Festival as cultural sharing device

Festival as cultural sharing device

Luc Champarnaud

^∗†

Amandine Ghintran

^‡

Fr´ed´eric Jouneau-Sion

^§

.

January 22, 2013

Abstract

In this article, we provide a theoretical analysis of spillovers between theaters that take part in a performance festival. We model a festival as a cooperative game where theaters coordinate their prices to encourage their audience to attend to the other performances of the festival. The model captures the relationship between usual and transferred attendance, taking into account heterogeneity of tastes and of reaction to tariff. We provide several interesting properties of festival games (in- cluding convexity) and we axiomatically justifies the use of the Shapley value as a solution concept to redistribute the wealth created by the festival. Then, we apply this model to the NEXT festival, for which we have collected data, and compute it’s Shapley value.

J.E.L. classification : C71, D62, Z11, C31.

Keywords : Cooperative game, Cultural economics, Festival.

1 Introduction

One well known application of cooperative game to the field of cultural economics has shown that a simple form of Shapley value could be used to share a pass revenues between museums. Attendance is shown to be the only legitimate data for assessing the value to the museums sharing the pass

¹

.

In the neighboring field of performing arts, festivals of music, theater or dance are a very usual form of supplying art to amateurs. Festival is an event which gathers a set of well defined performances supplied by a group of theaters behaving as a cartel during a

∗Corresponding author, e-mail : [email protected]

†EQUIPPE, Universite Lille Nord-de-France, Universite Lille 3 UFR Mathematiques et Sciences So- ciales, BP 60149, 59653 Villeneuve d’Ascq cedex.

‡EQUIPPE.

§EQUIPPE.

1Ginsburgh, V., Zang, I., Sharing the Income of a Museum Pass Program,Museum Management and Curatorship, Vol. 19, No. 4, p. 371-383, 2001.

short period of time, while remaining competitors the rest of time. Each performance is a singular good that is not observable before experimenting it, and a ’cartel’ in this context simply means that the festival as a whole looks for Pareto efficiency. One often stresses that, due to the singularity of each experience, good or bad surprises experimented in the recent past have a strong impact on the demand for next performances

²

.

During the period of a festival, attendance can thus be considered as a flow partly go- ing from theaters to theaters according to their positive and negative experiences within the festival

³

. Therefore, each member of the cartel shares with the others a snowball effect, i.e. the spillover effect of the attendance transferred from all performances to the one of the others. Looking at moves, each theater attracts its own usual atten- dance -considered as given for a short period of time-, and cooperates by low tariffs that reinforce the spillover effect.

We provided a cooperative game that describes this peculiar cooperation structure. It captures the relationship between usual and transferred attendance, taking into account heterogeneity of tastes and of reaction to tariff.

In Section 2 we set up the model, define a (cooperative) festival game.

⁴

We prove its convexity. We also provide a characterization of the Myerson-Shapley value in the set of festival games. In Section 3, we apply our model to the dance and theater NEXT festival for which we collected data. We estimate the demand functions for each performance and assess spillover effects as functions of explanatory variables. Finally we use this estimation to compute a value sharing of the festival. An appendix section provides some supplementary material about the NEXT festival.

2 A theoretical game for festivals

In this section, we set up a theoretical festival game. We then show that the general game admits a simple linear structure. More precisely we show that the payoff vector of any festival game is the linear combination of a limited number of ”elementary” festival games. Then we show that all elementary festival game are convex in the sense of Shapley. The convexity of any festival game follows. Finally we propose an axiomatic characterization of the Myerson-Shapley value in the set of festival games.

2.1 Setup

let n be an integer larger than 2. We consider a set N =

{

1, 2, . . . , n

}

of theaters in which cultural performances may be organized. We assume the program is already constrained so that theaters are indexed in the chronological order. The first performance

2Levy-Garboua, Louis and Montmarquette Claude. 1996.A Microeconometric Study of Theater De- mand,Journal of cultural economics, vol.20 :25-50.

3That makes our question thereafter rather similar to the question addressed by Ambec, S., Sprumont, Y., Sharing a river,Journal of Economic Theory, 107, 453-462 (2002).

4Pr. Mamoru Kaneko together with several co-authors studied in various papers a discrimination model in which a ”festival game” is described. Our model bears little -if any- resemblance to Pr.

Kaneko’s works. We are grateful to Pr. Nicolas Gravel for mentioning this potential source of confusion.

is organized in theater 1 (and nowhere else) , the second in theater 2, and so on. This may be justified for instance in the case of internationally famous music bands or theater companies. Another explanation is that the issue of the best program may be addressed in a first stage.

Before the i-th performance takes place, a queue of potential spectators is formed.

The size of this queue is q

_i

> 0. Then this potential demand “learns” the price p

_i

due to attend the i-th performance. Notice here the price representation is merely a simplification device. The model may be rewritten so that spectators are informed about the quality of the performance instead of the price.

The demand is assumed to be a strictly decreasing function of the price so that the audience for the i-th performance is S

_i

(p

_i

)q

_i

. Moreover, we assume S(0) = 1 so that everybody in the queue attend the show if it is free.

⁵

We assume each shows bears linear costs so that the payoff for show i is S

i

(p

i

)q

i

(p

i−

c

i

).

After the i-th performance a proportion ρ

_i+1 ∈

]0, 1] of the audience gather to form the next queue together with some newcomers e

_i+1≥

0 so that

q

_i+1

= ρ

_i+1

S(p

_i

)q

_i

+ e

_i+1

The spillover effect increases with ρ

_i+1

. In accordance with the interpretation of new- comers in the rest of the festival, we shall consider that q

0

= 0 so that the attenders of the first performance are all “newcomers”. We could also consider that q

₀

captured attenders who came just because the festival takes place. In any case our convention creates no extra difficulties in the remaining theoretical analysis.

Each theater is assumed to benefit from a quasi linear utility function adding the payoff and some side payment t

_i

. As usual we assume that the side payments sum up to zero, so that Pareto Efficiency among any set of players is obtained if the sum of the payoffs over this set is maximized. This maximization is assumed to rely only on the prices (this is the only variable for firms once the program of the festival is set). When they do not decide to coordinate their prices the theaters simply choose a monopoly pricing without taking downward spillovers into account. A set of theaters that coordinate their prices is called a coalition. The festival game is then defined as the cooperative game such that the value v(T ) of the coalition T

⊂

N is the total payoff of this coalition, with the convention v(

∅

) = 0.

We assume that prices are credible commitment among the member of the coalition.

This may be obtained by the “menu” device attached to the program of the festival or by formal conventions signed among festivals organizers. In the context of the apply exercise of Section 3 below, the convention exists since the NEXT festival is subsidized the E.C. Interreg Program. According to this convention, the costs induced by any performance is cover up to 50% by the Interreg Program, provided the other member of the festival all agree to add the performance in the NEXT program.

Whether collectively pricing rule may be harmful from a welfare viewpoint is be- yond the scope of this paper. To the best of our knowledge there exists a single case of

5In this paper,Sstands for ”survival” since it should be interpreted as the proportion of the potential demand that ”survives” the price’s revelation.

legal investigation against a festival grounded on the US antitrust legislation. The in- vestigation by Attorney General Lisa Madigan concerns exclusivity clauses which artists playing in the Chicago music festival Lollapalooza must sign. According to Hiller (2012) the number of venues in the cities covered by the clause significantly decrease.

It shall be stressed that the spectators are not informed of the suppliers agreements.

Hence, whether a theater decides to join a coalition or not, the audience’s reaction remains unchanged. In this respect, our model departs from Ambec and Sprumont [1]. In the river sharing model of Ambec and Sprumont, deviant players take all the remaining water. In our model, a deviant player may choose a prohibitive price but it is against its own interest. By playing in an ”efficiently selfish way” a deviant player passes part of its audience to downward players, some of which may be members of the coalition. In this sense, the model is more closely related to Ambec and Elhers [2].

Ambec and Elhers consider a river sharing model in which deviation cannot outsets some bliss point. A major difference is that the core may be empty in the model considered by Ambec and Ehlers. As we claimed in the introduction, we shall show that any festival game is convex.

We end up this section by giving a formal definition of a festival game. To this end, let

S

be the set of twice differentiable functions from R

⁺

to R

⁺⁺

such that S(0) = 1, S

^�

(0) = 0, S ”(0) < 0.

Definition 1

Let (v, N ) be a transferable utility game. It belongs to

F

if and only if there exists three sequences of positive real numbers e

^�

= (e

₁

, . . . , e

_n

)

^�

, c

^�

= (c

₁

, . . . , c

_n

)

^�

, and ρ

^�

= (ρ

₂

, . . . , ρ

_n

)

^�

and (S

₁

, . . . , S

_n

)

∈ Sⁿ

such that for all T

⊂

N we have

v(T ) = max

^�_i∈S

(p

i−

c

i

)d

i

wrt p

^�

= (p

₁

, . . . , p

_n

)

^�

s.t. d

_i

= (e

_i

+ ρ

_i

d

_i−1

)S

_i

(p

_i

) d

0

= 0

p

_i

= p

^m_i

= argmax

_pi∈R+

S

_i

(p

_i

)(p

_i−

c

_i

),

∀

i

�∈

S.

Let us show that v(.) is well defined. Notice first that S

_i ∈ S

implies argmax

_pi

S

_i

(p

_i

)(p

_i−

a) = p

^m_i

is unique and well defined. Moreover we have p

^m_i

> max

{

a, 0

}

. Indeed, we have

∂

∂pi{

S

_i

(p

_i

)(p

_i−

a)

}

= S

_i^�

(p

_i

)(p

_i−

a) + S

_i

(p

_i

)

∂²

∂p²_i {

S

i

(p

i

)(p

i−

a)

}

= S”

i

(p

i

)(p

i−

a) + 2S

_i^�

(p

i

) As S

^�

(0) = 0, S ”(0) < 0 we have S

^�

(p) < 0 whenever p > 0 and

_∂p^∂22

i {

S

_i

(p

_i

)(p

_i−

a)

}

<

0 when p

_i

> a. Now S

^�

(0) = 0 and S(0) = 1 implies

_∂p^∂

i{

S

_i

(p

_i

)(p

_i−

a)

}pi=0

> 0 which implies argmax

_pi∈R+

S

_i

(p

_i

)(p

_i −

a) > 0. Moreover, if a > 0 and p < a then S

i

(p

i

)(p

i−

a) < 0 which implies argmax

_pi∈R⁺

S

i

(p

i

)(p

i−

a) > a.

We will now show that when a theater leaves any (non empty) coalition S we are left

with a new festival game. This is trivial if n leaves a coalition. If n > 2 consider T

⊂

N,

1 < i < n and i

�∈

T we have p

i

= p

^m_i

so that d

i

= (e

i

+ ρ

i

d

i−1

)S

_i^m

where S

_i^m

= S

i

(p

^m_i

)

and we may write:

d

₁

= e

₁

S

₁

(p

₁

)

d

₂

= (e

₂

+ ρ

₂

d

₁

)S

₂

(p

₂

) . . . d

i−1

= (e

i−1

+ ρ

i−1

d

i−2

)S

i

(p

i−1

)

d

_i+1

= (e

_i+1

+ ρ

_i+1

e

_i

S

_i^m

+ ρ

_i+1

ρ

_i

S

_i^m

d

_i−1

)S

_i+1

(p

_i+1

) d

_i+2

= (e

_i+2

+ ρ

_i+2

d

_i+1

)S

_i+1

(p

_i+1

) . . .

d

n

= (e

n

+ ρ

n

d

n−1

)S

n

(p

n

)

(1)

Finally if 1 leaves coalition T we may write d

₁

= e

₁

S

₁

(p

^m₁

)

d

₂

= (e

₂

+ ρ

₂

e

₁

S

₁^m

S

₂

(p

₂

) d

₃

= (e

₃

+ ρ

₃

d

₂

)S

₃

(p

₃

) . . . d

_n

= (e

_n

+ ρ

_n

d

_n−1

)S

_n

(p

_n

)

Thus in any case, if theater i leaves a coalition, the remaining players enter a new festival game in which e

_i−1

and ρ

_i−1

are redefined. Also remark that e

_i−1

in this new game is some affine function of previous e

_i

and e

_i−1

. We conclude that if v(N ) is well defined for any game then so is v(T ) for any coalition in any festival game.

Remark if ρ

_i

= 0 then all the decisions taken before i have no impact for the payoffs of downstream coalitions. More precisely let A(i) =

{

1, . . . , i

−

1

}

be the set of (strict) upstream ancestors of i. Define v

_|A(i)

(T ) = v(T) whenever T

⊂

A(i) and v

_|A(i)

(T) = 0 otherwise. Define v

_|A(i)\N

accordingly. If ρ

_i

= 0 the program defining v(T) may be divided in two parts, and we may write v(T ) = v

_|A(i)

(T

∩

A(i)) + v

_|A(i)\N

(T

∩

A(i)

\

N ).

So if we can prove that v(N ) is well defined in the case min

_{i∈{2,...,n}}

ρ

i

> 0 we are done.

Also, assume e

₁

= 0 then clearly for any coalition we have v(T ) = v(T

\{

1

}

) (that is 1 is a dummy player). Hence, if we have to prove that v(N ) is well defined in the case e

1

> 0 and min

_{i∈{2,...,n}}

ρ

i

> 0. Notice in this case we have d

i

> S

1

(p

1

)

^�i_j=1⁻¹

ρ

j

> 0 for all i, whatever the choice of p

₁

.

We now proceed by induction on n. As we already noted S

₁ ∈ S

implies v(.) is well defined in the case n = 1. Assume that the v(N ) is well defined for all games up to size n

−

1 and consider adding a new theater. As e

_n

+ ρ

_n

d

_n−1

> 0 theater n then chooses the monopoly price so as to solve

max

pn

(e

_n

+ ρ

_n

d

_n−1

)S

_n

(p

_n

)(p

_n−

c

_n

).

Theater n

−

1 must then solve

max

pn−1

(e

_n−1

+ ρ

_n−1

d

_n−2

)S

_n−1

(p

_n−1

)(p

_n−1−

c

_n−1

) + (e

_n

+ ρ

_n

S

_n−1

(p

_n−1

))S

_n^m

(p

^m_n −

c

_n

).

As e

n−1

+ ρ

n−1

d

n−2

> 0 this amounts to solve

max

pn−1

(e

_n−1

+ ρ

_n−1

d

_n−2

)S

_n−1

(p

_n−1

)(p

_n−1−

c

_n−1

(N )) (2)

with

c

n−1

(N ) = c

n−1−

ρ

_n

S

_n^m

(p

^m_n −

c

_n

) e

_n−1

+ ρ

_n−1

d

_n−2

and the program (2) admits a unique solution as we previously shown. Then we may consider n

−

2 as we considered n

−

1 and we are done.

2.2 Properties of festival games

We now establish several properties of festival games that may be useful for their own right or will be used in the sequel. The first lemma has already been proven and used in the previous section

Lemma 1

For any (v, N)

∈ F

where n > 1 and i

∈

N we define the game (v

_|i

, N

\{

i

}

) with characteristic function v

_|i

(T ) = v(T) whenever i

�∈

T. Then (v

_|i

, N

\{

i

}

)

∈ F

. Moreover if (v, N ) is associated with the sequences e, c, ρ, D then (v

_|i

, N

\{

i

}

) is associated with the sequences

e

_|i

= e

₁

, . . . , e

_i−1

, e

_i+1

+ ρ

_i+1

e

_i

S

_i^m

, e

_i+2

, . . . e

_n

c

_|i

= c

₁

, . . . , c

_i−1

, c

_i+1

, . . . , c

_n

ρ

_|i

= ρ

₁

, . . . , ρ

_i−1

, ρ

_i

ρ

_i+1

S

_i^m

, ρ

_i+2

, . . . ρ

_n

S

_|i

= S

₁

, . . . , S

_i−1

, S

_i+1

, . . . , S

_n

This asserts that any sub-game of a festival game is also a festival game. Notice in particular that e

_|i

is an affine transformation of e.

Lemma 2

let (v, N )

∈ F

where n > 1. For any T

⊂

N and i

∈

S define S

i

(S) as the fraction of the queue q

_i

which enters theater i when coalition S forms. If R

⊂

T and i

∈

R we have

S

_i

(R)

≤

S

_i

(T )

Proof : First recognize argmax

_pi∈R⁺

S

_i

(p

_i

)(p

_i−

a) is a decreasing function of a. Next according to lemma 1, we only need to establish the result in the case T = N. Then let R =

{

i

₁

, i

₂

, . . . , i

_r}

.

If i is the last element of R then S

ir

(R) = S

_i^m_r

and this is smaller than S

ir

(N) since it is not in interest of theater i

_r

to choose a price larger than the monopoly price. Now consider theater i

_r−1

. When coalition R forms it chooses the price so as to maximize

S

_ir−1

(p

_ir−1

)(p

_ir−1−

c

_ir−1

)+S

_ir−1

(p

_ir−1

)ρ

_1+ir−1×

. . .

×

ρ

_ir

D

^m_1+ir−1×

. . .

×

D

^m_ir

= S

_ir−1

(p

_ir−1

)(p

_ir−1−

c

_ir−1

(R)) whereas when coalition N forms the same theater choose the price as to maximize

S

ir−1

(p

ir−1

)(p

ir−1−

c

ir−1

)+S

ir−1

(p

ir−1

)ρ

1+ir−1×

. . .

×

ρ

it

S

1+ir−1

(N )

×

. . .

×

S

it

(N) = S

ir−1

(p

ir−1

)(p

ir−1−

c

ir−1

(N )) and c

_ir−1

(N ) < c

_ir−1

(R) since S

_1+ir−1

(N )

×

. . .

×

S

_it

(N ) < D

^m_1+ir

−1×

. . .

×

D

_i^m_t

. Proceeding

recursively this way we get the result.

Q.E.D Let n > 1. We already noticed that if ρ

_i

= 0 then the festival game may be sepa- rated in two sub-festivals. The strict positiveness of all ρ

₁

, . . . ρ

_n

is also linked with the following property

Proposition 1

A festival game (v, N) is strictly super-additive — formally v(T

∪

R) >

v(R) + v(T ) for any disjoints coalitions R and T — if and only if min

_{i∈{2,...,n}}

ρ

i

> 0 and e

₁

> 0.

Proof : Necessity derive easily since ρ

_i

= 0 implies v(

{

i

−

1, i

}

) = v(

{

i

−

1

}

) + v(

{

i

}

) and e

1

= 0 implies v(

{

1

} ∪

T ) = v(T ). Sufficiency results from the same argument as in the proof that v(T ) is well defined.

Q.E.D Notice that (e, c, ρ, D) fully characterizes the game (v, N ). The converse is not true.

Consider N =

{

1

}

we have v = (0, e

₁

S

₁

(p

^m₁

)(p

^m₁ −

c

₁

)) and several choices for e

₁

, c

₁

, S

₁

lead to the same characteristic function. As a consequence we shall say that (e, c, ρ, D) is a representation of the game (v, N ).

The proposition 1 also shows that if the same game admits two representations (e, c, ρ, D) and (e

^�

, c

^�

, ρ

^�

, D

^�

) (say). Then e

₁

> 0 and min

i∈{2,...,n}

ρ

_i

> 0 implies e

^�₁

> 0 and min

_{i∈{2,...,n}}

ρ

^�_i

> 0.

The above arguments leads to the following definition of super-additive festivals.

Definition 2

(v, N)

∈ F

belongs to

F⁺

if and only if (v, N ) admits a representation such that e

₁

> 0 and ρ

_i

> 0 for all i

≥

2.

As the above discussion makes clear, for most of the properties to be established we may restrict our attention to

F⁺

.

If min

_{i∈{2,...,n}}

ρ

_i

> 0 then dummy players may easily be characterized.

Proposition 2

Let (v, N )

∈ F

. Also assume min

_{i∈{2,...,n}}

ρ

_i

> 0. Player i is dummy if and only if for all j

∈ {

1, . . . , i

}

e

_j

= 0.

Proof : Assume there exist j

≤

i such that e

_j

> 0 then d

_i−1

> 0 and v(

{

i

}

) > 0.

Conversely, if e

_j

= 0 for all j

∈ {

1, . . . , i

}

then the demand for theater 1 to i are all zero whatever the prices and V (T

∪ {

j

}

) = V (T ) for any T

⊂

N.

Q.E.D

Next a linear basis for

F

for a given sequence of demands and marginal costs may

be provided.

Proposition 3

Let u

1

, . . . , u

n

be the canonical basis of R

ⁿ

. Let (v, N )

∈ F

with repre- sentation (e, c, ρ, D) then for any T

⊂

N we have

v(T) =

�N i=1

e

_i

v

_i

(T ) where v

i

has representation (u

i

, c, D, ρ).

Proof : As we already noticed (1) implies that the quantity v(T ) may be computed as the value of the grand coalition of a festival game with a sequence (e

₁

(T), . . . , e

_t

(T )) in which every e

i

(T) is an affine transformation of the original sequence e

1

, . . . , e

n

. So if we can prove that v(N ) is a linear function of e

_i

then this is also true for v(T ).

According also to previous arguments we may restrict our analysis to the case e

₁

> 0 and min

_{i∈{2,...,n}}

ρ

i

> 0.

The program defining v(N ) may be written as

v(N ) = max

^�_i∈N

(p

_i−

c

_i

)d

_i

wrt p

₁

, . . . , p

_n

; d

₁

, . . . , d

_n

s.t. d

i

= (e

i

+ ρ

i

d

i−1

)S

i

(p

i

) d

₀

= 0

We already remarked when derived well-defineness of v.() that the first order con- dition provides a unique, strictly interior solution. Let λ

_i

be the Lagrange multiplier associated with constraint d

_i

= (e

_i

+ ρ

_i

d

_i−1

)S

_i

(p

_i

) the first order condition for this

program is

_





p

_i−

c

_i−

λ

_i

+ λ

_i+1

ρ

_i+1

S

_i+1

(p

_i+1

) = 0 d

i

= λ

i

(e

i

+ ρ

i

d

i−1

)S

^�

(p

i

)

d

_i

= (e

_i

+ ρ

_i

d

_i−1

)S

_i

(p

_i

)

As e

i

+ ρ

i

d

i−1

> 0 the two last equations implies λ

i

= S

i

(p

i

)/S

^�

(p

i

). Taken this into account in the first equation provides

p

_i−

c

_i−

S

_i

(p

_i

)

S

_i^�

(p

i

) + S

_i+1

(p

_i+1

)

S

_i+1^�

(p

i+1

) ρ

_i+1

S

_i+1

(p

_i+1

) = 0

Now for a given value of p

^�_i+1

this equation admits a single solution (since v(N ) is well defined) and this solution does not depend on e. As we already noticed, the last theater chooses a price independent from e hence so does all the other theaters.

This implies that v(N ) depends on e through d

₁

, . . . , d

_n

only. But according to d

_i

= (e

_i

+ ρ

_i

d

_i−1

)S

_i

(p

_i

) these are linear functions of e. Now consider the case e = u

_i

for i = n, n

−

1, . . . 1 and we get the result.

Q.E.D

Proposition 4

If (v, N ) is a festival game then (v, N) is convex.

Proof : As the previous arguments shows, we may considered the case e = u

n

and (v, N )

∈ F⁺

. Moreover the case n = 2 is easy since convexity amounts to monotonicity and this property is already established. Now assume the property is established for all festival games of size smaller or equal to n

−

1 and consider the case of game of size n.

We have to show that R

⊂

T and i

�∈

T imply

v(T

∪ {

i

}

)

−

v(T )

≥

V (R

∪ {

i

}

)

−

v(R).

But as we already mentioned, if a theater leaves a festival, the remaining players are involved in a new festival with one player less. Hence, by induction we only need to establish the previous inequality holds in the case T

∪ {

i

}

= N and R = N

\{

i, j

}

. We then need to show

v(N )

−

v(N

\{

i

}

)

≥

v(N

\{

j

}

)

−

v(N

\{

i, j

}

).

Notice that this inequality is equivalent to

v(N ) + v(N

\{

i, j

}

)

≥

v(N

\{

i

}

) + v(N

\{

j

}

).

we remark that by symmetry we may consider the case j > i.

First assume i = 1. We have

v(N )

−

v(N

\{

1

}

) =

−

v(N

\{

1

}

) + max

p1

�

S

1

(p

1

)(p

1−

c

1

) + S

1

(p

1

)

S

₁^m

v(N

\{

1

}

)

�

where S

₁^m

= S

₁

(p

^m₁

). Similarly, so we get (for j > 1) v(N

\{

j

}

)

−

v(N

\{

1, j

}

) =

−

v(N

\{

1, j

}

) + max

p1

�

S

1

(p

1

)(p

1−

c

1

) + S

1

(p

1

)

S

₁^m

v(N

\{

1, j

}

)

�

Consider the following function f

1

(x) =

−

x + max

p1

�

S

1

(p

1

)(p

1−

c

1

) + S

1

(p

1

) S

₁^m

x

�

The envelope theorem asserts that f

₁^�

(x) =

^S1(p_S¹m^(x))

1 −

1 > 0 where p

₁

(x) = argmax

_p1

�

S

₁

(p

₁

)(p

₁−

c

₁

) + S

1

(p

1

) S

₁^m

x

�

Now as we conclude using v(N

\{

1

}

)

≥

v(N

\{

1, j

}

).

f

₁

(v(N

\{

1

}

)) = v(N )

−

v(N

\{

1

}

)

≥

v(N

\{

j

}

)

−

v(N

\{

1, j

}

) = f

₁

(v(N

\{

1, j

}

)) Finally consider n

≥

3 and i > 1. This case is an elaboration of the case i = 1. More precisely let us write v(N )

−

v(N

\{

i

}

) as

max

_pi{

max

_{p1,p2,...pi−1}{

S

₁

(p

₁

)(p

₁−

c

₁

) + ρ

₂

S

₁

(p

₁

)S

₂

(p

₂

)(p

₂−

c

₂

) + . . .

. . . + ρ

₂×

. . .

×

ρ

_i−1

S

₁

(p

₁

)

×

. . .

×

S

_i−1

(p

_i−1

)(p

_i−1−

c

_i−1

) +ρ

2×

. . .

×

ρ

i−1

ρ

i

S

1

(p

1

)

×

. . .

×

S

i−1

(p

i−1

)

×

S(p

i

)(p

i−

c

i

) +

^S_S^i(pmⁱ⁾

i

S1(p1)×...×Si−1(pi−1)

S₁^m×...×S_i^m₋₁

v(N

\{

1, . . . , i

}

)

}

}

−

max

_{p1,p2,...pi−1}{

S

₁

(p

₁

)(p

₁−

c

₁

) + ρ

₂

S

₁

(p

₁

)S

₂

(p

₂

)(p

₂−

c

₂

) + . . .

. . . + ρ

₂×

. . .

×

ρ

_i−1

S

₁

(p

₁

)

×

. . .

×

S

_i−1

(p

_i−1

)(p

_i−1−

c

_i−1

) +

^S1(p1_S⁾m^{×...×Si−1(pi−1)}

1 ×...×S^m_i−1

v(N

\{

1, . . . , i

}

)

}

Accordingly, as we assumed j > i we may writes v(N

\{

j

}

)

−

v(N

\{

i, j

}

) as max

_pi{

max

_{p1,p2,...pi−1}{

S

₁

(p

₁

)(p

₁−

c

₁

) + ρ

₂

S

₁

(p

₁

)S

₂

(p

₂

)(p

₂−

c

₂

) + . . .

. . . + ρ

₂×

. . .

×

ρ

_i−1

S

₁

(p

₁

)

×

. . .

×

S

_i−1

(p

_i−1

)(p

_i−1−

c

_i−1

) +ρ

2×

. . .

×

ρ

i−1

ρ

i

S

1

(p

1

)

×

. . .

×

S

i−1

(p

i−1

)

×

S(p

i

)(p

i−

c

i

) +

^S_S^i(pmⁱ⁾

i

S1(p1)×...×Si−1(pi−1)

S₁^m×...×S_i^m₋₁

v(N

\{

1, . . . , i, j

}

)

}

}

−

max

_{p1,p2,...pi−1}{

S

₁

(p

₁

)(p

₁−

c

₁

) + ρ

₂

S

₁

(p

₁

)S

₂

(p

₂

)(p

₂−

c

₂

) + . . .

. . . + ρ

₂×

. . .

×

ρ

_i−1

S

₁

(p

₁

)

×

. . .

×

S

_i−1

(p

_i−1

)(p

_i−1−

c

_i−1

) +

^S1(p1_S^)×...×Sm ^{i−1(pi−1)}

1 ×...×S^m_i−1

v(N

\{

1, . . . , i, j

}

)

}

Following the previous argument let us define

g

i

(p

i

, x) = max

p1,p2,...pi−1{

S

1

(p

1

)(p

1−

c

1

) + ρ

2

S

1

(p

1

)S

2

(p

2

)(p

2−

c

2

) + . . .

. . . + ρ

₂×

. . .

×

ρ

_i−1

S

₁

(p

₁

)

×

. . .

×

S

_i−1

(p

_i−1

)(p

_i−1−

c

_i−1

) +ρ

₂×

. . .

×

ρ

_i−1

ρ

_i

S

₁

(p

₁

)

×

. . .

×

S

_i−1

(p

_i−1

)

×

S(p

_i

)(p

_i−

c

_i

) +

^Si_S^(pmⁱ⁾

i

S1(p1)×...×Si−1(p_i−1) S^m₁ ×...×S_i−1^m

x

}

h

_i

(x) = max

_{p1,p2,...pi−1}{

S

₁

(p

₁

)(p

₁−

c

₁

) + ρ

₂

S

₁

(p

₁

)S

₂

(p

₂

)(p

₂−

c

₂

) + . . .

. . . + ρ

₂×

. . .

×

ρ

_i−1

S

₁

(p

₁

)

×

. . .

×

S

_i−1

(p

_i−1

)(p

_i−1−

c

_i−1

) +

^S1(p1_S^)×...×Sm ^{i−1(pi−1)}

1 ×...×S_i−1^m

x

}

f

_i

(x) = max

_pi{

g

_i

(p

_i

, x)

} −

h(x)

We shall prove that f

_i^�

> 0 so that the same argument as before may be applied. The envelope theorem applies, but one must be cautious for the sequences p

₁

(x), . . . , p

_i−1

(x) for which the maximum of the function defining g

i

and h

i

are not the same. To this end, define for all given value of x the function p

_i

(x) as the value of p

_i

for which the func- tion defining g

_i

achieves its maximum and p

₁

(p

_i

(x), x), . . . , p

_i−1

(p

_i

(x), x) the remaining terms of this sequence whereas p

1

(x), . . . , p

i−1

(x) corresponds to the sequence of prices associated with the definition of h

_i

. We then have

f

_i^�

(x) =

^Si(p_Sⁱm^(x)) i

S1(p1(pi(x),x))×...×Si−1(pi−1(pi(x),x))

S₁^m×...×S^m_i−1 −^{S1(p1(x))×...×S}_S^{m i−1(pi−1(x))}

1 ×...×S_i^m₋₁

Now using lemma 2 we have S

_k

(p

_k

(x)) < S

_k

(p

_k

(p

_i

(x), x)) for all k

∈ {

1, i

−

1

}

and

Si(pi(x))

S_i^m

> 1 hence f

_i^�

> 0.

Q.E.D The previous property is a strong statement in favor of the cooperative approach. In particular, it is well known that the core of any convex game is non-empty and that the Shapley sharing rule cannot be blocked. We will now provide a characterization of the Shapley value in the set of festival games. To this end, we closely follow Myerson (1980).

The Myerson approach to Shapley value in graphs rests on a equalization of rewards and threats among coalition participants. Together with efficiency this property characterizes the Shapley rule of sharing in the set of Transferable Utility games. We now show this properties also lead to Shapley in the subset of festival games.

We then show the following property

Proposition 5

Let (v, N)

∈ F

and ψ(v)

^�

= (ψ

1

(v), . . . , ψ

n

(v))

^� ∈

R

ⁿ

be a solution concept such that

ψ

_i

(v)

−

ψ

_i

(v

_|j

) = ψ

_j

(v)

−

ψ

_j

(v

_|i

)

∀

i, j

∈

N

×

N

�

i∈N

ψ

_i

(v) = v(N ) Then ψ coincides with the Shapley value.

First consider the case n = 2. The two conditions on ψ

1

and ψ

2

write as ψ

₁

(v)

−

v(

{

1

}

) = ψ

₂

(v)

−

v(

{

2

}

)

ψ

₁

(v) + ψ

₂

(v) = v(

{

1, 2

}

)

Hence ψ(v) is uniquely defined and the result holds for n = 2. Now consider ψ coincides with the Shapley value for all (v, M)

∈ F

, with 2

≤

m < n and consider (v, N)

∈ F

. Using the recurrence hypothesis and the fact that removing one player leaves us with a festival game of size n

−

1, we have ψ

i

(v

_|j

) = φ

i

(v

_|j

) for all i, j (where φ() stands for the Shapley value). Now the linear system

ψ

_i

(v)

−

φ

_i

(v

_|j

) = ψ

_j

(v)

−

φ

_j

(v

_|i

)

∀

i, j

∈

N

×

N

�

i∈N

ψ

_i

(v) = v(N )

admits a single solution hence ψ coincides with the Shapley value on

F

.

Q.E.D

2.3 Multi events

Up to now we assumed that each theater organizes a single event. This will rarely be the case in applied situations. More formally consider k theaters organize a festival and denote N the set of events (again organized in chronological order). Let T

₁

, . . . , T

_k

be the partition of N such that T

₁

collects the indexes in N corresponding to the events orga- nized by theater 1, and so on. Myerson’s characterization of Shapley’s rule is commonly justified by balanced threat/rewards arguments. More precisely the requirement

ψ

_i

(v)

−

ψ

_i

(v

_|j

) = ψ

_j

(v)

−

ψ

_j

(v

_|i

)

amounts to balance the loss caused to i when j deviates with that caused to j when i deviates. However, when one theater organizes several events, it does not make much sense to balance the losses caused by deviations of i, j events if both events are organized by the same theater. If decision units are the theaters then we shall consider that deviations from the grand coalition should concern theater and not events. We thus need a variation of Myerson’s characterization of Shapley’s rule to cover this case.

More precisely, we establish the following property

Proposition 6

Let v, N

∈ F

and T

₁

, . . . , T

_k

be the partition of N There exists a single vector ψ

^�

= (ψ

₁

, . . . , ψ

_k

)

^�

such that

ψ

_k1

(v)

−

ψ(v

_k1|

T

_k2

) = ψ

_k2

(v)

−

ψ(v

_k2|

T

_k1

)

∀

k

1

, k

2 ∈ {

1, . . . , k

} × {

1, . . . , k

}

�k

l=1

ψ

_l

(v) = v(N )

The case k = 1 is trivial since it leads to ψ

₁

= v(N). Now let T

⊂

N. By repeated uses for all i

∈

T of the fact that removing one player leaves us with a (sub) festival game, we establish that (v

_|T

, N

\{

T

}

)

∈ F

(with straightforward notation).

Consider the case k = 2. Since v

_|T1

and v

_|T2

both belong to

F

we get ψ

₁

(v)

−

ψ

₁

(v

_|T2

) = ψ

₂

(v)

−

ψ

₂

(v

_|T1

)

ψ

1

(v) + ψ

2

(v) = v(N ) which is equivalent to

ψ

₁

(v)

−

v(T

₂

) = ψ

₂

(v)

−

v(T

₁

) ψ

₁

(v) + ψ

₂

(v) = v(N )

and this system admits a unique solution. Finally assume the property has been estab- lished up to k

−

1 theaters and proceed as in the characterization proof.

Q.E.D This proposition tells us that we can safely apply the Shapley sharing rule among the Theaters in the case of a multi-events program. Notice however that the above solution implicitly assume that once a theater leaves the coalition, it charges the monopoly price for all shows it organizes. This is not too harsh an assumption if the spillover between the shows organized in the same theater are small (for instance because it is the very same event or if different events are chronologically far from each other). But if the spillover among different events organized by the same theater are large, then when it deviates, these must be taken into account. In the following example we will compute these spillover effect and show they are very small.

3 An example

This section is devoted to the presentation of a concrete example. we present the available

data set, then we propose an applied version of the theoretical model finally we discuss

the estimations.

3.1 Data

We collected data related to a dance and theater festival that took place in the north of France and southern part of Belgium between 02/18/2012 and 03/03/2012. The festival is organized on a yearly basis. The theaters involved change slightly from one edition to the other. For the 2012 edition 9 theaters (5 French, 4 Belgian) were involved. The theaters display some heterogeneity : the largest total capacity is 800 and the smallest is 70, one theater is entirely new whereas the oldest theater is more than one century old, etc. Each of these theaters have their own program during the remaining part of the year, some of them are also involved in other festivals. One common feature is the “avant-garde” nature of the works presented. The program presents a selection of contemporaneous, ambitious, young and well-established performers, dancers and directors (the agenda is provided in the appendix section). The audience of this festival mainly consists in aware amateurs, professional and semi-professional artists, so that the spillover effects during the festival are likely large enough for the above model to be useful. Spillover effects are a major issue the organizers as the report for public subsides explicitly mentions audience mobility as a major objective of the festival. Organizers of this festival particularly promoted trans-borders mobility. To this end Dutch subtitles are displayed for many French speaking performances (and vice-versa) and a shuttle service has been organized.

For each performance, we observe the audience and the total capacity of the theater.

We used the presentation of the performance provided by the festival’s program to collect information about the nature of the performance (dance, theater or other type of performance). To measure relative notoriety we counted Google citations of each performances.

⁶

We also observed characteristics that are probably linked to the cost such as the number of artists and the duration of the show.

Of course, the program provides a full description of the prices. The full price is charged for the first ticket only. A 2

e

discount is offered for all other tickets. Finally less than 26 years old people benefits from a special tariff (7

e) for all performances.

We also have extra information about mobility. Surveys have been collected before and during most of the performances

⁷

Each survey consists in 44 questions (an English translation of the complete survey is displayed in appendix). Questions 34 to 42 explicitly measures the mobility of the audience. We use the answers to 476 surveys to compute the following Markov transition matrix.

Table 1

6We searched for exact citations of the artist(s) and/or the show. The mention ”NEXT” has been deleted for the search and we also restrict to dates before November 2012.

7Three performances out of 22 could not have been covered. One of them, -”On the Concept of the Face, regarding the Son of God” by Romeo Castellucci- experienced violent demonstrations from extreme Christian activists. As the survey have been conducted by students we prefer to cancel the interviews for this performance. Notice the demonstrations did not discourage audience, since both representations of this performance were sold out weeks before the performance. For two other performances, we ran out of survey conductors.