Dongshuang Hou
email: dshhou@126.com1

Theo Driessen
email: t.s.h.driessen@ewi.utwente.nl2

Aymeric Lardon

Convexity and the Shapley value in Bertrand oligopoly TU-games with Shubik's demand functions

Keywords: Bertrand Oligopoly situation, Bertrand Oligopoly Game, Convexity, Shapley Value, Total Balancedness

The Bertrand Oligopoly situation with Shubik's demand functions is modelled as a cooperative TU game. For that purpose two optimization problems are solved to arrive at the description of the worth of any coalition in the so-called Bertrand Oligopoly Game. Under certain circumstances, this Bertrand oligopoly game has clear affinities with the well-known notion in statistics called variance with respect to the distinct marginal costs. This Bertrand Oligopoly Game is shown to be totally balanced, but fails to be convex unless all the firms have the same marginal costs. Under the complementary circumstances, the Bertrand Oligopoly Game is shown to be convex and in addition, its Shapley value is fully determined on the basis of linearity applied to an appealing decomposition of the Bertrand Oligopoly Game into the difference between two convex games, besides two nonessential games. One of these two essential games concerns the square of one non-essential game.

I. INTRODUCTION

A central problem in oligopoly theory is the existence of collusive behaviors between firms, that is, situations in which firms are able to coordinate and to stabilize their strategies in order to increase their profits. The classical Cournot and Bertrand oligopoly situations are such examples where firms are better off through cooperation rather than by acting independently. A cartel operating successfully is the OPEC (Organization of the Petroleum Exporting Countries) cartel which restricts oil supply in order to control oil price market. Another example of a cartel which had operated is the agreement between multinational firms Saint-Gobain, Pilkington, Asahi and Soliver in flat glass industry. Their illegal agreement on the price of glass in the car industry from 1998 to 2003 had been fined by the European Commission in 2008. Non-cooperative game theory has provided the theoretical bases for the existence of collusive behaviors between firms by means of repeated games. Under this approach, each firm does not have any interest in defecting from the collusive behavior because it rationally anticipates future punishments in the periods following its defection. We refer to Pepall et al. (2008) for an overview on this topic. An alternative way to formalize the existence of collusive behaviors comes from cooperative game theory. Under this approach, firms are allowed to sign binding agreements in order to form cartels called coalitions. With such an assumption cooperative games called oligopoly TU(Transferable Utility)-games can be defined and the existence of collusive behaviors is then related to the non-emptiness of the core of such games. Aumann (1959) proposes two approaches in order to define cooperative game: according to the first, every cartel computes the total profit which it can guarantee itself regardless of what outsiders do; the second approach consists in computing the minimal profit for which outsiders can prevent the firms in the cartel from getting more. These two assumptions lead to consider the α and βcharacteristic functions respectively. In this article, we follow this cooperative approach to analyze collusive behaviors and we study a subclass of oligopoly TUgames in α and β-characteristic function forms. Many works have studied the core of oligopoly TU-games. As regards Cournot oligopoly TU-games with or without transferable technologies, Zhao (1999a,b) shows that the α and β-characteristic functions lead to the same class of Cournot oligopoly TU-games. When technologies are transferable, Zhao(1999a) provides a necessary and sufficient condition to establish the convexity property in case the inverse demand function and cost functions are linear. Although these games may fail to be convex in general, [START_REF] Norde | Oligopoly Games with and without Transferable Technologies[END_REF] show they are nevertheless totally balanced. When technologies are not transferable, Zhao(1999b) proves that the core of such games is nonempty if every individual profit function is continuous and concave. Furthermore, [START_REF] Norde | Oligopoly Games with and without Transferable Technologies[END_REF] show that these games are convex in case the inverse demand function and cost functions are linear, and [START_REF] Driessen | Convexity of Oligopoly Games without Transferable Technologies[END_REF] provide economically meaningful sufficient conditions to guarantee the convexity property in a more general case. As regards Bertrand oligopoly TU-games, Deneckere and Davison (1985) consider a Bertrand oligopoly situation with differentiated products in which the demand system is [START_REF] Shubik | Market Structure and Behavior[END_REF] and firms operate at a constant and identical marginal and average cost. They prove that these games have a superadditive property in the sense that a merger of two disjoint cartels results in a joint after-merger profit for them which is greater than the sum of their pre-merger profits. [START_REF] Lardon | Convexity of Bertrand Oligopoly TU-Games with differentiated products[END_REF] extends this result by considering the α and β-characteristic functions of these games. As for Cournot oligopoly TU-games, he shows that the α and β-characteristic functions lead to the same class of Bertrand oligopoly TU-games and proves that the convexity property holds for this class of games.

In this article, we study Bertrand oligopoly TU-games in α and β-characteristic function forms with Shubik's demand functions in which firms have possibly distinct marginal costs. We prove that the α and β-characteristic functions lead to the same class of Bertrand oligopoly TU-games, and so we extend Lardon 's result (2010) which holds only with identical marginal costs of firms. For this class of games, we show that if the intercept of demand is sufficiently small, then Bertrand oligopoly TU-games share clear similarities with a well-known notion in statistics called variance with respect to the distinct marginal costs. Such games are shown to be totally balanced, but fail to be convex unless all the firms have the same marginal costs. Moreover, we prove that if the intercept of demand is sufficiently large, then Bertrand oligopoly TU-games are convex and in addition, the Shapley value is fully determined on the basis of linearity applied to an appealing decomposition of Bertrand oligopoly TU-games into the difference between two convex games, besides two non-essential games. One of these two convex games is defined as the square of one of the two non-essential games. This article is structured as follows. In section 2, we introduce the model and show that Bertrand oligopoly TU-games in β-characteristic function form with distinct marginal costs are well-defined. Moreover, we prove that the α and β-characteristic functions lead to the same class of Bertrand oligopoly TU-games. In section 3, we show that if the intercept of demand is sufficiently large, then Bertrand oligopoly TU-games are convex. In section 4, we study and characterize the Shapley value on the class of Bertrand oligopoly TU-games. In section 5, we show that if the intercept of demand is sufficiently small, then Bertrand oligopoly TU-games fail to be convex but are nevertheless totally balanced. Section 6 gives some concluding remarks.

II. THE NON-SYMMETRIC BERTRAND OLIGOPOLY TU-GAME WITH SHUBIK'S DEMAND FUNCTIONS

A Bertrand oligopoly situation is described by a 3tuple N, (D i ) i∈N , (C i ) i∈N , where N = {1, 2, . . . , n} is the finite set of firms, such that, for every firm i ∈ N , the Shubik's demand function D i : R n + → R and the (linear) cost function C i : R + → R + with marginal cost c i ≥ 0 respectively are given by ( [START_REF] Shubik | Market Structure and Behavior[END_REF])

D i (p 1 , p 2 , . . . , p n ) = V -p i -r• p i -1 n • k∈N p k •x (1)
and C i (x) = c i for all x ∈ R + where p i is the price charged by firm i, the demand's intercept V ≥ 0 when all prices are zero, and let r > 0 be the substitutability parameter. When r approaches zero, products become unrelated, and when r approaches infinity, products become perfect substitutes. The quantity demanded of firm i's brand depends on its own price p i and the difference between its own price and the average price in the industry. The latter quantity is decreasing with respect to its own price p i and increasing with respect to any price p j , j = i. Notice that firms may operate at possibly different marginal costs c i ≥ 0, i ∈ N , and these marginal costs do not limit the non-negative prices p i ≥ 0, i ∈ N , of firms. The corresponding Bertrand Oligopoly Game in normal form N, (X i ) i∈N , (π i ) i∈N is given by player i's strategy set

X i = R + = [0, ∞) and individual profit function π i : Π k∈N X k → R such that π i (p 1 , p 2 , . . . , p n ) = (p i -c i ) • D i (p 1 , p 2 , . . . , p n )
So, for all i ∈ N ,

π i (p 1 , p 2 , . . . , p n ) = (p i -c i ) • V -(1 + r) • p i + r n • k∈N p k (2)
Denote for any T ⊆ N , T = ∅, the coalitional strategy set X T = Π k∈T X k and define the coalitional profit

function π T : X T × X N \T → R by π T (p T , p N \T ) = k∈T π k (p T , p N \T ), for all (p T , p N \T ) ∈ X T × X N \T .
The corresponding Bertrand Oligopoly Game in αand β-characteristic function form N, v α and N, v β are defined, for every coalition S ⊆ N , S = ∅, as follows:

v α (S) = max pS∈XS min pN\S∈XN\S π S (p S , p N \S ) (3)
v β (S) = min pN\S∈XN\S max pS∈XS π S (p S , p N \S ) (4)
π S (p S , p N \S ) = j∈S π j (p S , p N \S ) = j∈S (p j -c j ) • V -(1 + r) • p j + r n • k∈N p k (5)
Generally speaking, it always holds v α (S) ≤ v β (S) as well as v α (N ) = v β (N ). In the remainder we pay attention to the Bertrand oligopoly game in β-characteristic function form and let cS = (i) Concerning the maximization program max pS∈XS π S (p S , p N \S ), the first order conditions are given by ∂πS ∂pi (p S , p N \S ) = 0 for all i ∈ S and its unique solution pS i is given by

pS i = pS + ci 2 for all i ∈ S, where (6)
pS = 1 2 • n•V +r• k∈N \S pk n+r•(n-s) (7)
(ii) Concerning the quadratic minimization program min pN\S∈XN\S π S ((p S j ) j∈S , p N \S ), where S = N , its solution is given by

k∈N \S pk = max 0, 1 r • n+r•(n-s) •c S -n•V (8) We distinguish two cases. (iii) If n • V ≤ [n + r • (n -s)]
• cS , then the following holds: for all i ∈ S,

pS i = cS + c i 2 and pS = cS 2 (9) k∈N \S pk = 1 r • n + r • (n -s) • cS -n • V (10) v β (S) = 1+r 4 • j∈S (c j ) 2 -s • (c S ) 2 = 1+r 4 • j∈S c j -cS 2 (11) (iv) If n • V > [n + r • (n -s)]
• cS , then the following holds: for all i ∈ S

pS i = 1 2 • n•V n+r•(n-s) + c i > cS+ci 2 (12) k∈N \S pk = 0 (13) v β (S) = s 4•n • (n•V ) 2 n+r•(n-s) -V 2 • j∈S c j + + 1+r 4 • j∈S (c j ) 2 -r 4•n • j∈S c j 2 (14) = s 4•n • n•V -n+r•(n-s) •cS 2 n+r•(n-s) + + 1+r 4 • j∈S (c j ) 2 -s • (c S ) 2 (15)
Remark 2.1: With every collection C = (c i ) i∈N of marginal costs, there is associated the so-called Variance TU game N, V AR C of which the characteristic function V AR C : P(N ) → R is defined by V AR C (∅) = 0 and

V AR C (S) = j∈S c j -cS 2 for all S ⊆ N , S = ∅. (16)
The notion of variance is well-known in the field of statistics and it refers to the deviations by individuals from the average coalitional cost measured as the sum of squares of member's deviations. According to [START_REF] Zhao | A β-Core existence result and its application to oligopoly markets[END_REF], the Bertrand Oligopoly Game resembles the associated Variance TU game. For instance, if the marginal costs are given by c 1 = c 2 = c and c 3 = c + 1. Then the zero-normalized three-firm Variance Game is given by v

(S) = 0 if 3 ∈ S and v(S) = s-1 s if 3 ∈ S. That is, v({1, 2}) = 0, v({1, 3}) = v({2, 3}) = 1 2 and v(N ) = 2 3 .
Note that firms 1 and 2 are substitutes in this variance game. Generally speaking, if i ∈ N and S ⊆ N , S i, satisfy c i = cS , then it follows immediately c i = cS\{i} and in turn,

V AR C (S\{i}) = V AR C (S). Particularly, if i ∈ N satisfies c i = cN , then it holds V AR C (N ) - V AR C (N \{i}) = 0.
Because the latter expression represents an upper bound for the core of the zeronormalized Variance game, any core allocation to player i degenerates to zero, provided the marginal cost of firm i coincides with the average grand coalitional cost. Nevertheless, in Section V we prove the balancedness for Variance TU games (or equivalently, the non-emptiness of the core).

Corollary 2.1:

Consider the symmetric Bertrand Oligopoly Game N, v β in that c i = c > 0 for all i ∈ N . Fix coalition S ⊆ N , S = ∅. (i) If n • V ≤ [n + r • (n -s)] • c, then v β (S) = 0 due to pS i = c for all i ∈ S. (ii) Define the proportionally aggregate netto demand E through r •c•E = n•(V -c). If n•V > [n+r •(n-s)]•c, then v β (S) = s 4•n • n•V -n+r•(n-s) •c 2 n+r•(n-s) > 0 (17) or equivalently, if E > n -s, then v β (S) = s 4•n • (r • c) 2 • E-(n-s) 2 n+r•(n-s) > 0 (18)
The non-zero coalitional worth in the symmetric Bertrand Oligopoly Game depends on the validity of the constraint n

• V > [n + r • (n -s)] • c involving the intercept V or the equivalent constraint E > n -s involving the proportionally aggregate netto demand E.
In this setting, we interpret n • (V -c) as the aggregate netto demand when prices are zero. Obviously, if a coalition S of size s meets the constraint E ≤ n -s yielding zero worth v β (S) = 0, then any coalition of the same size s or less inherits the same constraint yielding zero worth. Similarly, if a coalition T of size t meets the inverse constraint E > n -t yielding non-zero worth v β (T ) > 0, then any coalition of the same size or more inherits the same inverse constraint yielding non-zero worth. According to (19), the per-capita worth vβ(S) s is strategically equivalent to the quotient of the square of a bankruptcy game (with estate E and unitary claims) and a linearly decreasing symmetric game (varying from levels (1 + r) • n down to level n).

The non-symmetric Bertrand Oligopoly Game consists of two types of coalitions because the validity of the relevant constraint

n • V > [n + r • (n -s)]
• cS involves the average coalitional cost cS . According to [START_REF] Zhao | A β-Core existence result and its application to oligopoly markets[END_REF], if the average coalitional cost is sufficiently large, then the coalitional worth is fully determined by the multiple 1+r 4 of the Variance TU Game induced by the distinct marginal costs. Otherwise, according to (14), if the average coalitional cost is sufficiently small, then the coalitional worth in the non-symmetric Bertrand Oligopoly Game counts, besides the associated variance, the non-zero worth in the symmetric Bertrand Oligopoly Game, with the understanding that the constant marginal cost is to be replaced by the average coalitional cost. The rather appealing alternative decomposition (14) into four types of games will be exploited in the Sections IV and V.

Proof of Theorem 2.1 Part 1. Let i ∈ S. The partial derivative of the coalitional profit function ( 5) is as follows:

∂π S ∂p i (p S , p N \S ) = V -(1 + r) • p i + r n • k∈N p k + + (p i -c i ) • -(1 + r) + r n + j∈S\{i} (p j -c j ) • r n = V -(1 + r) • (2 • p i -c i ) + + r n • j∈S (p j -c j ) + r n • k∈N p k
Consequently, the solution to the first order condition ∂πS ∂pi = 0 satisfies, for all i ∈ S,

p i = ci 2 + 1 2•(1+r) • V + r n • j∈S (p j -c j ) + k∈N p k .
So far, we conclude that for the solution to the first order conditions (associated with the maximization program) it holds that p i -ci 2 is constant for all i ∈ S, say p i -ci 2 = pS for every i ∈ S. Through substitution in the latter expression, we arrive at the following relationships:

2 • (1 + r) • pS = V + r n • k∈N \S p k + j∈S (2 • p j -c j ) = V + r n • k∈N \S p k + 2 • s • pS
Rewriting the latter equality yields

2 • (1 + r) -2•r•s n • pS = V + r n • k∈N \S p k and thus, pS = 1 2 • n•V +r• k∈N \S pk n+r•(n-s)
Recall that the solution pS i to the first order conditions is given by pS i = pS + ci 2 for all i ∈ S.

Part 2. First method.

For the sake of the minimization program min pN\S∈XN\S π S ((p S j ) j∈S , p N \S ), we derive from (5) that its objective function reduces as follows:

π S ((p S j ) j∈S , p N \S ) = j∈S (p S j -c j ) • V -(1 + r) • pS j + + r n • k∈S pS k + r n • k∈N \S p k (19)
Note that, by [START_REF] Norde | Oligopoly Games with and without Transferable Technologies[END_REF], each partial derivative of pS i , i ∈ S, is the same, given by

∂ pS i ∂p = r 2 • n + r • (n -s) = δ s for all ∈ N \S.
Hence, by differentiating (19),

∂π S ∂p ((p S j ) j∈S , p N \S ) = j∈S δ s • V -(1 + r) • pS j + r n • k∈S pS k + r n • k∈N \S p k + j∈S (p S j -c j ) • -(1 + r) • δ s + r n • s • δ s + r n = δ s • s • V + r•s n -(1 + r) • k∈S pS k + r•s n • k∈N \S p k + j∈S pS j -s • cS • δ s • r•s n -(1 + r) + r n = δ s • s • V + r•s n • k∈N \S p k -r 2•n k∈S pS k + j∈S pS j -s • cS • -r 2 • n + r n = δ s • s • V + r•s n • k∈N \S p k -r•s 2•n • cS
Concerning the solution to the minimization problem, it follows from the first order conditions that for all ∈ N \S

∂π S ∂p ((p S j ) j∈S , p N \S ) = 0 iff δ s • V + r n • k∈N \S p k -r 2•n • cS = 0 iff r • k∈N \S p k = r 2 • cS δs -n • V
Finally take care about the non-negativity constraint for prices. This completes the proof of ( 8) according to the first method.

Part 2. Second method.

For the sake of the minimization program min pN\S∈XN\S π S ((p S j ) j∈S , p N \S ), we derive from (5) that its objective function partially reduces as follows: for all j ∈ S

V -(1 + r) • pS j + r n • k∈N pk = V -(1 + r) • pS + cj 2 + r n • k∈S pS + ck 2 + r n • k∈N \S p k = V -1 + r -s•r n • pS -(1 + r) • cj 2 + r•s 2•n • cS + r n • k∈N \S p k = -1 n • n + r • (n -s) • pS -(1 + r) • cj 2 + r•s 2•n • cS + 2 n • n + r • (n -s) • pS = 1 n • n + r • (n -s) • pS -(1 + r) • cj 2 + r•s 2•n • cS
by using [START_REF] Norde | Oligopoly Games with and without Transferable Technologies[END_REF]. Thus, by treating pS as the variable, the objective function is described by

π S ((p S j ) j∈S , p N \S ) = j∈S (p S j -c j ) • V -(1 + r) • pS j + r n • k∈N pk = j∈S (p S -cj 2 ) • 1 n • n + r • (n -s) • pS -(1 + r) • cj 2 + r•s 2•n • cS
Obviously, by the latter equality, the objective function π S ((p j ) j∈S , p N \S ) is a quadratic function of pS , say of the form

α 2 • (p S ) 2 + α 1 • pS + α 0 . Clearly, the coefficient of the quadratic term (p S ) 2 equals α 2 = s n • n + r • (n -
s) > 0, while the coefficient of the term pS equals 2 . This quadratic function attains its minimum for pS =

α 1 = - j∈S c j • 1 2•n • n + r • (n -s) -1+r 2 • j∈S c j + r•s 2 2•n • cS = -c S 2 • n • s • n + r • (n -s) + (1 + r) • s • n -r • s 2 = -s n • n + r • (n -s) • cS and the constant term α 0 = 1+r 4 • j∈S (c j ) 2 -r•s 2 4•n •(c S )
-α1 2•α2 = cS 2
or equivalently,

\S p k = 1 r • n + r • (n -s) • cS -n • V .
Finally take care about the non-negative constraint for prices. This completes the proof of (8) according to the second method.

Part 3. First assumption. Suppose n • V ≤ [n + r • (n -s)] • cS .
Under these circumstances, we derive from [START_REF] Shapley | A value for n-person games. Contributions to the theory of games II[END_REF] and in turn, through substitution in [START_REF] Norde | Oligopoly Games with and without Transferable Technologies[END_REF],

r • k∈N \S pk = n + r • (n -s) • cS -n • V
as well asp S = cS 2 and so,

pS i = pS + c i 2 = cS + c i 2
as well as pS i -c i = cS-ci 2 for all i ∈ S. Further, for all j ∈ S, the objective function partially reduces as follows:

V -(1 + r) • pS j + r n • k∈N pk = V -(1 + r) • cS + c j 2 + r n • j∈S cS+cj 2 + 1 n • n + r • (n -s) • cS -n • V = -(1 + r) • cS + c j 2 + r•s n • cS + 1 n • n + r • (n -s) • cS
Through substitution in [START_REF] Lardon | Convexity of Bertrand Oligopoly TU-Games with differentiated products[END_REF], the coalitional profit reduces as follows:

v β (S) = π S ((p S j ) j∈S , pN\S ) = j∈S cS -c j 2 • -(1 + r) • cS + c j 2 + r•s n • cS + 1 n • n + r • (n -s) • cS = -1+r 4 • j∈S (c S -c j ) • c j = 1+r 4 • j∈S (c j ) 2 -s • (c S ) 2
This proves [START_REF] Zhao | A β-Core existence result and its application to oligopoly markets[END_REF] concerning the coalitional worth v β (S).

Part 3. Second assumption. Suppose n • V > [n + r • (n -s)] • cS .
Under these circumstances, we derive from (8) that k∈N \S pk = 0, and in turn, through substitution in [START_REF] Norde | Oligopoly Games with and without Transferable Technologies[END_REF],

pS = 1 2 • n•V [n+r•(n-s)]
Recall that pS i = pS + ci 2 for all i ∈ S. Further, for all j ∈ S, the objective function partially reduces as follows:

V -(1 + r) • pS j + r n • k∈N pk = V -(1 + r) • (p S + cj 2 ) + r n • k∈S (p S + ck 2 ) = V -1 + r -s•r n • pS + 1 2 • -(1 + r) • c j + r n • k∈S c k = V 2 + 1 2 • -(1 + r) • c j + r n • k∈S c k = 1 2 • V -(1 + r) • c j + r•s n • cS
Recall that pS j -c j = pS -cj 2 for all j ∈ S. Through substitution in [START_REF] Lardon | Convexity of Bertrand Oligopoly TU-Games with differentiated products[END_REF], the coalitional profit reduces as follows:

v β (S) = π S ((p S j ) j∈S , pN\S ) = 1 2 • j∈S (p S -cj 2 ) • V -(1 + r) • c j + r•s n • cS = s • pS 2 • V + r•s n • cS -(1 + r) • cS -V + r•s n • cS • s•cS 4 + 1+r 4 • j∈S (c j ) 2 = s•V 4 • 1 [n+r•(n-s)] • n • V -n + r • (n -s) • cS -n • V + r • s • cS • s • cS 4 • n + 1+r 4 • j∈S (c j ) 2 = s 4•n • (n•V ) 2 [n+r•(n-s)] -V 2 • j∈S c j + 1+r 4 • j∈S (c j ) 2 -r 4•n • j∈S c j 2
This proves (14) concerning the coalitional worth v β (S) as well as the equivalent formula (15).

In the context of (15), we remark, without going into details, that, concerning the first component, and sufficiently small r, the sum of individual worths approaches i∈N (V -c i ) 2 which always exceeds n • (V -cN ) 2 being the worth of the grand coalition. In other words, for r approaching to zero, the imputation set of the first component is empty. Nevertheless, the imputation set of the Bertrand Oligopoly Game is compensated by the imputation set of the second zero-normalized game being the Variance game.

Corollary 2.2: The αand β-characteristic function forms N, v α and N, v β coincide, that is v α (S) = v β (S) for all S ⊆ N .

Proof. Fix the coalition S ⊆ N , S = ∅. It remains to prove the inequality v α (S) ≥ v β (S). We claim the following chain of (in)equalities:

v α (S) = max pS∈XS min pN\S∈XN\S π S (p S , p N \S ) ≥ min pN\S∈XN\S π S ((p S j ) j∈S , p N \S ) = π S ((p S j ) j∈S , pN\S ) = v β (S)
The last equality holds by ( 6) because of the construction of both vectors (p S j ) j∈S and pN\S , whereas the last equality but one is due by ( 8) because pN\S is a minimizer of the minimization program min pN\S∈XN\S π S ((p S j ) j∈S , p N \S ).

III. THE CONVEXITY OF THE BERTRAND OLIGOPOLY TU-GAME

Let P(N ) = {S | S ⊆ N } denote the power set of the player set N , consisting of all subsets of N . A cooperative TU game N, w is called convex (supermodular) if its characteristic function w : P(N ) → R satisfies one of the following equivalent conditions ( [START_REF] Shapley | Cores of Convex Games[END_REF]): for all S, T ⊆ N , w(S) + w(T ) ≤ w(S ∪ T ) + w(S ∩ T ) or for all i ∈ N and all S ⊆ T ⊆ N \{i}, w(S ∪ {i}) -w(S) ≤ w(T ∪ {i}) -w(T ) or for all i, j ∈ N , i = j, and all S ⊆ N \{i, j}, w(S ∪ {i}) -w(S) ≤ w(S ∪ {i, j}) -w(S ∪ {j}) (20) First of all, the next example illustrates that Bertrand Oligopoly Games of form [START_REF] Zhao | A β-Core existence result and its application to oligopoly markets[END_REF] needs not to be convex.

v({1, 2}) = v({2, 3}) = 1 2 and v({1, 3}) = v({1, 2, 3}) = 2. This game is not convex since v(N ) -v({1, 3}) = 0 < 1 2 = v({1, 2}) -v({1}).
v({1, 2}) = v({2, 3}) = c 2 2 , v({1, 3}) = v({1, 2, 3}) = 2 • c 2 .
Notice that this variance game equals the former example multiplied by c 2 .

A TU game N, w is said to be non-essential (additive) if its characteristic function w : P(N ) → R satisfies w(S) = j∈S w({j}) for all S ⊆ N , S = ∅.

Obviously, non-essential games are convex since all convexity conditions are met as equalities. The main goal of this section is to prove the convexity (supermodularity) for Bertrand Oligopoly Games of the form (14).

For that purpose we assume that n•V > [n+r•(n-1)]• max k∈N c k in order to guarantee that the constraint n • V > [n+r •(n-s)]•c S is met by every coalition S ⊆ N , S = ∅. According to the equivalent description ( 14), this type of Bertrand Oligopoly Game can be decomposed into four types of TU games as follows: for all S ⊆ N , S = ∅,

v β (S) = -V 2 • w 1 (S) -r 4•n • w 2 (S) + 1+r 4 • w 3 (S) + V 2 4•(1+r) • w 4 (S) (21)
Here both games N, w k , k = 1, 3, are non-essential arising from the distinct marginal costs and their squares respectively, that is w 1 (S) = j∈S c j and w 3 (S) = j∈S (c j ) 2 for all S ⊆ N . In fact, both non-essential games are redundant for the convexity property. The game N, w 2 is the square of the first non-essential game N, w 1 in that w 2 (S) = (w 1 (S)) 2 for all S ⊆ N . Generally speaking, the square of an non-essential game is convex too because the marginal contribution of a fixed player i with respect to variable coalitions S ⊆ N \{i} are non-decreasing with respect to set inclusion, that is

w 2 (S ∪ {i}) -w 2 (S) = w 1 (S ∪ {i}) 2 -w 1 (S) 2 = w 1 (S) + w 1 ({i}) 2 -w 1 (S) 2 = 2 • w 1 ({i}) • w 1 (S) + (w 1 ({i})) 2 and so w 2 (S ∪ {i, j}) -w 2 (S ∪ {j}) -w 2 (S ∪ {i}) -w 2 (S) = 2 • w 1 ({i}) • w 1 ({j}) ≥ 0 (22)
for all i, j ∈ N , i = j, and all S ⊆ N \{i, j} Lemma 3.1: Given the substitutability parameter r > 0, define the n-person game N, w 4 and the real-valued function f : [0, 1 rn ) → R by

w 4 (S) = s•n•(1+r) n+r•(n-s)
for all S ⊆ N , ( 23)

f (x) = x 1 -r n • x for all x ∈ [0, 1 rn ), ( 24
)
where

r n = r n•(1+r)
Then the following holds:

(i) w 4 (S) = f (s) for all S ⊆ N with size s, s = 0, 1, 2, . . . , n and f (n

) = n • (1 + r) (ii)
The function f : R → R is strictly increasing and strictly convex on the interval [0, 1 rn ) and consequently, the game N, w 4 is strictly convex in that

f (s + 2) -f (s + 1) > f (s + 1) -f (s) (25)
for all s = 0, 1, 2, . . . , n -2 (iii) The marginal returns of the function f satisfy

f (s + 1) -f (s) = n 2 • (1 + r) 2 [n + r • (n -s)] • [n + r • (n -1 -s)] (26)
(iv) Applying (26) twice yields

f (s + 2) -f (s + 1) -f (s + 1) -f (s) = 2 • r • n 2 • (1 + r) 2 • [n + r • (n -s)] -1 • [n + r • (n -1 -s)] -1 • [n + r • (n -2 -s)] -1 (27) (v) v β (S) = 1+r 4 • V 1+r •f (s)-s•cS 2 f (s) + 1+r 4 • V AR C (S) provided f (s) > (1+r)•s V • cS Proof.
Let S ⊆ N be of size s, s = 0, 1, 2, . . . , n. From ( 23)-( 24), we derive

f (s) = s 1 -r n • s = s 1 -r•s n•(1+r) = s • n • (1 + r) n • (1 + r) -r • s = s • n • (1 + r) n + r • (n -s) = w 4 (S) So, V 1+r • f (s) -s • cS = s•n•V n+r•(n-s) -s • cS = s • n • V n + r • (n -s) -cS
Together with (15)-( 16), this completes the proof of the coalitional worth v β (S). It is left to the reader to verify that the first and second derivative of the differentiable function f (x) are given by f

(x) = 1 (1-rn•x) 2 > 0 as well as f (x) = 2•rn (1-rn•x) 3 > 0.
Recall that a differentiable function is convex if and only if the second derivative is non-negative.

In summary, so far, all four games N, w k , k = 1, 2, 3, 4, are convex (where the two non-essential games are redundant for the convexity property). Because the Bertrand Oligopoly Game of the form ( 21) is the difference of two convex games, it may fail to be convex itself. According to the proof of the next main theorem, convexity still holds for the Bertrand Oligopoly Game due to the existence of the underlying constraints.

Theorem 3.1:

Suppose n • V > [n + r • (n -1)] • max k∈N c k . Then the Bertrand Oligopoly Game N, v β of the form (21) is convex (supermodular).
Proof. In view of the decomposition (21), the Bertrand Oligopoly Game is convex if and only if the game N, n • V 2 • w 4 -r • (1 + r) • w 2 is convex. By Lemma 3.1(i), w 4 (S) = f (s) for all S ⊆ N , whereas (22) holds in the setting of the game N, w 2 satisfying w 2 ({k}) = c k for all k ∈ N . In summary, the convexity property (20) applies to the Bertrand Oligopoly Game if and only if the following holds: for all i, j ∈ N , i = j, and all s = 0, 1, 2, . . . , n -2,

f (s + 2) -f (s + 1) -f (s + 1) -f (s) ≥ r • (1 + r) n • V 2 • 2 • c i • c j (28)
By assumption,

c k ≤ n•V n+r•(n-1)
for k ∈ {i, j} and thus, it suffices to prove

f (s + 2) -f (s + 1) -f (s + 1) -f (s) ≥ 2 • n • r • (1 + r) [n + r • (n -1)] 2 (29)
Recall the function f (x) = x•n•(1+r) n+r•(n-x) and the results (26)-( 27). Note that the expression at the right hand of ( 27) is non-decreasing in the variable coalition size s, attaining its minimum at s = 0. It follows that

f (s + 2) -f (s + 1) -f (s + 1) -f (s) ≥ 2 • r • n 2 • (1 + r) 2 [n + r • n] • [n + r • (n -1)] • [n + r • (n -2)] = 2 • r • n • (1 + r) [n + r • (n -1)] • [n + r • (n -2)] ≥ 2 • n • r • (1 + r) [n + r • (n -1)] 2
This completes the proof of convexity for the Bertrand Oligopoly Game.

With the specific choice (24) of the convex function f in mind, (29) is a sufficient condition for the convexity notion applied to the Bertrand Oligopoly Game of the form (21). Under the transformation g(x) = r n • f (x), we have that (29) is fully equivalent to g(s + 2) -g(s + 1) -g(s + 1) -g(s) ≥ 2 • (g(1)) 2 (30) Hence, in the framework of any not yet specified function g : R → R, the convexity of g together with (30), are sufficient for the convexity of the Bertrand Oligopoly Game of the form

v β (S) = -r 4•n • w 2 (S) + V 2 4 • n r • g(s) for all S ⊆ N .
Proposition 3.1: The Variance game N, V AR C of (15) is not convex, unless all the firms have the same marginal costs yielding the zero game.

Proof. Suppose that at least two firms (but not all the firms) have the same marginal cost, say cost c. Let E denote the set of firms with the constant marginal cost c, that is E = {i ∈ N | c i = c}. By assumption, 2 ≤ |E| < n. Obviously, cS = c and thus, V AR C (S) = 0 for every S ⊆ E. For every S ⊆ N satisfying S ⊆ E, it holds

c(S) = 1 s • k∈S c k = 1 s • |S ∩ E| • c + k∈S\E c k = c + 1 s • k∈S\E (c k -c
) and so,

V AR C (S) = k∈S∩E c -cS 2 + k∈S\E c k -cS 2 = k∈S∩E (-δ S ) 2 + k∈S\E c k -c -δ S 2 (where δ S = 1 s • k∈S\E (c k -c)) = k∈S∩E (δ S ) 2 + k∈S\E c k -c 2 - k∈S\E 2 • δ S • c k -c + k∈S\E (δ S ) 2 = s • (δ S ) 2 + k∈S\E c k -c 2 -2 • s • (δ S ) 2 = k∈S\E c k -c 2 -s • (δ S ) 2 = k∈S\E c k -c 2 -1 s • k∈S\E (c k -c) 2
For all i ∈ E, and all S ⊆ N \{i} satisfying S ⊆ E, it holds that S ∪ {i} ⊆ E as well as (S ∪ {i})\E = S\E. From this, together with the latter equality, we derive on the one hand

V AR C (S ∪ {i}) -V AR C (S) = -1 s + 1 + 1 s • k∈S\E (c k -c) 2 = 1 s•(s+1) • k∈S\E (c k -c) 2
Similarly, for all i, j ∈ E, and all S ⊆ N \{i, j} satisfying S ⊆ E, it holds that S ∪ {i, j} ⊆ E as well as (S ∪ {i, j})\E = (S ∪ {j})\E = S\E and so, we derive on the other

V AR C (S ∪ {i, j}) -V AR C (S ∪ {j}) = 1 (s+1)•(s+2) • k∈S\E (c k -c) 2
To complete the proof, choose i, j ∈ E. i = j, as well as any ∈ N \E. The former two equalities apply to S = { } since S ⊆ E. Then the strict inequalty V AR C (S ∪ {i, j})-V AR C (S ∪{j}) < V AR C (S ∪{i})-V AR C (S) holds due to (s + 1) • (s + 2) > s • (s + 1) as well as

k∈S\E (c k -c) 2 = c -c 2 > 0.

IV. THE SHAPLEY VALUE OF THE BERTRAND OLIGOPOLY TU-GAME

The decomposition (21) of the Bertrand Oligopoly Game N, v β into four types of games permits to determine its Shapley value Sh(N, v β ) on basis of its linearity, efficiency, symmetry, and strategic equivalence. Generally speaking, the Shapley value Sh(N, w) = (Sh i (N, w)) i∈N of an arbitrary game N, w is given by an appropriate weighted, probabilistic sum of player's marginal contributions of the form w(S ∪ {i}) -w(S), S ⊆ N \{i}, that is ( [START_REF] Shapley | A value for n-person games. Contributions to the theory of games II[END_REF]),for all i ∈ N Sh i (N, w) = S⊆N \{i} p n (s)• w(S ∪{i})-w(S) (31) where p n (s) =

1 n•( n-1 s )
for all s = 0, 1, 2, . . . , n -1. Due to its probabilistic interpretation, the Shapley value of any non-essential game N, w equals the individual's worth w({i}), i ∈ N . Moreover, because of anonimity and efficiency, the Shapley value of the symmetric game N, w 4 is fully determined by Sh i (N, w 4 ) = w4(N ) n = 1 + r for all i ∈ N . Fourthly, the computation of the Shapley value of the game N, w 2 proceeds by using (22) yielding for all i ∈ N Sh

i (N, w 2 ) = S⊆N \{i} p n (s) • (c i ) 2 + 2 • c i • j∈S c j = (c i ) 2 + 2 • c i • p n (s) • S⊆N \{i} j∈S c j Through the inverse order j∈N \{i} c j • S⊆N \{i}. S j p n (s),
together with two combinatorial steps, we arrive at

Sh i (N, w 2 ) = c i • j∈N c j = n • c i • cN .
Sh i (N, v β ) = -V 2 • c i -r 4•n • n • c i • cN + 1+r 4 • (c i ) 2 + V 2 4 (32) In short, Sh i (N, v β ) = (V -ci) 2 4 + r 4 • c i • (c i -cN ) for all i ∈ N .
In words, the Shapley value of the Bertrand Oligopoly Game involves two types of payoffs to each firm i, i ∈ N , namely the square of the netto demand intercept Vc i , as well as a proportional part c i of the firm's deviation c i -cN from the average grand coalitional cost.

Sh i (N, V AR C ) = a n • c+i n -b n • c-i n where (33) c+i n = j∈N \{i} (c i -c j ) 2 n -1 (34) c-i n = {j,k}⊆N \{i}, j =k (c j -c k ) 2 n-1 2 (35)
a n = n -H n n and b n = n+1 2 -H n n (36)
H n = n k=1 1 k (37)
Lemma 4.1: The game representation of the Variance TU game N, V AR C of the form (15) with respect to the basis of unanimity games {(N, u T ) | T ⊆ N, T = ∅} is given by

V AR C = T ⊆N, T =∅ λ V AR T • u T (38)
where

λ V AR T = (-1) t 2 • {j,k}⊆T , j =k (cj-ck) 2 ( t 2 )
if t ≥ 2, while λ V AR {j} = 0 for all j ∈ N . Proof of Lemma 4.1 Through substitution of λ V AR T , T ⊆ N , T = ∅, we show that for all S ⊆ N , S = ∅,

V AR C (S) = T ⊆N, T =∅ λ V AR T • u T (S) (39)
or equivalently,V AR

C (S) = T ⊆S, t≥2 λ V AR T .
Let S ⊆ N , S = ∅. For one-person coalitions S = {j}, j ∈ N , the latter equality holds because of λ V AR {j} = 0 = V AR C ({j}). In the remainder, let s ≥ 2. On the one hand, we obtain the following chain of equations.

T ⊆S, t≥2 λ V AR T = T ⊆S, t≥2 (-1) t t • (t -1) • {j,k}⊆T , j =k (c j -c k ) 2 = {j,k}⊆S, j =k (c j -c k ) 2 • T ⊆S, {j,k}⊆T (-1) t t • (t -1) = {j,k}⊆S, j =k (c j -c k ) 2 • s t=2 s -2 t -2 • (-1) t t • (t -1) = {j,k}⊆S, j =k (c j -c k ) 2 • s t=2 s t • (-1) t • 1 s • (s -1) = 1 s•(s-1) • {j,k}⊆S, j =k (c j -c k ) 2 • s t=0 s t • (-1) t - s 0 + s 1 = 1 s • (s -1) • {j,k}⊆S, j =k (c j -c k ) 2 • ((-1) + 1) s -1 + s = 1 s • {j,k}⊆S, j =k (c j -c k ) 2
On the other hand,

V AR C (S) = j∈S c j -c(S) 2 = j∈S (c j ) 2 + (c(S)) 2 -2 • c j • c(S) = j∈S (c j ) 2 -s • (c(S)) 2 = j∈S (c j ) 2 -1 s • j∈S c j 2 = j∈S (c j ) 2 -1 s • j∈S (c j ) 2 + 2 • {j,k}⊆S, j =k c j • c k = s-1 s • j∈S (c j ) 2 -2 s • {j,k}⊆S, j =k c j • c k = 1 s • j∈S (s -1) • (c j ) 2 -2 s • {j,k}⊆S, j =k c j • c k = 1 s • {j,k}⊆S, j =k (c j ) 2 + (c k ) 2 -2 • c j • c k = 1 s • {j,k}⊆S, j =k (c j -c k ) 2
Both parts yield (39) for all S ⊆ N , S = ∅, and consequently, (38) holds.

Proof of Theorem 4.2 Recall the well-known fact that the Shapley value of any unanimity game is determined by Sh i (N, u T ) = 0 for all i ∈ N \T , whereas Sh i (N, u T ) = 1 t for all i ∈ T . Together with the linearity of the Shapley value and the game representation (38) of the Variance TU game, it follows that its Shapley value is given by, for all i ∈ N ,

Sh i (N, V AR C ) = T ⊆N, T =∅ λ V AR T • Sh i (N, u T ) = T ⊆N, i∈T λ V AR T t = T ⊆N,t≥2, i∈T (-1) t 2 • t • {j,k}⊆T , j =k (c j -c k ) 2 t 2 = T ⊆N,t≥2, i∈T (-1) t t 2 • (t -1) • {j,k}⊆T , j =k (c j -c k ) 2 = {j,k}⊆N, j =k (c j -c k ) 2 • T ⊆N,i∈T , {j,k}⊆T (-1) t t 2 • (t -1)
Given any pair {j, k} ⊆ N , and player i ∈ N , we distinguish two cases. If i ∈ {j, k}, then

T ⊆N,i∈T , {j,k}⊆T (-1) t t 2 • (t -1) = T ⊆N, {j,k}⊆T (-1) t t 2 • (t -1) = n t=2 n -2 t -2 • (-1) t t 2 • (t -1) = 1 n • (n -1) • n t=2 n t • (-1) t t = 1 n -1 + 1 n • (n -1) • n t=1 n t • (-1) t t = 1 n -1 + 1 n • (n -1) • α n = n + α n n • (n -1) where α n = n t=1 n t • (-1) t t . If i ∈ {j, k}, then T ⊆N,i∈T , {j,k}⊆T (-1) t t 2 • (t -1) = T ⊆N, {i,j,k}⊆T (-1) t t 2 • (t -1) = n t=3 n -3 t -3 • (-1) t t 2 • (t -1) = 1 n • (n -1) • (n -2) • n t=3 n t • t -2 t • (-1) t = -1 (n -1) • (n -2) + 1 n • (n -1) • (n -2) • n t=1 n t • t -2 t • (-1) t = -1 (n -1) • (n -2) - 1 n • (n -1) • (n -2) + 1 n • (n -1) • (n -2) • -2 • n t=1 n t • (-1) t t + n t=0 n t • (-1) t = -(n + 1) n • (n -1) • (n -2) + -2 • α n + ((-1) + 1) n n • (n -1) • (n -2) = -(n + 1) -2 • α n n • (n -1) • (n -2)
We arrive at

Sh i (N, V AR C ) = {j,k}⊆N,j =k, i∈{j,k} (c j -c k ) 2 • n + α n n • (n -1) - {j,k}⊆N,j =k, i ∈{j,k} (c j -c k ) 2 • (n + 1) + 2 • α n n • (n -1) • (n -2)
It remains to pove that the sequence α n =

λ S • cN -cS • 2 • c i -cN -cS = V AR C (N ) + S∈B λ S • (c N -cS ) • i∈S 2 • c i -cS -cN = V AR C (N ) + S∈B λ S • (c N -cS ) • 2 • s • cS -s • cS -s • cN = V AR C (N ) + S∈B λ S • (c N -cS ) • s • (c S -cN ) = V AR C (N ) - S∈B λ S • s • cN -cS 2 ≤ V AR C (N )
Because every subgame of the Bertrand Oligopoly Game is of the same type (associated with the same data restricted to the new player set), the Bertrand Oligopoly Game is called totally balanced (and so, every subgame has a non-emepty core).

VI. CONCLUDING REMARKS

The Bertrand Oligopoly situation with Shubik's demand functions has been modelled as a cooperative TU-game in [START_REF] Lardon | Convexity of Bertrand Oligopoly TU-Games with differentiated products[END_REF], but only with reference to identical marginal costs for all firms. The current paper continues to study the general situation with distinct marginal costs. The complexity of the description of the associated cooperative game, as a result of solving two subsequent optimization problems, is compensated by decomposing the Bertrand Oligopoly Game into four types of games, namely two non-essential games, one symmetric game, and the square of one of these non-essential games. Although it concerns the difference of two convex games, it is shown that the Bertrand Oligopoly Game is convex too. Its current proof technique by decomposition differs from Lardon's proof of convexity for the symmetrical Bertrand Oligopoly Game. Surprisingly, under certain circumstances, the Bertrand Oligopoly Game agrees with the fundamental notion in statistics called Variance with respect to the distinct marginal costs. In the symmetric Bertrand Oligopoly Game this variance degenerates into the trivial zero game.

1 s • j∈S c j denote 2 . 1 :

 21 the average coalitional cost of coalition S ⊆ N , S = ∅.Theorem Fix coalition S ⊆ N , S = ∅, and strategy profile p N \S ∈ X N \S .

Example 3 . 1 :

 31 We consider two three-firm Variance Games (i) Consider three almost equal marginal costs c 1 = c -1, c 2 = c and c 3 = c + 1. The zero-normalized three-firm Variance Game is given by

  The variance game is neither average convex since the constraint i∈S [v(S) -v(S\{i})] ≤ i∈S [v(T ) -v(T \{i})] for all S ⊆ T ⊆ N fails to hold whenever S = {1, 3} and T = N . (ii) Consider three very distinct marginal costs c 1 = 1, c 2 = c + 1 and c 3 = 2 • c + 1. The zero-normalized three-firm Variance Game is given by

  Finally, by applying the linearity of the Shapley value, we conclude the following: Theorem 4.1: The Shapley value of the Bertrand Oligopoly Game of the form (21) is given by

Theorem 4 . 2 :

 42 The Shapley value of the Variance TU game N, V AR C of the form (15) is given by

1 k 5 . 1 :λ 2 = 2 + 2 • 2 +•

 1512222 . It is left for the reader to check that the recursive relationship α n+1 = α n -1 n+1 holds by using the combinatorial relationship n+1 t = n t + n t-1 for all 0 ≤ t ≤ n. Together with α 1 = -1, the claim follows immediately. This completes the full proof of formula (34) for the Shapley value of the Variance TU game.V. TOTALLY BALANCEDNESS OF THE BERTRAND OLIGOPOLY TU-GAMEAn arbitrary game N, w with characteristic function w : P(N ) → R is said to be balanced if it holds S∈B λ S •w(S) ≤ w(N ) for every balanced collection B ⊆ P(N ) of coalitions and corresponding weights (λ S ) S∈B satisfying λ S > 0 for all S ∈ B and S∈B, S i λ S = 1 for all i ∈ N .Theorem The Variance TU game N, V AR C of the form (15) is balanced (and hence, its core is nonempty)Proof. Let B be a balanced collection of coalitions S, S ∈ B, with corresponding weights λ S > 0, S ∈ B, satisfying S∈B, S i λ S = 1 for all i ∈ N . By various appropriate computations we obtain the following chain of equalities:S∈B λ S • V AR C (S) = S∈B S • (c i -cN ) + (c N -cS ) i∈N S∈B, S i λ S • (c i -cN ) 2 + (c N -cS ) (c i -cN ) • (c N -cS ) = i∈N S∈B, S i λ S • c i -cN -cS ) 2 + 2 • (c i -cN ) • (c N -cS ) cN -cS • (c N -cS ) + 2 • (c i -cN )


Theo S H Driessen
email: t.s.h.driessen@ewi.utwente.nl

Dongshuang Hou
email: dshhou@126.com

Aymeric Lardon
email: aymeric.lardon@univ-st-etienne.fr

‡ A Lardon

Stackelberg oligopoly TU-games: characterization of the core and 1-concavity of the dual game

Keywords: Stackelberg oligopoly TU-game, Dual game, 1-concavity

 in which any deviating coalition produces an output at a first period as a leader and outsiders simultaneously and independently play a quantity at a second period as followers. We assume that the inverse demand function is linear and that firms operate at constant but possibly distinct marginal costs. Generally speaking, for any TU-game we show that the 1-concavity property of its dual game is a necessary and sufficient condition under which the core of the initial game is non-empty and coincides with the set of imputations. The dual game of a Stackelberg oligopoly TU-game is of great interest since it describes the marginal contribution of followers to join the grand coalition by turning leaders. The aim is to provide a necessary and sufficient condition which ensures that the dual game of a Stackelberg oligopoly TU-game satisfies the 1-concavity property. Moreover, we prove that this condition depends on the heterogeneity of firms' marginal costs, i.e., the dual game is 1-concave if and only if firms' marginal costs are not too heterogeneous. This last result extends Marini and Currarini's core non-emptiness result (2003) for oligopoly situations.

Introduction

Usually, oligopoly situations are modeled by means of non-cooperative games. Every profit-maximizing firm pursues Nash strategies and the resulting outcome is not Pareto optimal. Yet, it is known that firms are better off by forming cartels and that Pareto efficiency is achieved when all the firms merge together. A problem faced by the members of a cartel is the stability of the agreement and non-cooperative game theory predicts that the cartel members always have an incentive to deviate from the agreed-upon output decision.

However, in some oligopoly situations firms don't always behave non-cooperatively and if sufficient communication is feasible it may be possible for firms to sign agreements. A question is then whether it is possible for firms to agree all together and coordinate their decision to achieve Pareto efficiency. For that, we consider a fully cooperative approach for oligopoly situations. Under this approach, firms are allowed to sign binding agreements in order to form cartels called coalitions. Under such an assumption cooperative games called oligopoly TU(Transferable Utility)games can be defined and the existence of stable collusive behaviors is then related to the non-emptiness of the core of such games. [START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF] proposes two approaches in order to define cooperative games: according to the first, every cartel computes the total profit which it can guarantee itself regardless of what outsiders do; the second approach consists in computing the minimal profit for which outsiders can prevent the cartel members from getting more. These two assumptions lead to consider the α and β-characteristic functions respectively. However, these two approaches can be questioned since outsiders probably cause substantial damages upon themselves by minimizing the profit of the cartel. This is why [START_REF] Chander | The core of an economy with multilateral environmental externalities[END_REF] propose an alternative blocking rule where external firms choose their strategy individually as a best reply to the cartel action. This leads to consider the "partial agreement characteristic function" or, for short, the γ-characteristic function.

As regards Cournot oligopoly TU-games in α and β-characteristic function forms with or without transferable technologies, 1 Zhao (1999a,b) shows that the α and βcharacteristic functions lead to the same set of Cournot oligopoly TU-games. When technologies are transferable, Zhao (1999a) provides a necessary and sufficient condition to establish the convexity property in case the inverse demand function and cost functions are linear. Although these games may fail to be convex in general, [START_REF] Norde | Oligopoly games with and without transferable technologies[END_REF] show they are nevertheless totally balanced. When technologies are not transferable, Zhao (1999b) proves that the core of such games is non-empty if every individual profit function is continuous and concave.2 Furthermore, [START_REF] Norde | Oligopoly games with and without transferable technologies[END_REF] show that these games are convex in case the inverse demand function and cost functions are linear, and [START_REF] Driessen | Convexity of oligopoly games without transferable technologies[END_REF] provide economically meaningful sufficient conditions to guarantee the convexity property in a more general case.

As regards Cournot oligopoly TU-games in γ-characteristic function form without transferable technologies, [START_REF] Lardon | The γ-core of cournot oligopoly games with capacity constraints[END_REF] shows that the differentiability of the inverse demand function ensures that these games are well-defined and provides two core non-emptiness results. The first result establishes that such games are balanced, and therefore have a non-empty core, if every individual profit function is continuous and concave. When cost functions are linear, the second result provides a single-valued allocation rule in the core, called NP(Nash Pro rata)-value, which is characterized by four axioms: efficiency, null firm, monotonicity and noncooperative fairness. 3 In a more general framework, [START_REF] Lardon | Cournot oligopoly interval games[END_REF] shows that if the inverse demand function is continuous but not necessarily differentiable, it is always possible to define a Cournot oligopoly interval game4 in γ-characteristic function form. This set of Cournot oligopoly interval games includes the set of Cournot oligopoly TU-games in γ-characteristic function form. Lardon considers two extensions of the core, the interval core and the standard core, and provides a necessary and sufficient condition for the non-emptiness of each of these two core solution concepts. For general TU-games, [START_REF] Marini | A sequential approach to the characteristic function and the core in games with externalities[END_REF] associate a two-stage structure with the γ-characteristic function. In this temporal sequence every deviating coalition possesses a first-mover advantage by acting as a leader while outsiders play their individual best reply strategies as followers. Under standard assumptions on the payoff functions,5 they prove that if the players and externalities are symmetric6 and the game has strategic complementarities then the equal division solution belongs to the core. They also study a quantity competition setting where strategies are substitutes for which they show that the same mechanism underlying their core result determines the non-emptiness of the core.

Marini and Currarini's core non-emptiness result ( 2003) raises two questions. The first concerns the core structure of TU-games in sequential form since they only provide a single-valued allocation rule (the equal division solution) in the core. The second turns on the role of the symmetric players assumption on the nonemptiness of the core. In this article we answer both questions by considering the two-stage structure associated with the γ-characteristic function as suggested by [START_REF] Marini | A sequential approach to the characteristic function and the core in games with externalities[END_REF] for oligopoly situations. The introduction of this temporal sequence leads to consider a specific set of cooperative oligopoly games, i.e., the set of Stackelberg oligopoly TU-games in γ-characteristic function form. Thus, contrary to Cournot oligopoly TU-games in γ-characteristic function form in which all the firms simultaneously choose their strategies, every deviating coalition produces an output at a first period as a leader and outsiders simultaneously and independently play a quantity at a second period as followers. We assume that the inverse demand function is linear and firms operate at constant but possibly distinct marginal costs. Thus, contrary to [START_REF] Marini | A sequential approach to the characteristic function and the core in games with externalities[END_REF] the payoff (profit) functions are not necessarily identical. The 1-concavity property of the dual game of a Stackelberg oligopoly TU-game answers the above two questions raised by Marini and Currarini's core non-emptiness result (2003). Indeed, generally speaking, for any TU-game we show that the 1-concavity property of its dual game is a necessary and sufficient condition under which the core of the initial game is non-empty and coincides with the set of imputations. Particularly, the nucleolus of such a 1concave dual game agrees with the center of gravity of the set of imputations. The dual game of a Stackelberg oligopoly TU-game is of great interest since it gives the marginal contribution of followers to join the grand coalition by turning leaders. In order to establish the 1-concavity of the dual game we first characterize the core by proving that it is equal to the set of imputations which answers the first question. The reason is that the first-mover advantage gives too much power to singletons so that the worth of every deviating coalition is less than or equal to the sum of its members' individual worths except for the grand coalition. Then, we provide a necessary and sufficient condition under which the core is non-empty. Finally, we prove that this condition depends on the heterogeneity of firms' marginal costs, i.e., for a fixed number of firms the dual game is 1-concave if and only if firms' marginal costs are not too heterogeneous. The more the number of firms, the less the heterogeneity of firms' marginal costs must be in order to ensure the 1-concavity of the dual game, and so the non-emptiness of the core, which answers the second question. Surprisingly, in case the inverse demand function is strictly concave, we provide an example in which the opposite result holds, i.e., when the heterogeneity of firms' marginal costs increases the core becomes larger.

The remainder of this article is structured as follows. In section 2 we recall some basic definitions in cooperative game theory and show that Stackelberg oligopoly TU-games in γ-characteristic function form are well-defined. The aim of Section 3 is to establish the 1-concavity of the dual game. Section 4 gives some concluding remarks.

The model

Before introducing Stackelberg oligopoly TU-games in γ-characteristic function form [START_REF] Chander | The core of an economy with multilateral environmental externalities[END_REF], we recall some basic definitions in cooperative game theory. Given a set of players N , we call a subset S ∈ 2 N \{∅}, a coalition. The size s = |S| of coalition S is the number of players in S. A TU-game (N, v) is a set function v : 2 N -→ R with the convention that v(∅) = 0, which assigns a number v(S) ∈ R to every coalition S ∈ 2 N \{∅}. The number v(S) is the worth of coalition S and represents the maximal amount of monetary benefits due to the mutual cooperation among the members of the coalition, on the understanding that there are no benefits by absence of players. We denote by G the set of TU-games where (N, v) is a representative element of G.

With every TU-game (N, v) ∈ G there is associated its dual game (N, v * ) defined by v * (S) = v(N ) -v(N \S) for all S ∈ 2 N . That is, the worth of any coalition in the dual game is given by the coalitionally marginal contribution with respect to the formation of the grand coalition N in the original game. Particularly, v * (∅) = 0, v * (N ) = v(N ), and so, the marginal contribution

m v * i = v * (N ) -v * (N \{i}) = v({i}) for all i ∈ N .
In the framework of the division problem of the benefits v(N ) of the grand coalition N among the potential players, any allocation scheme of the form x = (x i ) i∈N ∈ R n is supposed to meet, besides the efficiency principle i∈N x i = v(N ), the so-called individual rationality condition in that each player is allocated at least the individual worth, i.e., x i ≥ v({i}) for all i ∈ N . For any TU-game (N, v) ∈ G, these two conditions lead to consider the set of imputations:

I(N, v) = x ∈ R n : i∈N x i = v(N ) and ∀i ∈ N , x i ≥ v({i}) .
The best known set-valued solution concept called core requires the group rationality condition in that the aggregate allocation to the members of any coalition is at least its coalitional worth:

C(N, v) = x ∈ I(N, v) : ∀S ∈ 2 N \{∅, N }, i∈S x i ≥ v(S) (1)
Of significant importance, is the upper core bound composed of the marginal contributions m v i = v(N ) -v(N \{i}), i ∈ N , with respect to the formation of the grand coalition N in the game (N, v). Obviously, x i ≤ m v i for all i ∈ N and all x ∈ C(N, v). In this context, we focus on the following core catcher called Core-Cover:

CC(N, v) = x ∈ R n : i∈N x i = v(N ) and ∀i ∈ N , x i ≤ m v i .
Now, we introduce the Stackelberg oligopoly model. A Stackelberg oligopoly situation is a quintuplet (L, F, (q i , C i ) i∈N , p) defined as:

1. the disjoint finite sets of leaders and followers L and F respectively where L ∪ F = {1, 2, . . . , n} is the set of firms denoted by N ;

2. for every i ∈ N , a capacity constraint q i ∈ R + ;

3. for every i ∈ N , an individual cost function C i : R + -→ R + ;

4. an inverse demand function p : R + -→ R which assigns to any aggregate quantity X ∈ R + the unit price p(X).

Throughout this article, we assume that:

(a) firms have no capacity constraint: ∀i ∈ N , q i = +∞;

(b) firms operate at constant but possibly distinct marginal costs:

∀i ∈ N ,∃c i ∈ R + : C i (y i ) = c i y i ,
where c i is firm i's marginal cost, and y i ∈ R + is the quantity produced by firm i;

(c) firms face the linear inverse demand function:

p(X) = a -X,
where X ∈ R + is the total production of the industry and a ∈ R + is the prohibitive price (the intercept) of the inverse demand function p such that a ≥ 2n × max{c i : i ∈ N }.

Given assumptions (a), (b) and (c), a Stackelberg oligopoly situation is summarized by the 4-tuple (L, F, (c i ) i∈N , a). Without loss of generality we assume that the firms are ranked according to their marginal costs, i.e., c 1 ≤ . . . ≤ c n . For notational convenience, for any set of firms S ∈ 2 N \{∅} we denote the minimal coalitional cost by c S = min{c i : i ∈ S} and by i S ∈ S the firm in S with the smallest index that operates at marginal cost c S .

The strategic Stackelberg oligopoly game associated with the Stackelberg oligopoly situation (L, F, (c i ) i∈N , a) is a quadruplet Γ so = (L, F, (X i , π i ) i∈N ) defined as:

1. the disjoint finite sets of leaders and followers L and F respectively where N = L ∪ F is the set of firms;

2. for every k ∈ N , an individual strategy set X k such that:

-for every leader i ∈ L, X i = R + where x i ∈ X i represents the quantity produced by leader i. We denote by X L = i∈L X i the set of strategy profiles of the leaders where

x L = (x i ) i∈L is a representative element of X L ;
-for every follower j ∈ F , X j is the set of mappings x j : X L -→ R + where x j (x L ) represents the quantity produced by follower j given leaders' strategy profile x L ∈ X L . We denote by X F = j∈F X j the set of strategy profiles of the followers where x F = (x j ) j∈F is a representative element of X F ;

3. for every k ∈ N , an individual profit function π k : X L × X F -→ R + such that:

-for every i ∈ L, π i : X L × X F -→ R + is defined as:

π i (x L , x F (x L )) = p(X)x i -c i x i ;
-for every j ∈ F , π j : X L × X F -→ R + is defined as:

π j (x L , x F (x L )) = p(X)x j (x L ) -c j x j (x L ),
where X = i∈L x i + j∈F x j (x L ) is the total production.

Given a strategic Stackelberg oligopoly game Γ so = (L, F, (X i , π i ) i∈N ), every leader i ∈ L simultaneously and independently produces an output x i ∈ X i at a first period while every follower j ∈ F simultaneously and independently plays a quantity x j (x L ) ∈ X j at a second period given leaders' strategy profile x L ∈ X L . We denote by G so the set of strategic Stackelberg oligopoly games.

In case there is a single leader and multiple followers, [START_REF] Sherali | Stackelberg-Nash-Cournot Equilibria: Characterizations and Computations[END_REF] prove the existence and uniqueness of the Nash equilibrium in strategic Stackelberg oligopoly games under standard assumptions on the inverse demand function and the individual cost functions, i.e., the inverse demand function is twice differentiable, strictly decreasing and satisfies:

∀X ∈ R + , dp dX (X) + X d 2 p
dX 2 (X) ≤ 0, and the individual cost functions are twice differentiable and convex. In particular, they show that the convexity of followers' reaction functions with respect to leader's output is crucial for the uniqueness of the Nash equilibrium. Assumptions (a), (b) and (c) ensure that Sherali et al.'s result (1983) holds on G so so that any strategic Stackelberg oligopoly game Γ so = (L, F, (X i , π i ) i∈N ) ∈ G so such that |L| = 1 admits a unique Nash equilibrium. Now, given a strategic Stackelberg oligopoly game Γ so = (L, F, (X i , π i ) i∈N ) ∈ G so , we want to associate a Stackelberg oligopoly TU-game in γ-characteristic function form. In a dynamic oligopoly "à la Stackelberg," this assumption implies that the coalition members produce an output at a first period, thus anticipating outsiders' reaction who simultaneously and independently play a quantity at a second period. For any coalition S ∈ 2 N \{∅} where S = L and N \S = F , the coalition profit function π S : X S × X N \S -→ R is defined as:

π S (x S , x N \S (x S )) = i∈S π i (x S , x N \S (x S )).
Moreover, followers' individual best reply strategies xN\S : X S -→ X N \S is defined as:

∀j ∈ N \S, ∀x S ∈ X S , xj (x S ) ∈ arg max x j (x S )∈X j π j (x S , xN\(S∪{j}) (x S ), x j (x S )).
For any coalition S ∈ 2 N \{∅}, given the strategic Stackelberg oligopoly game

Γ so = (S, N \S, (X i , π i ) i∈N ) ∈ G so , a strategy profile (x * S , xN\S (x * S )) ∈ X S × X N \S is a partial agreement equilibrium under S if: ∀x S ∈ X S , π S (x * S , xN\S (x * S )) ≥ π S (x S , xN\S (x S )), and
∀j ∈ N \S, ∀x j ∈ X j , π j (x * S , xN\S (x * S )) ≥ π j (x * S , xN\(S∪{j}) (x * S ), x j (x * S )).
For any coalition S ∈ 2 N \{∅} and the induced strategic Stackelberg oligopoly game Γ so = (S, N \S, (X i , π i ) i∈N ) ∈ G so , the associated Stackelberg oligopoly TUgame in γ-characteristic function form, denoted by (N, v γ ), is defined as:

v γ (S) = π S (x * S , xN\S (x * S )
), where (x * S , xN\S (x * S )) ∈ X S × X N \S is a partial agreement equilibrium under S. We denote by G γ so ⊆ G the set of Stackelberg oligopoly TU-games in γcharacteristic function form.

Given a strategic Stackelberg oligopoly game Γ so = (S, N \S, (X i , π i ) i∈N ) ∈ G so , by assumptions (a), (b) and (c), and by the definition of the partial agreement equilibrium, any deviating coalition S ∈ 2 N \{∅} can be represented by firm i S ∈ S (the firm in S with the smallest marginal cost) acting as a single leader while the other firms in coalition S play a zero output. It follows from Sherali et al.'s result (1983) that the induced strategic Stackelberg oligopoly game Γ so = ({i S }, N \S, (X i , π i ) i∈{i S }∪N \S ) ∈ G so has a unique Nash equilibrium, and so the strategic Stackelberg oligopoly game Γ so admits a partial agreement equilibrium under S. Indeed, in case there are at least two firms operating at the minimal marginal cost c S , the most efficient firms in coalition S can coordinate their output decision and reallocate the Nash equilibrium output of firm i S among themselves. We conclude that there can exist several partial agreement equilibria under S which support the unique worth v γ (S). Hence, the γ-characteristic function is well-defined. The following proposition goes further by expressing the worth of any deviating coalition.

Proposition 2.1 For any coalition S ∈ 2 N \{∅} and the associated strategic Stackelberg oligopoly game Γ so = (S, N \S, (X i , π i ) i∈N ) ∈ G so , it holds that:

v γ (S) = 1 4(n -s + 1) a + j∈N \S c j -c S (n -s + 1) 2 .
Proof: Take any coalition S ∈ 2 N \{∅} and consider the associated strategic Stackelberg oligopoly game Γ so = (S, N \S, (X i , π i ) i∈N ) ∈ G so . In order to compute the worth v γ (S) of coalition S, we have to successively solve the maximization problems derived from the definition of the partial agreement equilibrium. First, consider the profit maximization program of any follower j ∈ N \S at the second period:

∀x S ∈ X S ,∀x N \(S∪{j}) (x S ) ∈ X N \(S∪{j}) , max x j (x S )∈X j π j (x S , x N \(S∪{j}) (x S ), x j (x S )).
The first-order conditions for a maximum are:

∀j ∈ N \S,∀x N \{j} ∈ X N \{j} , ∂π j ∂x j (x j , x N \{j} ) = 0,
and imply that the unique maximizers xj (x S ), j ∈ N \S, satisfy:

∀j ∈ N \S,∀x S ∈ X S , xj (x S ) = 1 2 a - i∈S x i - k∈N \(S∪{j}) xk (x S ) -c j .
By solving the above system of equations, we deduce that followers' individual best reply strategies at the second period are given by:

∀j ∈ N \S,∀x S ∈ X S , xj (x S ) = 1 (n -s + 1) a - i∈S x i + k∈N \S c k -c j (2)
Then, given xN\S (x S ) ∈ X N \S consider the profit maximization program of coalition S at the first period:

max x S ∈X S π S (x S , xN\S (x S )).
Since the firms have no capacity constraint, it follows that the above profit maximization program of coalition S is equivalent to the profit maximization program of firm i S ∈ S given that the other members in S play a zero output:

max x i S ∈X i S π i S (x i S , 0 S\{i S } , xN\S (x i S , 0 S\{i S } )).
The first-order condition for a maximum is:

∂π i S ∂x i S (x i S , 0 S\{i S } , xN\S (x i S , 0 S\{i S } )) = 0,
and implies that the unique maximizer x * i S ∈ X i S is given by:

x * i S = 1 2 a + j∈N \S c j -c S (n -s + 1) (3)
By ( 2) and ( 3), for any j ∈ N \S it holds that:

xj (x * S ) = xj (x * i S , 0 S\{i S } ) = 1 2(n -s + 1) a + k∈N \S c k + c S (n -s + 1) -c j (4)
By ( 3) and ( 4), we deduce that:

v γ (S) = π S (x * S , xN\S (x * S )) = π i S ((x * i S , 0 S\{i S } ), xN\S (x * i S , 0 S\{i S } )) = 1 4(n -s + 1) a + j∈N \S c j -c S (n -s + 1) 2 ,
which completes the proof.

Thus, the worth of any deviating coalition is increasing with respect to outsiders' marginal costs and decreasing with respect to the smallest marginal cost among its members. Note that the condition a ≥ 2n × max{c i : i ∈ N } (assumption (c)) ensures that the equilibrium outputs in (3) and (4) are positive.

3 Characterization of the core and 1-concavity property

In this section, we first introduce the gap function and the 1-concavity property.

Generally speaking, we show that the 1-concavity property of the dual game is a necessary and sufficient condition for the core of the initial TU-game to coincide with the set of imputations and to be non-empty. The dual game of a Stackelberg oligopoly TU-game is of great interest since it gives the marginal contribution of followers to join the grand coalition by turning leaders. Based on the description of the γ-characteristic function in a Stackelberg oligopoly TU-game, the aim is to establish the 1-concavity property of its dual game. To this end, we first characterize the core by proving that it always coincides with the set of imputations. Then, we provide a necessary and sufficient condition under which the core is non-empty, and so the dual game is 1-concave. Finally, we prove that this condition depends on the heterogeneity of firms' marginal costs, i.e., the dual game is 1-concave if and only if firms' marginal costs are not too heterogeneous. This last result extends Marini and Currarini's core non-emptiness result (2003) for oligopoly situations.

In the setting of the core, a helpful tool appears to be the so-called gap function

g v : 2 N -→ R defined by g v (S) = i∈S m v i -v(S) for all S ∈ 2 N \{∅} with m v i = v(N ) -v(N \{i})
, where g v (∅) = 0. So, the gap g v (S) of any coalition S measures how much the coalitional worth v(S) differs from the aggregate allocation based on the individually marginal contributions. The interrelationship between the gap function and the general inclusion C(N, v) ⊆ CC(N, v) is the following equivalence:

C(N, v) = CC(N, v) ⇐⇒ 0 ≤ g v (N ) ≤ g w (S) for all S ∈ 2 N \{∅} (5)
In words, the core catcher CC(N, v) coincides with the core C(N, v) if and only if the non-negative gap function g v attains its minimum at the grand coalition N . If the latter property (5) holds, the savings game (N, v) ∈ G is said to be 1-convex. We arrive at the first corollary.

Corollary 3.1 Three equivalent statements for any TU-game (N, v) ∈ G are the following:

(i) I(N, v) = ∅; (ii) v(N ) ≥ i∈N v({i}); (iii) g v * (N ) ≤ 0.
In fact, the dual game (N, v * ) ∈ G of any TU-game (N, v) ∈ G is treated as a cost game such that the core equality C(N, v * ) = C(N, v) holds, on the understanding that the core of any cost game is defined through the reversed inequalities of (1). Thus, x ∈ C(N, v * ) if and only if x ∈ C(N, v). As the counterpart to 1-convex TU-games (with non-negative gap functions), we deal with so-called 1-concave TUgames (with non-positive gap functions). A TU-game (N, v) ∈ G is said to be 1concave if its non-positive gap function attains its maximum at the grand coalition N :

g v (S) ≤ g v (N ) ≤ 0 for all S ∈ 2 N \{∅} (6)
Corollary 3.2 Three equivalent statements for any TU-game (N, v) ∈ G are the following:

(i) The dual game (N, v * ) is 1-concave, that is (6) applied to (N, v * ) holds;

(ii) It holds that:

v(N ) ≥ i∈N v({i}) and v(S) ≤ i∈S v({i}) for all S ∈ 2 N \{∅, N } (7) (iii) I(N, v) = ∅ and C(N, v) = I(N, v).
Proof: In view of Corollary 3.1, together with C(N, v) ⊆ I(N, v), it remains to prove the implication (iii) =⇒ (ii). By contraposition, suppose (ii) does not hold in that there exists S ∈ 2 N \{∅, N } with v(S) > i∈S v({i}). Define the allocation x ∈ R n by x i = v({i}) for all i ∈ S and

x i = v({i}) + (1/(n -s)) v(N ) -j∈N v({j}) for all i ∈ N \S. Obviously, x ∈ I(N, v)\C(N, v).
Now, we want to provide a necessary and sufficient condition under which the dual game of a Stackelberg oligopoly TU-game in γ-characteristic function form is 1concave. In order to establish the 1-concavity of the dual game (N, v * γ ) ∈ G, we want to show (iii) of Corollary 3.2. First, the following proposition provides a characterization of the core.

Proposition 3.3 Let (N, v γ ) ∈ G γ
so be a Stackelberg oligopoly TU-game. Then, it holds that:

C(N, v γ ) = I(N, v γ ).
In order to establish the proof of Proposition 3.3 we first need the following lemma. Given a set of marginal costs {c i } i∈N and any coalition S ∈ 2 N \{∅} let α(S) = j∈S\{i S } (c S -c j ) 2 and denote by:

A 1 (S) = 1 2 j∈S\{i S } k∈S\{i S } (c j -c k ) 2 ; B 1 (S) = (s -1) α(S) -A 1 (S) ; C 1 (S) = -(s -1) sα(S) + A 1 (S) ; D 1 (S) = -(s -1) α(S) + A 1 (S) .
We define the functions f 1 : N × 2 N \{∅} -→ R and f 2 : N × 2 N \{∅} -→ R as:

f 1 (n, S) = 3A 1 (S)n 2 + 3A 1 (S) + 2B 1 (S) n + A 1 (S) + B 1 (S) + C 1 (S); f 2 (n, S) = A 1 (S)n 3 + B 1 (S)n 2 + C 1 (S)n + D 1 (S).
Lemma 3.4 Let {c i } i∈N be a set of marginal costs. Then, for any n ≥ 3 and any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , n -1} it holds that (i) f 1 (n, S) ≥ 0, and (ii) f 2 (n, S) ≥ 0.

Proof: First we show point (i). For any n ≥ 3 and any coalition S ∈ 2 N \{∅} such that s = n -1 it holds that:

f 1 (n, S) = (n 2 -4)α(S) + (n 2 + 5n + 5)A 1 (S) ≥ 0 (8)
Then, we show that for any n ≥ 3 and any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , n -1}, f 1 (n, S) ≥ 0. We proceed by a double induction on the number of firms n ≥ 3 and the size s ∈ {2, . . . , n -1} of coalition S respectively. Initialisation: assume that n = 3 and take any coalition S ∈ 2 N \{∅} such that s = 2. By (8) it holds that f 1 (3, S) ≥ 0. Induction hypothesis: assume that for any n ≤ k and for any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , n -1}, f 1 (n, S) ≥ 0.

Induction step: we want to show that for n = k + 1 and for any coalition

S ∈ 2 N \{∅} such that s ∈ {2, . . . , k}, f 1 (k + 1, S) ≥ 0. It follows from (8) that for any coalition S ∈ 2 N \{∅} such that s = k, f 1 (k + 1, S) ≥ 0. It remains to show that for any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , k -1}, f 1 (k + 1, S) ≥ 0. Take any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , k -1}.
Then it follows from the definition of f 1 and the induction hypothesis that:

f 1 (k + 1, S) = f 1 (k, S) + 6A 1 (S)k + 6A 1 (S) + 2B 1 (S) = f 1 (k, S) + A 1 (S)(6k -2s + 8) + 2(s -1)α(S) ≥ 0,
which concludes the proof of point (i).

Then, we show point (ii). For any n ≥ 3 and any coalition S ∈ 2 N \{∅} such that s = n -1 it holds that:

f 2 (n, S) = (n 2 -3n + 2)α(S) + (n 2 + n + 2)A 1 (S) ≥ 0 (9)
Then, we show that for any n ≥ 3 and any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , n -1}, f 2 (n, S) ≥ 0. We proceed by a double induction on the number of firms n ≥ 3 and the size s ∈ {2, . . . , n -1} of coalition S respectively. Initialisation: assume that n = 3 and take any coalition S ∈ 2 N \{∅} such that s = 2. By (9) it holds that f 2 (3, S) ≥ 0. Induction hypothesis: assume that for any n ≤ k and for any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , n -1}, f 2 (n, S) ≥ 0.

Induction step: we want to show that for n = k + 1 and for any coalition

S ∈ 2 N \{∅} such that s ∈ {2, . . . , k}, f 2 (k + 1, S) ≥ 0. It follows from (9) that for any coalition S ∈ 2 N \{∅} such that s = k, f 2 (k + 1, S) ≥ 0. It remains to show that for any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , k -1}, f 2 (k + 1, S) ≥ 0. Take any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , k -1}.
Then it follows from the definitions of f 1 and f 2 , the induction hypothesis and point (i) of Lemma 3.4 that:

f 2 (k + 1, S) = f 2 (k, S) + f 1 (k, S) ≥ 0,
which concludes the proof of point (ii). Now, we are ready to establish the proof of Proposition 3.3 which consists in showing that the first-mover advantage gives too much power to singletons so that the worth of any deviating coalition is less than or equal to the sum of its members' individual worths except for the grand coalition. Given a set of marginal costs {c i } i∈N and any coalition S ∈ 2 N \{∅} we denote by:

A 2 (S) = (n -s)(s -1) 4n(n -s + 1) (note that for s ∈ {2, . . . , n -1}, A 2 (S) > 0); B 2 (S) = 1 2n i∈S j∈N \{i} c j -nc i - 1 2(n -s + 1) j∈N \S c j -c S (n -s + 1) ; C 2 (S) = 1 4n i∈S j∈N \{i} c j -nc i 2 - 1 4(n -s + 1) j∈N \S c j -c S (n -s + 1) 2 .
These quantities will be used in the following proof.

Proof (of Proposition 3.3): First, assume that n = 2. By the definitions of the core and the set of imputations it holds that C(N, v γ ) = I(N, v γ ). Then, assume that n ≥ 3. The core is equal to the set of imputations if and only if:

∀S ∈ 2 N \{∅} : s ∈ {2, . . . , n -1}, v γ (S) ≤ i∈S v γ ({i}).
In order to prove the above condition, take any coalition S ∈ 2 N \{∅} such that s ∈ {2, . . . , n -1}. By Proposition 2.1 we deduce that:

i∈S v γ ({i}) -v γ (S) = A 2 (S)a 2 + B 2 (S)a + C 2 (S).
Now, we define the mapping P S : R -→ R as:

P S (y) = A 2 (S)y 2 + B 2 (S)y + C 2 (S), so that P S (a) = i∈S v γ ({i}) -v γ (S)
. We want to show that for any y ∈ R, P S (y) ≥ 0. It follows from A 2 (S) > 0 that the minimum of P S is obtained at point y * ∈ R such that:

y * = - B 2 (S)
2A 2 (S) .

After some calculation steps, we obtain that the minimum of P S is equal to:

P S (y * ) = 1 4n(n -s)(s -1) f 2 (n, S),
where f 2 is defined as in Lemma 3.4. Hence, it follows from point (ii) of Lemma 3.4 that P S (y * ) ≥ 0, which implies that for any y ∈ R, P S (y) ≥ 0. In particular, we conclude that P S (a) ≥ 0, and so i∈S v γ ({i}) -v γ (S) ≥ 0.

Now, we provide a necessary and sufficient condition for the non-emptiness of the core of Stackelberg oligopoly TU-games in γ-characteristic function form as enunciated in the following proposition.

Theorem 3.5 Let (N, v γ ) ∈ G γ so be a Stackelberg oligopoly TU-game. Then, its dual game (N, v * γ ) ∈ G is 1-concave if and only if:

2a i∈N c i -nc N ≥ i∈N j∈N \{i} c j -nc i 2 -nc 2 N ( 10
)
or equivalently

2a(c N -c N ) ≥ (n + 1) 2 n j∈N c 2 j - (n + 2) n j∈N c j 2 -c 2 N ( 11
)
where c N = i∈N c i /n is the average cost of the grand coalition.

Proof: It follows from Proposition 3.3 that the core is non-empty if and only if i∈N v γ ({i}) ≤ v γ (N ). By Proposition 2.1 it holds that:

i∈N v γ ({i}) = 1 4n i∈N a + j∈N \{i} c j -nc i 2 = 1 4n i∈N a + j∈N \{i} c j -nc i -c N + c N 2 = 1 4n i∈N a -c N 2 + 2(a -c N ) j∈N \{i} c j -nc i + c N + j∈N \{i} c j -nc i + c N 2 = v γ (N ) + 1 4n 2a nc N - i∈N c i + i∈N j∈N \{i} c j -nc i 2 -nc 2 N = v γ (N ) + 1 4n 2a nc N -nc N + (n + 1) 2 j∈N c 2 j -(n + 2) j∈N c j 2 -nc 2 N ,
which permits to conclude that i∈N v γ ({i}) ≤ v γ (N ) if and only if inequalities (10) or (11) holds.

In case all the firms have the same marginal cost, the both sides of inequality (10) are equal to zero which implies that i∈N v γ ({i}) = v γ (N ), and so (v γ ({i})) i∈N = (v γ (N )/n) i∈N is the unique core element which coincides with Marini and Currarini's core allocation result (2003). [START_REF] Driessen | One-concavity of data cost games[END_REF] show that the nucleolus of any 1-concave TU-game coincides with the center of gravity of the core. Hence, for any Stackelberg oligopoly TUgame (N, v γ ) ∈ G γ so , the dual game (N, v * γ ) is 1-concave if and only if inequality (10) holds, implying that the nucleolus of (N, v * γ ) is the center of gravity of the core of (N, v * γ ).

The following theorem gives a more relevant expression of inequality (10). When the difference between any two successive marginal costs is constant, it provides an upper bound on the heterogeneity of firms' marginal costs below which inequality (10) holds.

Theorem 3.6 Let (N, v γ ) ∈ G γ so be a Stackelberg oligopoly TU-game such that:

∃δ ∈ R + : ∀i ∈ {1, . . . , n -1}, c i+1 = c i + δ (12)
Then, inequality (10) holds if and only if:

δ ≤ δ * (n) = 12(a -c N )(n -1) n 4 + 2n 3 + 3n 2 -8n + 2 (13)
Proof: It follows from ( 12) that inequality (10) can be expressed as:

i∈N δ (n 2 -2ni + n) 2 -c i 2 -nc 2 N ≤ an(n -1)δ.
By noting that:

i∈N c 2 i -nc 2 N = δc N n(n -1) + δ 2 n(n -1)(2n -1) 6 ,
we deduce that the above inequality is equivalent to:

δ i∈N n 2 -2ni + n 2 2 + n(n -1)(2n -1) 6 ≤ (a -c N )n(n -1) + i∈N (n 2 -2ni + n)c i (14)
It remains to compute the two sums in inequality ( 14). First, it holds that:

i∈N n 2 -2ni + n 2 2 = n 3 (n + 1) 2 4 -n 2 (n + 1) i∈N i + n 2 i∈N i 2 = 1 12 2n 4 (n + 1) -n 3 (n + 1) 2 (15)
Then, it holds that:

i∈N c i = nc N + δ n-1 i=1 i = n c N + δ (n -1) 2 ,
and

i∈N ic i = c N n i=1 i + δ n i=1 i(i -1) = c N n(n + 1) 2 + δ n(n + 1)(2n -2) 6 .
Hence, we deduce that:

i∈N (n 2 -2ni + n)c i = (n 2 + n) i∈N c i -2n i∈N ic i = 1 6 δn 2 (n + 1)(1 -n) (16)
By ( 15) and ( 16), we conclude that inequality ( 14) is equivalent to (13).

By noting that:

dδ * dn (n) = - 36(a -c N )(n 2 + 2n + 2) (n 3 + 3n 2 + 6n -2) 2 < 0, and
d 2 δ * dn 2 (n) = 72(a -c N )(2n 4 + 8n 3 + 15n 2 + 20n + 14) (n 3 + 3n 2 + 6n -2) 3 > 0,
we deduce that the bound δ * (n) is strictly decreasing and strictly convex with respect to the number of firms n. Moreover, when n tends to infinity its limit is equal to 0. So, the more the number of firms is, the less the heterogeneity of firms' marginal costs must be in order to ensure the non-emptiness of the core. This result extends Marini and Currarini's core allocation result ( 2003) and shows that their result crucially depends on the symmetric players assumption.

We saw that when the heterogeneity of firms' marginal costs increases the core becomes smaller. Surprisingly, in case the inverse demand function is strictly concave, the following example shows that the opposite result may holds, i.e., when the heterogeneity of firms' marginal costs increases the core becomes larger.

Example 3.7

Consider the three Stackelberg oligopoly TU-games Thus, it holds that ∅ = C(N, v 1 γ ) ⊂ C(N, v 2 γ ) ⊂ C(N, v 3 γ ), and so when the heterogeneity of firms' marginal costs increases the core becomes larger.

(N, v 1 γ ) ∈ G γ so , (N, v 2 γ ) ∈ G γ so and (N, v 3 γ ) ∈ G γ so associated with the Stackelberg oligopoly situations (L, F, (c 1 , c 1 2 ), p), (L, F, (c 1 , c 2 2 ), p) and (L, F, (c 1 , c 3 2 ), p) respectively where N = {1, 2}, c 1 = c 1 2 = 2, c 2 2 = 4, c 3 2 = 5 and p = 10 -X 2 .

Concluding remarks

In this article we have considered Stackelberg oligopoly TU-games in γ-characteristic function form [START_REF] Chander | The core of an economy with multilateral environmental externalities[END_REF] in which every deviating coalition produces an output at a first period and outsiders simultaneously and independently play a quantity at a second period. We have assumed that the inverse demand function is linear and that firms operate at constant but possibly distinct marginal and average costs. Thus, contrary to [START_REF] Marini | A sequential approach to the characteristic function and the core in games with externalities[END_REF] the payoff (profit) functions are not necessarily identical. Generally speaking, for any TU-game we have showed that the 1-concavity property of its dual game is a necessary and sufficient condition under which the core of the initial TU-game coincides with the set of imputations and is non-empty. Particularly, the nucleolus of such a 1-concave dual game agrees with the center of gravity of the set of imputations. The dual game of a Stackelberg oligopoly TU-game is of great interest since it describes the marginal contribution of followers to join the grand coalition by turning leaders. Based on the description of the γ-characteristic function in a Stackelberg oligopoly TU-game, the aim was to establish the 1-concavity property of its dual game. To this end, we have first proved that the core always coincides with the set of imputations. Indeed, the first-mover advantage gives too much power to singletons so that the worth of every deviating coalition is less than or equal to the sum of its members' individual worths except for the grand coalition. Then, we have provided a necessary and sufficient condition under which the core is non-empty. Finally, we have proved that this condition depends on the heterogeneity of firms' marginal costs, i.e., the dual game is 1-concave if and only if firms' marginal costs are not too heterogeneous. The more the number of firms, the less the heterogeneity of firms' marginal costs must be in order to ensure the 1-concavity property of the dual game. This last result extends Marini and Currarini's core non-emptiness result (2003) for oligopoly situations and shows that their result crucially depends on their symmetric players assumption. Surprisingly, in case the inverse demand function is strictly concave, we have provided an example in which the opposite result holds, i.e., when the heterogeneity of firms' marginal costs increases the core becomes larger. Instead of quantity competition, we can associate a two-stage structure with the γ-characteristic function in a price competition setting. Marini and Currarini's core non-emptiness result (2003) applies to this setting and they provide examples in which the core of the sequential Bertrand oligopoly TU-games is included in the core of the classical Bertrand oligopoly TU-games. A question concerns the effect of the heterogeneity of firms' marginal costs on the non-emptiness of the core. This is left for future work.

  The worths of any coalition S ∈ 2 N \{∅} are given in the following table:

	S	{1} {2} {1, 2}
	v 1 γ (S) 4.70 4.70 8.71
	v 2 γ (S) 6.19 2.02 8.71
	v 3 γ (S) 7.05 1.05 8.71

We refer to[START_REF] Norde | Oligopoly games with and without transferable technologies[END_REF] for a detailed discussion of this distinction.

Zhao shows that the core is non-empty for general TU-games in β-characteristic function form in which every strategy set is compact and convex, every utility function is continuous and concave, and satisfying the strong separability condition that requires that the utility function of a coalition and each of its members' utility functions have the same minimizers. Zhao proves that Cournot oligopoly TU-games satisfy this latter condition.

We refer to[START_REF] Lardon | The γ-core of cournot oligopoly games with capacity constraints[END_REF] for a precise description of these axioms.

Interval games are introduced by[START_REF] Branzei | Shapley-like values for interval bankruptcy games[END_REF].

Every player's payoff function is assumed twice differentiable and strictly concave on its own strategy set.

The players are symmetric if they have identical payoff functions and strategy sets. Externalities are symmetric if they are either positive or negative.