deﬁne a museum pass program by considering a set of participating
museums, a set of pass buyers and, for each pass buyer, the group of museums visited
by the pass buyer. We consider natural axioms for museum pass programs. Some of
these axioms have been informally discussed in Ginsburgh and Zang [2]. Combining
some of these axioms allows to characterize two allocation rules for sharing the total
income of museum pass programs. The ﬁrst one consists in redistributing the income
of each pass equally to the museums visited by the pass buyer. This allocation rule
highlights the inﬂuence of each coalition of museums in the sharing process. The second
one attributes an identical share of the total income to each participating museum. These
two characterizations are comparable in the sense that one can switch from the ﬁrst to
the second by dropping one axiom and extending the principle of another axiom. In the
last section, we study this problem from a game-theoretic point of view by associating to
each museum pass program the TU-game considered by Ginsburgh and Zang [1]. It turns
out that the above allocation rules coincide with the Shapley value and the equal division
solution of the associated TU-game respectively. As such, our results are in the same
spirit as the comparable characterizations of the Shapley value and the equal division
solution obtained by van den Brink [4] on the class of all TU-games.
Financial support by IXXI (Lyon Complex Systems Institute) – research project “Cooperation, allo-
cation and complexity” – and the National Agency for Research (ANR) – research program “Models of
Inﬂuence and Network Theory” – is gratefully acknowledged.
yCorresponding author. Université de Saint-Etienne, CNRS UMR 5824 GATE Lyon Saint-Etienne,
France, sylvain.beal@univ-st-etienne.fr. Tel: (+33)(0)4.77.42.19.68. Fax: (+33)(0)4.77.42.19.50.
zUniversité de Saint-Etienne, CNRS UMR 5824 GATE Lyon Saint-Etienne, France,
philippe.solal@univ-st-etienne.fr. Tel: (+33)(0)4.77.42.19.61. Fax: (+33)(0)4.77.42.19.50.
12 Axiomatic study of museum pass programs
Let N = f1;:::;ng be a ﬁnite set of museums and X  N be the nonempty universe of
pass buyers. For each pass buyer k 2 X the nonempty set of museums visited by k is
denoted by Mk  N. The price of each pass is normalized to 1. A museum pass program
on N is given by a (possibly empty) subset X  X of pass buyers. Let 2X denote the set
of all museum pass programs on N. A payoﬀ vector y 2 Rn is an n-dimensional vector
giving a payoﬀ yi 2 R to each museum i 2 N. An allocation rule on 2X is a function
f : 2X  ! Rn that assigns a payoﬀ vector f(X) 2 Rn to each X 2 2X.
We consider ﬁve axioms that describe an allocation rule for museum pass programs in
a natural way. Full distribution states that the total income of the museum pass program
is fully redistributed to the participating museums.
Full distribution. For each X 2 2X, it holds that
P
i2N fi(X) = jXj.
Independence of external visitors states that the payoﬀ of a museum is not aﬀected if
one more customer buys a pass but does not visit the museum.
Independence of external visitors. For each X 2 2X, each k 2 X and each i 2 N
such that i 62 Mk, it holds that fi(X) = fi(Xnfkg).
Fair treatment requires that the change in payoﬀ resulting from the sale of an extra
pass is the same for any two museums.
Fair treatment. For each X 2 2X, each k 2 X and each i;j 2 N, it holds that
fi(X)   fi(Xnfkg) = fj(X)   fj(Xnfkg):
Fair treatment for visited museums consists in applying fair treatment only to muse-
ums visited by the extra customer.
Fair treatment for visited museums. For each X 2 2X, each k 2 X and each i;j 2 N
such that i;j 2 Mk, it holds that
fi(X)   fi(Xnfkg) = fj(X)   fj(Xnfkg):
Finally, positivity requires that no museum incurs a loss from sharing the total income
of the museum pass program.
Positivity. For each X 2 2X and each i 2 N, it holds that fi(X)  0.
We are now ready to study the consequences of combining some of these axioms. Note
that fair treatment implies fair treatment for visited museums.
Theorem 1 The unique allocation rule g on 2X that satisﬁes full distribution, indepen-
dence of external visitors, fair treatment for visited museums and positivity assigns to
each X 2 2X the payoﬀ vector
gi(X) =
X
SN:i2S
jfk 2 X : Mk = Sgj
jSj
, i 2 N: (1)
2Proof. [Uniqueness] Consider an allocation rule f on 2X that satisﬁes the four axioms.
The proof that f is uniquely determined is by induction on the number jXj of pass buyers.
Initial step: assume jXj = 0, i.e. X = ;. Then full distribution implies
P
i2N fi(;) = 0.
Together with positivity, we obtain fi(;) = 0 for each i 2 N. Next, assume that jXj = 1,
i.e. X = fkg for some k 2 X. Pick any museum i 2 N such that i 62 Mk if such a
museum exists. By independence of external visitors, we have fi(fkg) = fi(;) = 0. Full
distribution yields
P
i2Mk fi(fkg) = 1. Now, pick any i;j 2 N such that i;j 2 Mk. By fair
treatment for visited museums, we get fi(fkg)   fi(;) = fj(fkg)   fj(;) or equivalently
fi(fkg) = fj(fkg). Therefore, fi(fkg) = 1=jMkj for each i 2 Mk.
Induction hypothesis: let x 2 N be any number of pass buyers and assume that f is
uniquely determined for each X 2 2X such that jXj  x   1.
Induction step: consider any X 2 2X such that jXj = x. Pick any k 2 X. From
independence of external visitors, we get fi(X) = fi(Xnfkg) for each i 62 Mk. Together
with full distribution, this implies that
X
i2N
fi(X)  
X
i2N
fi(Xnfkg) =
X
i2Mk
fi(X)  
X
i2Mk
fi(Xnfkg) = 1: (2)
In addition, for each i;j 2 Mk, by fair treatment for visited museums we have
fi(X)   fi(Xnfkg) = fj(X)   fj(Xnfkg):
Summing both sides on museums j in Mk, we get
X
j2Mk

fi(X)   fi(Xnfkg)

=
X
j2Mk

fj(X)   fj(Xnfkg)

;
which is equivalent to jMkj(fi(X) fi(Xnfkg)) = 1 by (2). We obtain fi(X) = fi(Xnfkg)+
1=jMkj for each i 2 Mk. Since jXnfkgj = x   1, the induction hypothesis implies that
fi(Xnfkg) is uniquely determined. We conclude that f is uniquely determined for X.
[Existence] Consider the allocation rule g given by (1). For each X 2 2X, we have
X
i2N
gi(X) =
X
i2N
X
SN:i2S
jfk 2 X : Mk = Sgj
jSj
=
X
SN
jfk 2 X : Mk = Sgj
= jfk 2 X : Mk  Ngj
= jXj;
which proves that g satisﬁes full distribution. Next, pick any k 2 X and any i 2 N such
that i 62 Mk. Then
gi(X) =
X
SN:i2S
jfh 2 X : Mh = Sgj
jSj
=
X
SN:i2S
jfh 2 Xnfkg : Mh = Sgj
jSj
= gi(Xnfkg)
so that independence of external visitors holds. Now, for any k 2 X and any i;j 2 Mk,
we have
gi(X)   gi(Xnfkg) =
X
SN:i2S
jfh 2 X : Mh = Sgj
jSj
 
X
SN:i2S
jfh 2 Xnfkg : Mh = Sgj
jSj
=
1
jMkj
= gj(X)   gj(Xnfkg)
3which implies that g satisﬁes fair treatment for visited museums. Finally, g satisﬁes
positivity as a sum of non-negative rational numbers. 
The allocation rule g consists in redistributing the income of each pass equally to the
museums visited by the pass buyer. Replacing fair treatment for visited museums and
independence of external visitors by fair treatment yields a characterization of another
allocation rule that consists in sharing equally the total income of a museum pass program.
Theorem 2 The unique allocation rule e on 2X that satisﬁes full distribution, fair treat-
ment and positivity assigns to each X 2 2X and each i 2 N the payoﬀ ei(X) = jXj=n.
Proof. [Uniqueness] Consider an allocation rule f on 2X that satisﬁes the three
axioms. The proof that f is uniquely determined is by induction on the number jXj of
pass buyers.
Initial step: assume jXj = 0, i.e. X = ;. Then full distribution and positivity imply
that fi(;) = 0 for each i 2 N.
Induction hypothesis: let x 2 N be any number of pass buyers and assume that f is
uniquely determined for each X 2 2X such that jXj  x   1.
Induction step: consider any X 2 2X such that jXj = x. Applying fair treatment for
any k 2 X and any i;j 2 N, we have
fi(X)   fi(Xnfkg) = fj(X)   fj(Xnfkg):
Summing both sides on N, we get
X
j2N

fi(X)   fi(Xnfkg)

=
X
j2N

fj(X)   fj(Xnfkg)

:
By full distribution we have n(fi(X)   fi(Xnfkg)) = 1, which implies that fi(X) =
fi(Xnfkg)+1=n for each i 2 N. Since fi(Xnfkg) is uniquely determined by assumption,
we conclude that f is uniquely determined for X.
[Existence] It is obvious that e satisﬁes full distribution and positivity. Next, con-
sider any X 2 2X, X 6= ;, any k 2 X and any i;j 2 N. We have
ei(X)   ei(Xnfkg) =
jXj
n
 
jXj   1
n
= ej(X)   ej(Xnfkg)
which completes the proof. 
Logical independence of the axioms in Theorems 1 and 2 is shown by the following
allocation rules on 2X.
 The null allocation rule that assigns to each X 2 2X and each i 2 N a null payoﬀ
satisﬁes independence of external visitors, fair treatment for visited museums, fair
treatment and positivity. It does not satisfy full distribution.
 The allocation rule e satisﬁes full distribution, fair treatment for visited museums
and positivity. It does not satisfy independence of external visitors.
4 The allocation rule fg that assigns to each X 2 2X, the payoﬀs f
g
i (X) = 1 n+gi(X)
for some i 2 N and f
g
j (X) = 1 + gj(X) if j 2 Nnfig satisﬁes full distribution,
independence of external visitors and fair treatment for visited museums. It does
not satisfy positivity.
 The allocation rule fmax that assigns to each X 2 2X the payoﬀ vector
f
max
i (X) =
X
SN:i=maxj2S j
jfk 2 X : Mk = Sgj, i 2 N
satisﬁes full distribution, independence of external visitors and positivity. It does
not satisfy fair treatment for visited museums and fair treatment.
 The allocation rule fe that assigns to each X 2 2X, the payoﬀs fe
i (X) = 1 n+ei(X)
for some i 2 N and fe
j(X) = 1 + ej(x) if j 2 Nnfig satisﬁes full distribution and
fair treatment. It does not satisfy positivity.
3 Concluding remarks
In this section, we associate to each museum pass program X 2 2X the TU-game vX on N
such that for each coalition of museums S  N, vX(S) is the total income of passes that
would have been sold if only admissions for the subgroup of museums S were possible, i.e.
vX(S) = jfk 2 X : Mk  Sgj:
The game vX was introduced by Ginsburgh and Zang [1]. It is easy to check that
for each X 2 2X, the payoﬀ vector e(X) coincides with the equal division solution of
the museum TU-game vX. Furthermore, the Proposition in Ginsburgh and Zang [1]
establishes that for each museum pass program X 2 2X, the payoﬀ vector g(X) given by
(1) is the Shapley value (Shapley [3]) of the TU-game vX.
References
[1] V. Ginsburgh and I. Zang – “The Museum Pass Game and its Value”, Games and
Economic Behavior 43 (2003), pp. 322–325.
[2] — , “Sharing the Income of a Museum Pass Program”, Museum Management and
Curatorship 19 (2004), pp. 371–383.
[3] L. S. Shapley – “A Value for n-person Games”, Contribution to the Theory of
Games vol. II (H.W. Kuhn and A.W. Tucker eds), Annals of Mathematics Studies 28,
Princeton University Press, Princeton, 1953.
[4] R. van den Brink – “Null or Nullifying Players: The Diﬀerence Between the Shap-
ley Value and Equal Division Solutions”, Journal of Economic Theory 136 (2007),
pp. 767–775.
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