Francesca Busetto

Giulio Codognato

Sayantan Ghosal

Ludovic Julien

Simone Tonin

Existence and Optimality of Cournot-Nash Equilibria in a Bilateral Oligopoly with Atoms and an Atomless Part

Keywords:

, that a Cournot-Nash allocation is Pareto optimal if and only if it is a Walras allocation.

1 Introduction [START_REF] Gabszewicz | Oligopoly equilibrium in exchange economies[END_REF] introduced the so-called model of bilateral oligopoly, which consists of a two-commodity exchange economy where each 1 trader holds only one of the two commodities available for trade. In this framework, strategic interaction among traders was modeled as in strategic market games à la Shapley and Shubik (see [START_REF] Elgar | Strategic market games: an introduction[END_REF] for a survey of this literature). This model was analyzed, in the case of a finite number of traders, by [START_REF] Bloch | Stable trading structures in bilateral oligopolies[END_REF], [START_REF] Bloch | Trade fragmentation and coordination in strategic market games[END_REF], [START_REF] Dickson | The strategic Marshallian cross[END_REF], [START_REF] Amir | Comparative statics in a simple class of strategic market games[END_REF], among others.

In this paper, we consider the mixed bilateral oligopoly model introduced by Codognato et al. (2015). Following [START_REF] Shitovitz | Oligopoly in markets with a continuum of traders[END_REF], this model analyzes a mixed economy with large traders, represented as atoms, and small traders, represented by an atomless part. Noncooperative exchange is formalized as in the Shapley window model, a strategic market game which was first proposed informally by Lloys S. Shapley and further analyzed by [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF], [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF], [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders[END_REF], [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF], among others.

The first goal of the paper is to prove the existence of a Cournot-Nash equilibrium for the mixed bilateral oligopoly version of the Shapley window model. [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders[END_REF] provided an existence proof for the mixed version of the Shapley window model with any finite number of commodities. Their proof is based on the same assumptions as the proof provided by [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF] for the case of exchange economies with a finite number of traders. In particular, it requires that there are at least two atoms with strictly positive endowments, continuously differentiable utility functions, and indifference curves contained in the strict interior of the commodity space: these restrictions are stated by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders[END_REF] in their Assumption 4. Clearly, this proof does not apply to the bilateral oligopoly case where all atoms hold only one of the two commodities. [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF] proposed an alternative existence proof which is essentially based on restrictions on endowments and preference of the atomless part of the economy rather than on atoms. In particular, they kept all the assumptions made by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders[END_REF] with the exception of their Assumption 4, which was replaced by a new restriction requiring that the set of commodities is strongly connected through the characteristics of traders in the atomless part. The existence proof in Busetto et al. (2017) used a theorem which shows that any sequence of prices corresponding to a sequence of Cournot-Nash equilibria has a subsequence which converges to a strictly positive price vector and it requires that each commodity is held by a subset of the atomless part with positive measure. An appealing feature of our existence result is that it allows a commodity to be held only by atoms. In order to cover this case, we cannot directly use the price convergence theorem shown by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF] but we have to combine that proof with the price convergence result proved by [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF], which holds for a strategic market game with a finite number of traders, i.e., for a purely atomic exchange economy. This is one reason why our existence proof is not just a two-commodity case of that provided by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF]. There is one more difference between the two results. A step in the proof of both existence theorems consists in showing that the aggregate bid matrix, obtained as the limit of a sequence of perturbed Cournot-Nash equilibria, is irreducible. In [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF], obtaining this result requires that the two ordered pairs generated by the two traded commodities are connected through traders' characteristics. Here, instead, we require that only one of the two ordered pairs be so connected.

In the second main theorem of the paper, we provide a characterization of the Pareto optimality of Cournot-Nash allocations. The issue of Pareto optimality in strategic market games was raised since the seminal paper by [START_REF] Shapley | Trade using one commodity as a means of payment[END_REF]. This first analysis was mainly an intuitive discussion of Pareto optimality in the Edgeworth box. Then, more formal results on this issue were obtained by [START_REF] Dubey | Nash equilibria of market games: finiteness and inefficiency[END_REF], [START_REF] Dubey | Efficiency properties of strategic market games: an axiomatic approach[END_REF], [START_REF] Aghion | On the generic inefficiency of differentiable market games[END_REF], [START_REF] Dubey | Inefficiency of smooth market mechanisms[END_REF], among others. These results were obtained using the approach to general equilibrium based on differential topology and hold generically. Our second theorem is a general result obtained adding, to the same assumptions of the existence theorem, the further restriction that both pairs of commodities are connected through the characteristics of the same subset of traders. It states that a Cournot-Nash allocation is Pareto optimal if and only if it is a Walras allocation, thereby establishing a relationship among the Cournotian tradition of oligopoly, the Walrasian tradition of perfect competition, and the Paretian analysis of optimality. Some examples computed by [START_REF] Codognato | Atomic Cournotian traders may be Walrasian[END_REF] provide evidence that this characterization holds non-vacuously.

The paper is organized as follows. In Section 2, we introduce the mathematical model. In section 3, we prove the existence of a Cournot-Nash equilibrium. In Section 4, we characterize the Pareto optimality of Cournot-Nash equilibria. In Section 5, we discuss the model. In Section 6, we draw some conclusions and we sketch some further lines of research.

Mathematical model

We consider a pure exchange economy with large traders, represented as atoms, and small traders, represented by an atomless part. The space of traders is denoted by the measure space (T, T , µ), where T is the set of traders, T is the σ-algebra of all µ-measurable subsets of T , and µ is a real valued, non-negative, countably additive measure defined on T . We assume that (T, T , µ) is finite, i.e., µ(T ) < ∞. This implies that the measure space (T, T , µ) contains at most countably many atoms. Let T 1 denote the set of atoms and T 0 the atomless part of T . We assume that µ(T 1 ) > 0 and µ(T 0 ) > 0.1 A null set of traders is a set of measure 0. Null sets of traders are systematically ignored throughout the paper. Thus, a statement asserted for "each" trader in a certain set is to be understood to hold for all such traders except possibly for a null set of traders. A coalition is a nonnull element of T . The word "integrable" is to be understood in the sense of Lebesgue.

In the exchange economy, there are 2 different commodities. A commodity bundle is a point in R2

+ . An assignment (of commodity bundles to traders) is an integrable function x: T → R 2 + . There is a fixed initial assignment w, satisfying the following assumption.

Assumption 1. There is a coalition S such that µ(S ∩ T ) > 0, µ(S c ∩ T ) > 0, w 1 (t) > 0, w 2 (t) = 0, for each t ∈ S, w 1 (t) = 0, w 2 (t) > 0, for each t ∈ S c . Moreover, card(S ∩ T 1 ) ≥ 2, whenever µ(S ∩ T 0 ) = 0, and card(S c ∩ T 1 ) ≥ 2, whenever µ(S c ∩ T 0 ) = 0. 2 An allocation is an assignment x such that T x(t) dµ = T w(t) dµ. The preferences of each trader t ∈ T are described by a utility function u t : R 2 + → R, satisfying the following assumptions. Assumption 2. u t : R 2 + → R is continuous, strongly monotone, and quasiconcave, for each t ∈ T .

Let B denote the Borel σ-algebra of R 2 + . Moreover, let T B denote the σ-algebra generated by the sets E × F such that E ∈ T and F ∈ B.

Assumption 3. u : T × R 2 + → R, given by u(t, x) = u t (x), for each t ∈ T and for each x ∈ R 2 + , is T B-measurable.
In order to state our last assumption, we need a preliminary definition. We say that commodities i, j stand in relation Q if there is a coalition T i such that T i ⊂ {t ∈ T 0 : w i (t) > 0, w j (t) = 0}, u t (•) is differentiable, additively separable, i.e., u t (x) = v i t (x i ) + v j t (x j ), for each x ∈ R 2 + , and

dv j t (0)
dx j = +∞, for each t ∈ T i . 3 We can therefore state our last assumption as follows.

Assumption 4. Commodities 1 and 2 or commodities 2 and 1 stand in relation Q.

A price vector is a nonnull vector p ∈ R 2 + . We say that a price vector p

is normalized if p ∈ ∆ where ∆ = {p ∈ R 2 + : 2 i=1 p i = 1}. Let X 0 : T 0 × R 2 ++ → P(R 2 +
) be a correspondence such that, for each t ∈ T 0 and for each p ∈ R 2 ++ , X 0 (t, p) = argmax{u t (x) : x ∈ R 2 + and px ≤ pw(t)}. For each p ∈ R 2 ++ , let T 0 X 0 (t, p) dµ = { T 0 x 0 (t, p) dµ : x 0 (•, p) is integrable and x 0 (t, p) ∈ X 0 (t, p), for each t ∈ T 0 }. Finally, let Z 0 : R 2 ++ → P(R 2 ) be a correspondence which associates with each p ∈ R 2 ++ the Minkowski difference between the set T 0 X 0 (t, p) dµ and the set { T 0 w(t) dµ}. 4 According to [START_REF] Debreu | Existence of competitive equilibrium[END_REF]

, let |x| = 2 i=1 |x i |, for each x ∈ R 2 , and let d[0, V ] = inf x∈V |x|, for each V ⊂ R 2 .
The next proposition, which we shall use in the proof of the existence theorem in Section 3, is based on Property (iv) in [START_REF] Debreu | Existence of competitive equilibrium[END_REF], p. 728. Proposition 1. Under Assumptions 1, 2, and 3, let {p n } be a sequence of normalized price vectors such that p n 0, for each n = 1, 2, . . ., which converges to a normalized price vector p. Then, p1 = 0 and µ(S c ∩ T 0 ) > 0, or, p2 = 0 and µ(S ∩T 0 ) > 0, imply that the sequence {d[0, Z 0 (p n )]} diverges to +∞.

Proof. Let {p n } be a sequence of normalized price vectors such that p n 0, for each n = 1, 2, . . ., which converges to a normalized price vector p. Suppose that p1 = 0 and µ(S c ∩ T 0 ) > 0. Then, we have that p2 = 1. But then, the sequence {d[0, X 0 (t, p n )]} diverges to +∞ as p2 w 2 (t) > 0, for each t ∈ S c ∩ T 0 , by Lemma 4 in [START_REF] Debreu | Existence of competitive equilibrium[END_REF], p. 721. Therefore, {d[0, Z 0 (p n )]} diverges to +∞ as µ(S c ∩ T 0 ) > 0, by the argument used in the proof of Property (iv) in [START_REF] Debreu | Existence of competitive equilibrium[END_REF], p. 728. Suppose that p2 = 0 and µ(S ∩ T 0 ) > 0. Then, {d[0, Z 0 (p n )]} diverges to +∞, by using symmetrically the previous argument. Hence, p1 = 0 and µ(S c ∩ T 0 ) > 0, or, p2 = 0 and µ(S ∩ T 0 ) > 0, imply that the sequence {d[0, Z 0 (p n )]} diverges to +∞.

A Walras equilibrium is a pair (p, x), consisting of a price vector p and an allocation x, such that px(t) = pw(t) and u t (x(t)) ≥ u t (y), for all y ∈ {x ∈ R 2 + : px = pw(t)}, for each t ∈ T . A Walras allocation is an allocation x for which there exists a price vector p such that the pair (p, x) is a Walras equilibrium.

Borrowing from [START_REF] Codognato | Atomic Cournotian traders may be Walrasian[END_REF], we introduce now the two-commodity version of the Shapley window model. A strategy correspondence is a correspondence B :

T → P(R 4 + ) such that, for each t ∈ T , B(t) = {(b ij ) ∈ R 4 + : 2 j=1 b ij ≤ w i (t), i = 1, 2}.
With some abuse of notation, we denote by b(t) ∈ B(t) a strategy of trader t, where b ij (t), i, j = 1, 2, represents the amount of commodity i that trader t offers in exchange for commodity j. Consider the following three further definitions (see [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF]).

A strategy selection is an integrable function

b : T → R 4 + , such that, for each t ∈ T , b(t) ∈ B(t).
Definition 1. A nonnegative square matrix A is said to be irreducible if, for every pair (i, j), with i = j, there is a positive integer k such that a

(k) ij > 0, where a (k) ij denotes the ij-th entry of the k-th power A k of A.
Definition 2. A nonnegative square matrix A is said to be completely reducible if, after a permutation of indices, it can be written in a blockdiagonal form such that each diagonal block is irreducible. Definition 3. Given a strategy selection b, a price vector p is said to be market clearing if

p ∈ R 2 ++ , 2 i=1 p i bij = p j ( 2 i=1 bji ), j = 1, 2. (1)
By Lemma 1 in Sahi and Yao (1989), there is a unique, up to a scalar multiple, price vector p satisfying (1) if and only if B is irreducible. Then, we denote by p(b) a function which associates with each strategy selection b the unique, up to a scalar multiple, price vector p satisfying (1), if B is irreducible, and is equal to 0, otherwise. For each strategy selection b such that p(b) 0, we assume that the price vector p(b) is normalized. Given a strategy selection b and a price vector p, consider the assignment determined as follows:

x j (t, b(t), p) = w j (t) - 2 i=1 b ji (t) + 2 i=1 b ij (t) p i p j , if p ∈ R 2 ++ , x j (t, b(t), p) = w j (t), otherwise, j = 1, 2, for each t ∈ T .
Given a strategy selection b and the function p(b), traders' final holdings are determined according to this rule and consequently expressed by the assignment

x(t) = x(t, b(t), p(b)),
for each t ∈ T . 5 It is straightforward to show that this assignment is an allocation.

We are now able to define a notion of Cournot-Nash equilibrium for this reformulation of the Shapley window model (see [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF] and Busetto et al. ( 2011)).

Definition 4. A strategy selection b such that B is irreducible is a Cournot- Nash equilibrium if u t (x(t, b(t), p( b))) ≥ u t (x(t, b(t), p( b \ b(t)))),
for each b(t) ∈ B(t) and for each t ∈ T .

A Cournot-Nash allocation is an allocation x such that x(t) = x(t, b(t), p( b)), for each t ∈ T , where b is a Cournot-Nash equilibrium.

Finally, we introduce the same type of perturbation of the strategic market game which was used by [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF] and [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders[END_REF] to prove their existence theorems. Given > 0 and a strategy selection b, we define the aggregate matrix B as the matrix such that b ij = ( bij + ), i, j = 1, 2. Clearly, the matrix B is irreducible. The interpretation is that an outside agency places fixed bids of for each pair of commodities 1 and 2. Given > 0, we denote by p (b) the function which associates, with each strategy selection b, the unique, up to a scalar multiple, price vector which satisfies

2 i=1 p i ( bij + ) = p j ( 2 i=1 ( bji + ), j = 1, 2.
(

For each strategy selection b, we assume that the price vector p (b) is normalized.

Definition 5. Given > 0, a strategy selection b is an -Cournot-Nash equilibrium if

u t (x(t, b (t), p ( b ))) ≥ u t (x(t, b(t), p ( b \ b(t)))),
for each b(t) ∈ B(t) and for each t ∈ T .

Existence

In this section, we state and prove our first main theorem, establishing the existence of a Cournot-Nash equilibrium for the two-commodity version of the Shapley window model presented above. Then, we have to show that there exists the limit of a sequence of -Cournot-Nash equilibria and that this limit is a Cournot-Nash equilibrium. Let n = 1 n , n = 1, 2, . . .. By Lemma 1, for each n = 1, 2, . . ., there is an -Cournot-Nash equilibrium b n . The next step consists in showing that any sequence of normalized prices generated by the sequence of -Cournot-Nash equilibria corresponding to the sequence { n } has a convergent subsequence whose limit is a strictly positive normalized price vector. In order to prove this result, we cannot use the price convergence theorem proved by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF] as the proof of this theorem requires that each commodity is held by a subset of the atomless part with positive measure whereas, in our framework, one of the two commodities may be held only by atoms. Therefore, we have to combine the price convergence proof provided by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF] with another price convergence proof, proposed by [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF], which holds for a purely atomic exchange economy. To this end, we need to introduce the following preliminary lemma, based on the uniform monotonicity lemma proved by [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF].

Lemma 2. Consider an atom τ ∈ T 1 and a commodity j ∈ {1, 2}. For each real number H > 0, there is a real number 0 < h(u τ (•), j, H) < 1, depending on u τ (•), j, and

H, such that if x ∈ R 2 + , x ≤ H, y ∈ R 2 + and y -x ≤ h(u τ (•), j, H), then u τ (y + e j ) > u τ (x). 6
Proof. It is an immediate consequence of Lemma C (the uniform monotonicity lemma) in [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF] as u τ (•) is continuous and strongly monotone, by Assumption 2.

We can now state and prove the price convergence lemma. Lemma 3. Let {p n } be a sequence of normalized prices such that {p n } = p( b n ) where b n is an -Cournot-Nash equilibrium, for each n = 1, 2, . . .. Then, there exists a subsequence {p kn } of the sequence {p n } which converges to a normalized price vector p 0.

Proof. Let {p n } be a sequence of normalized prices such that {p n } = p n ( b n ), where b n is an -Cournot-Nash equilibrium, for each n = 1, 2, . . .. Then, there is a subsequence {p kn } of the sequence {p n } which converges to a price vector p ∈ ∆, as the unit simplex ∆ is a compact set. Consider the case where µ(S ∩ T 0 ) > 0 and µ(S c ∩ T 0 ) > 0. Suppose, without loss of generality, that p1 = 0. Then, the sequence {d[0, Z 0 (p kn )]} diverges to +∞ as µ(S c ∩ T 0 ) > 0, by Proposition 1. We adapt now to our framework the argument used by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF] to prove their Theorem 1. Let x n (t) = x(t, b n (t), p n ( b n )), for each t ∈ T , and for each n = 1, 2, . . .. Then, x n (t) ∈ X 0 (t, p n ), for each t ∈ T 0 , and for each n = 1, 2, . . ., by the same argument used by [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF] to prove their Theorem 2. But then, ( T 0 x n (t) dµ -T 0 w(t) dµ) ∈ Z 0 (p n ), for each n = 1, 2, . . .. We have that

T 0 x n (t) dµ ≤ T 0 w(t) dµ + T 1 w(t) dµ + e 1 + e 2
as T x n (t) dµ ≤ T w(t) dµ + n e 1 + n e 2 , for each n = 1, 2, . . .. Then,

T 0 xi n (t) dµ - T 0 w i (t) dµ ≤ T 0 w i (t) dµ + T 1 w i (t) dµ + 1 as -T 1 w i (t) dµ -1 ≤ T 0 xi n (t) dµ ≤ 2 T 0 w i (t) dµ + T 1 w i (t) dµ + 1, i = 1, 2, for each n = 1, 2, . . .. But then, 2 i=1 T 0 xi n (t) dµ - T 0 w i (t) dµ ≤ 2 i=1 T 0 w i (t) dµ + T 1 w i (t) dµ + 1 ,
for each n = 1, 2, . . .. Moreover, there exists an n 0 such that

d[0, Z 0 (p kn )] > 2 i=1 T 0 w i (t) dµ + T 1 w i (t) dµ + 1 ,
for each n ≥ n 0 , as the sequence {d[0, Z 0 (p kn )]} diverges to +∞. Then,

2 i=1 T 0 xi kn (t) dµ - T 0 w i (t) dµ > 2 i=1 T 0 w i (t) dµ + T 1 w i (t) dµ + 1 as 2 i=1 | T 0 xi kn (t) dµ -T 0 w i (t) dµ| ≥ d[0, Z 0 (p kn )],
for each n ≥ n 0 , a contradiction. Therefore, we must have that p1 > 0. Consider the case where µ(S ∩ T 0 ) = 0 or µ(S c ∩ T 0 ) = 0. Suppose, without loss of generality, that µ(S ∩ T 0 ) = 0. Then, we have that µ(S c ∩ T 0 ) > 0 as µ(T 0 ) > 0. Moreover, there are at least two atoms τ, ρ ∈ S ∩ T 1 , by Assumption 1. We have that p1 > 0, as µ(S c ∩ T 0 ) > 0, by the same argument used in the proof of the previous case. In order to prove that p2 > 0, we now show that there is a real number η > 0 such that

p2 n p1 n > η, (3)
for each n = 1, 2, . . .. To this end, we adapt to our framework the proof of Lemma 2 in [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF]. In what follows, we shall use the fact that (2) and the normalization rule imply straightforwardly that (3) holds if and only if b

n 12 + n b n 21 + n > η,
where the inequality is strict for the second component by [START_REF] Artstein | A note on Fatou's lemma in several dimensions[END_REF]. Let H = √ 2 max{ T w 1 (t) dµ+1, T w 2 (t) dµ+1} and let

y = x(τ, b n (τ ), p n ( b n ))+ z. It is straightforward to verify that x(τ, b n (τ ), p n ( b n )) ∈ R 2 + and x(τ, b n (τ ), p n ( b n )) ≤ H. Suppose that y ∈ R 2
+ and z ≤ h(u τ (•), 2, H). Then, by Lemma 2, we obtain that

u τ (x(τ, b n (τ ), p n ( b n )) + z + e 2 ) > u τ (x(τ, b n (τ ), p n ( b n ))).
But then, we have that

u τ (x(τ, b n (τ ), p n ( b n )) + 1 2 b n 21 + n b n 12 + n γ(z + e 2 )) ≥ u τ (x(τ, b n (τ ), p n ( b n ))), as 0 < 1 2 b n 21 + n b n 12 + n
γ < 1 and the function u τ (•) is quasi-concave, by Assumption 2. Therefore, it follows that

u τ (x(τ, b γ (τ ), p n ( b n \ b γ (τ )))) > u τ (x(τ, b n (τ ), p n ( b n ))), as (5)
holds strictly for its second component and u τ (•) is strongly monotone, by Assumption 2, a contradiction. Thus, it must be that y /

∈ R 2 + or z > h(u τ (•), 2, H). Suppose that y / ∈ R 2 + .
Then, it follows that

x 1 (τ, b n (τ ), p n ( b n )) -2 b n 12 + n b n 21 + n < 0, as y = x(τ, b n (τ ), p n ( b n )) + z and x(τ, b n (τ ), p n ( b n )) ∈ R 2 + . But then, it must be that b n 12 + n b n 21 + n > w 1 (τ ) 4 , as x 1 (τ, b n (τ ), p n ( b n )) = w 1 (τ ) -b n 12 (τ ) ≥ w 1 (τ ) 2 . Suppose that z > h(u τ (•), 2, H). Then, we have that b n 12 + n b n 21 + n > h(u τ (•), 2, H) 2 . Suppose now that w 1 (τ ) -b n 12 (τ ) < w 1 (τ ) 2 . Then, we have that b n 12 (τ ) > w 1 (τ )
2 . But then, it must be that b

n 12 + n b n 21 + n > w 1 (τ ) 2( T w 2 (t) dµ + 1)
.

of the set {(b ij ) ∈ R 4 + : b ij ≤ T w i (t) dµ, i, j = 1, 2} and a subsequence {p kn } of the sequence {p n } which converges to a price vector p ∈ ∆, with p 0, by Lemma 3. Since the sequence { b kn } satisfies the assumptions of Theorem A in [START_REF] Artstein | A note on Fatou's lemma in several dimensions[END_REF], by this theorem there is a function b such that b(t) is a limit point of the sequence { b kn (t)}, for each t ∈ T , and such that the sequence { B kn } converges to B. Moreover, p and B satisfy (1) as p kn and B kn kn satisfy (2), for each n = 1, 2, . . ., the sequence {p kn } converges to p, the sequence { B kn } converges to B, and the sequence { kn } converges to 0. Then, the matrix B is completely reducible, by Lemma 1 in Sahi and Yao (1989), as p 0. We want now to prove that B must be irreducible. Suppose, without loss of generality, that commodities 2 and 1 stand in the relation Q. We now show that b21 > 0. Suppose that b21 = 0. Then, we have that T 2 b21 (t) dµ = 0 as µ(T 2 ) > 0. Consider a trader τ ∈ T 2 . We can suppose that b21 (τ ) = 0 as we ignore null sets. Since b(τ ) is a limit point of the sequence { b kn (τ )}, there is a subsequence { b h kn (τ )} of this sequence which converges to b(τ ). Let x n (τ ) = x(τ, b n (τ ), p n ( b n )), for each n = 1, 2, . . ., and x(τ ) = x(τ, b(τ ), p). Then, the subsequence {x h kn (τ )} of the sequence {x n (τ )} converges to x(τ ) as the sequence { b h kn (τ )} converges to b(τ ) and the sequence {p h kn } converges to p, with p h kn 0, for each n = 1, 2, . . ., and p 0. But then, it must be that x1 (τ ) = 0 as b21 (τ ) = 0 and x(τ ) ∈ X 0 (τ, p) as x h kn (τ ) ∈ X 0 (τ, p h kn ), for each n = 1, 2, . . ., and the correspondence X 0 (τ, •) is upper hemicontinuous, by the argument used in [START_REF] Debreu | Existence of competitive equilibrium[END_REF], p. 721. Therefore, we have that ∂uτ (x(τ ))

Optimality

In this section, we study the Pareto optimality properties of a Cournot-Nash allocation for the two-commodity version of the Shaley window model. [START_REF] Shapley | Trade using one commodity as a means of payment[END_REF] first raised the question of the Pareto optimality features of Cournot-Nash allocations in the context of the prototypical strategic market games they proposed in that work. Nevertheless, their analysis was mainly based on examples drawn in an Edgeworth box. Then, some more general results about the Pareto optimality properties of Cournot-Nash allocations of strategic market games were obtained, both for exchange economies with a finite number of traders and with an atomless continuum of traders, by [START_REF] Dubey | Nash equilibria of market games: finiteness and inefficiency[END_REF], [START_REF] Dubey | Efficiency properties of strategic market games: an axiomatic approach[END_REF], [START_REF] Aghion | On the generic inefficiency of differentiable market games[END_REF], [START_REF] Dubey | Inefficiency of smooth market mechanisms[END_REF], among others. Their theorems were obtained in a framework of differential topology and hold generically. 7 Here, we extend the Pareto optimality analysis to our mixed version of the Shapley window model and we obtain a general result which characterizes Pareto optimal Cournot-Nash allocations as Walras allocations. To this end, we need to introduce the following further definitions. An allocation x is said to be individually rational if u t (x(t)) ≥ u t (w(t)), for each t ∈ T . An allocation x is said to be Pareto optimal if there is no allocation y such that u t (y(t)) > u t (x(t)), for each t ∈ T . An efficiency equilibrium is a pair (p, x), consisting of a price vector p and an allocation x, such that u t (x(t)) ≥ u t (y), for all y ∈ {x ∈ R 2 + : px = px(t)}, for each t ∈ T . Moreover, we need to impose the following assumption. Assumption 4 . There is a coalition T , with T ⊂ T 0 , such that u t (•) is differentiable, additively separable, and

dv j t (0) dx j = +∞, j = 1, 2, for each t ∈ T .
It is straightforward to verify that Assumption 4 is stricter than Assumption 4 as Assumption 4 implies Assumption 4 but the converse does not hold. This restriction is needed to guarantee that, at a Pareto optimal Cournot-Nash allocation, Cournot-Nash equilibrium prices are equal, up to a scalar multiple, to efficiency equilibrium prices.

We can now state and prove our optimality theorem, which establishes an equivalence between the set of Pareto optimal Cournot-Nash allocations and the set of Cournot-Nash allocations, whenever the latter are also Walrasian. Proof. Let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Suppose that x is Pareto optimal. We adapt to our framework the argument used by [START_REF] Shitovitz | Oligopoly in markets with a continuum of traders[END_REF] to prove the corollary to his Lemma 2. It is straightforward to verify that x is individually rational. Let Ĝ → P(R 2 ) be a correspondence such that Ĝ(t) = {x -x(t) :

x ∈ R 2 + and u t (x) > u t (x(t))}, for each t ∈ T . Moreover, let T Ĝ(t) dµ = { T ĝ(t) dµ : ĝ(t) is integrable and ĝ(t) ∈ Ĝ(t), for each t ∈ T }. The set {x ∈ R 2 + : u t (x) ≥ u t (x)} is convex as u t (•) is quasi-concave, by Assumption 2, for each t ∈ T 1 .
Then, it is straightforward to verify that the set Ĝ(t) is convex, for each t ∈ T 1 . But then, T Ĝ(t) dµ is convex, by Theorem 1 in [START_REF] Shitovitz | Oligopoly in markets with a continuum of traders[END_REF]. We now prove that 0 / ∈ T Ĝ(t) dµ. Suppose that 0 ∈ T Ĝ(t) dµ. Then, there is an assignment y such that u t (y(t)) > u t (x(t)), for each t ∈ T , which is an allocation as T y(t) dµ = T x(t) dµ = T w(t) dµ. But then, x is not Pareto optimal, a contradiction. Therefore, it must be that 0 / ∈ T Ĝ(t) dµ. Then, there exists a vector p such that p ∈ R 2 , (p = 0), and p T Ĝ(t) dµ ≥ 0, by the supporting hyperplane theorem. But then, the pair (p, x) is an efficiency equilibrium, by Lemma 2 in [START_REF] Shitovitz | Oligopoly in markets with a continuum of traders[END_REF]. We have that x(t) ∈ X 0 (t, p), by the same argument used by [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF] to prove their Theorem 2, for each t ∈ T 0 . Consider a trader τ ∈ T and suppose that either x1 (t) = 0 or x2 (t) = 0. Then, the necessary Kuhn-Tucker conditions lead, mutatis mutandis, to the same contradiction as in the proof of our Theorem 1, by Assumption 4 . But then, we have that x(t) 0. Therefore, it must be that

∂ut(x(t)) ∂x 1 ∂ut(x(t)) ∂x 2 = p1 p2 ,
for each t ∈ T . It must also be that

∂ut(x(t)) ∂x 1 ∂ut(x(t)) ∂x 2 = p1 p2 ,
as the pair (p, x) is an efficiency equilibrium, for each t ∈ T . Then, there exists a real number θ > 0 such that p1 = θ p1 and p2 = θ p2 . But then, x is such that px(t) = pw(t) and u t (x(t)) ≥ u t (y), for all y ∈ {x ∈ R 2 + : px = pw(t)}, for each t ∈ T . Therefore, the pair (p, x) is a Walras equilibrium. Suppose now that the pair (p, x) is a Walras equilibrium. Then, x is Pareto optimal, by the first fundamental theorem of welfare economics. Hence, x is Pareto optimal if and only if the pair (p, x) is a Walras equilibrium.

We study now the relationship between the set of Cournot-Nash allocations, the core, and the set of Walras allocations.

We say that an allocation y dominates an allocation x via a coalition S if u t (y(t)) > u t (x(t)), for each t ∈ S, and S y(t) dµ = S w(t) dµ. The core is the set of all allocations which are not dominated via any coalition. The following corollary is a straightforward consequence of Theorem 2.

Corollary 1. Under Assumptions 1, 2, 3, and 4 , let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Then, x is in the core if and only if the pair (p, x) is a Walras equilibrium.

Proof. Let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Suppose that x is in the core. Then, x is Pareto optimal. But then, the pair (p, x) is a Walras equilibrium, by Theorem 2. Suppose that the pair (p, x) is a Walras equilibrium. Then, x is in the core, by the same argument used by [START_REF] Aumann | Markets with a continuum of traders[END_REF] in the proof of his main theorem. Hence, x is in the core if and only if the pair (p, x) is a Walras equilibrium.

The next proposition provides a characterization of Pareto optimal Cournot-Nash allocations. To prove it, we use a result obtained in [START_REF] Codognato | Atomic Cournotian traders may be Walrasian[END_REF] which provides a necessary and sufficient condition for a Cournot-Nash allocation to be a Walras allocation. This characterization result requires the following assumption. Assumption 5. u t : R 2 + → R is differentiable, for each t ∈ T 1 . Our characterization of Pareto Optimal Cournot-Nash allocations is the following.

Proposition 2. Under Assumptions 1, 2, 3, 4 , and 5, let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Then, x is Pareto optimal if and only if x1 (t) = 0 or x2 (t) = 0, for each t ∈ T 1 .

Proof. Let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Suppose that x is Pareto optimal. Then, the pair (p, x) is a Walras equilibrium, by Theorem 2. But then, x1 (t) = 0 or x2 (t) = 0, for each t ∈ T 1 , by Theorem 4 in [START_REF] Codognato | Atomic Cournotian traders may be Walrasian[END_REF]. Suppose that x1 (t) = 0 or x2 (t) = 0, for each t ∈ T 1 . Then, the pair (p, x) is a Walras equilibrium, by Theorem 4 in [START_REF] Codognato | Atomic Cournotian traders may be Walrasian[END_REF]. But then, x is Pareto optimal, by Theorem 2. Hence, x is Pareto optimal if and only if x1 (t) = 0 or x2 (t) = 0, for each t ∈ T 1 .

The following corollary provides a characterization of Cournot-Nash allocations which are in the core.

Discussion of the model

This section is devoted to a discussion of some issues related to the existence and optimality of Cournot-Nash equilibria.

It is straightforward to show, using Theorem 5 in Codognato et al. (2015), that, in our mixed bilateral oligopoly framework, under Assumptions 1, 2, and 3, the set of Cournot-Nash allocations of the Shapley window model coincides with the set of the Cournot-Nash allocations of the model with commodity money proposed by [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF] and of its generalization to complete markets proposed by [START_REF] Amir | A strategic market game with complete markets[END_REF]. Therefore, all the results obtained in this paper also hold for these models. Let us further discuss now some features of the models in this class, when considered in a bilateral oligopoly framework, and their relationships with the results obtained in this paper. To be applied to our bilateral framework, Assumption 1 in Busetto et al. ( 2017) could be restated as follows.

Assumption 1 . There is a coalition S such that µ(S ∩T ) > 0, µ(S c ∩T ) > 0, w 1 (t) > 0, w 2 (t) = 0, for each t ∈ S, w 1 (t) = 0, w 2 (t) > 0, for each t ∈ S c . Moreover, T 0 w(t) dµ 0.

It is clear that if an initial assignment w satisfies Assumption 1 , then it also satisfies our Assumption 1. Assumption 1 rules out the cases where µ(S∩T 0 ) = 0 or µ(S c ∩T 0 ) = 0, that is those in which only the atoms hold one of the two commodities. Therefore, the price convergence theorem proved by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF], their Theorem 1, that holds only under Assumption 1 , cannot be used in the proof of our existence theorem. In order to cover the case where one of the two commodities is held only by atoms, we have proved a price convergence lemma which combines the argument used by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF] in the proof of their Theorem 1 with another argument used by [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF] in the proof of their convergence result for the purely atomic case, their Lemma 2. Moreover, we had to extend the proof of [START_REF] Dubey | The noncooperative equilibria of a closed trading economy with market supply and bidding strategies[END_REF], which holds under the assumption of concave utility functions, to cover the case of quasi-concave utility functions, as required by our Assumption 2.

Assumption 4 in Busetto et al. (2017) imposes a relation between commodities, their relation C, which is equivalent, in our bilateral model, to our relation Q. It requires that the set of commodities is strongly connected through traders' characteristics and it could be restated, in the bilateral framework, as follows.

Assumption 4 . Commodities 1 and 2 and commodities 2 and 1 stand in relation Q.

It is clear that Assumption 4 implies our Assumption 4 but the converse does not hold. Therefore, in order to show the irreducibility of the matrix B, in the proof of Theorem 1, bilateral oligopoly allows us to impose condition Q only for one of the two ordered pairs generated by commodities 1 and 2.

Therefore, we can conclude that our existence theorem holds under less restrictive conditions than the existence theorem in [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF].

Let us consider a further point. So far we have kept the assumption that µ(T 1 ) > 0 and µ(T 0 ) > 0. Suppose first that µ(T 1 ) = 0. We obtain this way an atomless bilateral framework. The next proposition shows the existence of a Cournot-Nash equilibrium in this context by adapting to this case the equivalence result proved by [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF]. Proof. Under Assumptions 1, 2, 3, and 4, there exists a Walras equilibrium (p, x), by Theorem 9 in [START_REF] Debreu | Existence of competitive equilibrium[END_REF], which is a generalization of an existence theorem for exchange economies with an atomless continuum of traders proved by [START_REF] Aumann | Existence of competitive equilibria in markets with a continuum of traders[END_REF]. Then, there exists a strategy selection b such that p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T , which is a Cournot-Nash equilibrium, by the argument used by [START_REF] Codognato | Cournot-Nash equilibria in limit exchange economies with complete markets and consistent prices[END_REF] to prove their Theorem 2, replacing their Assumption 4, which requires that the set of commodities is a net, with our Assumption 4.

Suppose now that µ(T 0 ) = 0. We obtain this way a purely atomic bilateral framework. Then, it is possible to adapt to this context the existence result proved by [START_REF] Bloch | Trade fragmentation and coordination in strategic market games[END_REF]. To this end, we have to introduce the following restrictions of our Assumptions 2 and 5, respectively. Assumption 2 . u t : R 2 + → R is continuous, strongly monotone, and strictly concave, for each t ∈ T 1 .

Assumption 5 . u t (•) is differentiable, additively separable, and

dv j t (0) dx j = +∞, j = 1, 2, for each t ∈ T 1 .
We can then state the existence result proved by [START_REF] Bloch | Trade fragmentation and coordination in strategic market games[END_REF]. Proof. We have that card(S ∩ T 1 ) ≥ 2 and card(S c ∩ T 1 ) ≥ 2 as µ(T 0 ) = 0, by Assumption 1. Then, there exists a Cournot-Nash equilibrium b, by the same argument used by [START_REF] Bloch | Trade fragmentation and coordination in strategic market games[END_REF] to prove their Proposition 2, which can also be extended to the case where T 1 contains countably infinite atoms by means of the product topology.

It could be worth investigating if Proposition 4 holds under weaker assumptions. 8We have already noticed, in Section 4, that Assumption 4 is more restrictive than Assumption 4. We leave for further research a possible answer to the question whether Theorem 2 holds under less restrictive or alternative assumptions.

There is a relationship also between our work and the paper by [START_REF] Dubey | Inefficiency of smooth market mechanisms[END_REF]. These authors showed that, for strategic market games with a finite number of traders, Cournot-Nash equilibria are generically not Pareto optimal in utility functions. In future work, we could verify if this result would hold in our bilateral oligopoly framework as it would imply, by Theorem 2, that the "probability" that a Cournot-Nash allocation is a Walras allocation is null.

Finally, let us notice that, in Section 2, we have provided a definition of a Cournot-Nash equilibrium referring explicitly to irreducible matrices. This definition applies only to active Cournot-Nash equilibria according to the definition of [START_REF] Sahi | The noncooperative equilibria of a trading economy with complete markets and consistent prices[END_REF]. Nevertheless, in the Shapley window model, as in all other strategic market games, the strategy selection b such that b(t) = 0, for each t ∈ T , is a Cournot-Nash equilibrium, usually called trivial equilibrium. This raises the question whether, under Assumptions 1-4, the allocation corresponding to the trivial Cournot-Nash equilibrium, namely the initial assignment w, may be Pareto optimal. The following proposition provides a negative answer to this question. Proposition 5. Under Assumptions 1, 2, 3, and 4, the allocation w is not Pareto optimal.

Proof. Suppose that w is Pareto optimal. Then, there exists a price vector p such that the pair (p, w) is an efficiency equilibrium, by the same argument used in the proof of Theorem 2. But then, the pair (p, w) is a Walras equilibrium. Therefore, for commodities which stand in the relation Q, the necessary Kuhn-Tucker conditions lead to the same contradiction as in the proof of Theorem 1. Hence, the allocation w is not Pareto optimal.

Conclusion

In Theorem 1, we have shown the existence of a Cournot-Nash equilibrium for the mixed bilateral oligopoly version of the Shapley window model first analyzed by [START_REF] Codognato | Atomic Cournotian traders may be Walrasian[END_REF]. Then, in Theorem 2, we have proved that a Cournot-Nash allocation is Pareto optimal if and only if it is a Walras allocation. The proof of this theorem is crucially based on a corollary in [START_REF] Shitovitz | Oligopoly in markets with a continuum of traders[END_REF], showing that the first and second welfare theorem still hold in mixed exchange economies. In their main theorem, [START_REF] Codognato | Atomic Cournotian traders may be Walrasian[END_REF] proved that, under a further differentiability assumption on atoms' utility functions, the condition which characterizes the nonempty intersection of the sets of Walras and Cournot-Nash allocations requires that each atom demands a null amount of one commodity. Combining this result with our Theorem 2 we have obtained, as a proposition, a characterization of the optimality property of Cournot-Nash equilibria, which requires that at a Pareto optimal Cournot-Nash equilibrium each atom demands a null amount of one commodity. Recasting antitrust analysis in the bilateral oligopoly framework, we could use these results in further research as a first step to analyze competition policy in a general equilibrium framework.

In the previous section, we have already stressed that the results we have obtained for the bilateral version of the Shapley window model also hold for other prototypes of strategic market games in the line inspired by [START_REF] Shapley | Trade using one commodity as a means of payment[END_REF]. Moreover, we have pointed out some issues connected with existence and Pareto optimality which in our opinion deserve to be considered for future research. A further question we propose to answer in forthcoming work is if the results obtained in this paper hold for another type of strategic market game, that is the one with fiat money first introduced by [START_REF] Postlewaite | Approximate efficiency of non-Walrasian Nash equilibrium[END_REF] and further analyzed by [START_REF] Peck | The market game: existence and structure of equilibrium[END_REF], [START_REF] Koutsougeras | A three way equivalence[END_REF], [START_REF] Koutsougeras | Convergence of strategic behavior to price taking[END_REF], among others.

  Given a strategy selection b, we call aggregate matrix the matrix B such that bij = ( T b ij (t) dµ), i, j = 1, 2. Moreover, we denote by b \ b(t) the strategy selection obtained from b by replacing b(t) with b(t) ∈ B(t) and by B \ b(t) the corresponding aggregate matrix.

Theorem 1 .

 1 Under Assumptions 1, 2, 3, and 4, there exists a Cournot-Nash equilibrium b. Proof. The first step in the proof of Theorem 1 requires that we show the existence of an -Cournot-Nash equilibrium. To this end, we use a result already proved by Busetto et al. (2011) by applying the Kakutani-Fan-Glicksberg theorem. It is stated in the following lemma. Lemma 1. For each > 0, there exists an -Cournot-Nash equilibrium b . Proof. See the proof of Lemma 3 in Busetto et al. (2011).

∂x 1 = 1 ≤ 2 = 2 =

 1122 +∞ as 2 and 1 stand in the relation Q and ∂uτ (x(τ )) ∂x λp1 , by the necessary conditions of the Kuhn-Tucker theorem. Moreover, it must be that x2 (τ ) = w 2 (τ ) > 0 as u τ (•) is strongly monotone, by Assumption 2, and pw(τ ) > 0. Then, ∂uτ (x(τ )) ∂x λp 2 , by the necessary conditions of the Kuhn-Tucker theorem.But then, ∂uτ (x(τ ))∂x +∞ as λ = +∞, contradicting the assumption that u τ (•) is continuously differentiable. Therefore, we can conclude that b21 > 0.Then, we must also have that b12 > 0 as B is completely reducible. But then, B is irreducible. Consider a trader τ ∈ T 1 . The matrix B \ b(τ ) is irreducible as b21 \ b(τ ) > 0, by the previous argument. Consider a trader τ ∈ T 0 . The matrix B \ b(τ ) is irreducible as B = B \ b(τ ). Then, the matrix B \ b(t) is irreducible, for each t ∈ T . But then, from the same argument used by Busetto et al. (2011) in their existence proof (Cases 1 and 3), it follows that u t (x(t, b(t), p( b))) ≥ u t (x(t, b(t), p( b\b(t)))), for each b(t) ∈ B(t) and for each t ∈ T . Hence, b is a Cournot-Nash equilibrium.

Theorem 2 .

 2 Under Assumptions 1, 2, 3, and 4 , let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Then, x is Pareto optimal if and only if the pair (p, x) is a Walras equilibrium.

Corollary 2 .

 2 Under Assumptions 1, 2, 3, 4 , and 5, let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Then, x is the core if and only if x1 (t) = 0 or x2 (t) = 0, for each t ∈ T 1 . Proof. Let b be a Cournot-Nash equilibrium and let p = p( b) and x(t) = x(t, b(t), p( b)), for each t ∈ T . Suppose that x is in the core. Then, x is Pareto optimal. But then, x1 (t) = 0 or x2 (t) = 0, for each t ∈ T 1 , by Proposition 2. Suppose that x1 (t) = 0 or x2 (t) = 0, for each t ∈ T 1 . Then the pair (p, x) is a Walras equilibrium, by Theorem 4 in Codognato et al. (2015). But then, x is in the core, by the same argument used by Aumann (1964) in the proof of his main theorem. Examples 6, 7, 8, and 9 in Codognato et al. (2015) show that Theorem 2, Proposition 2, and Corollaries 1 and 2 hold non-vacuously.

  Busetto et al. (2017) considered a mixed version of the Shapley window model for exchange economies with any finite number of commodities and, in their Theorem 2, they proved the existence of a Cournot-Nash equilibrium. In this paper, we have considered a bilateral oligopoly version of the model analyzed by Busetto et al. (2017). Nevertheless, our existence result, Theorem 1, is not just a two-commodity case of Theorem 2 in Busetto et al. (2017). While Assumptions 2 and 3 are the same in Busetto et al. (2017) and here, Assumptions 1 and 4 differ. Let us analyze in more detail the difference between the two versions of Assumptions 1 and 4 and the role they play in the two existence proofs.

Proposition 3 .

 3 Let µ(T 1 ) = 0. Under Assumptions 1, 2, 3, and 4, there exists a Cournot-Nash equilibrium b.

Proposition 4 .

 4 Let µ(T 0 ) = 0. Under Assumptions 1, 2 , 3, and 5 , there exists a Cournot-Nash equilibrium b.

The symbol 0 denotes the origin of R

+ as well as the real number zero: no confusion will result.2 card(A) denotes the cardinality of a set A.

In this definition, differentiability means continuous differentiability and is to be understood to include the case of infinite partial derivatives along the boundary of the consumption set (for a discussion of this case, see, for instance, Kreps (2012), p. 58).

[START_REF] Artstein | A note on Fatou's lemma in several dimensions[END_REF] For a discussion of the properties of the correspondences introduced above and their proofs see, for instance,[START_REF] Debreu | Existence of competitive equilibrium[END_REF], Section 4.

In order to save in notation, with some abuse we denote by x both the function x(t) and the function x(t, b(t), p(b)).

• denotes the Euclidean norm and e j denotes the vector in R l + whose jth coordinate is 1 and whose other coordinates vanish.

For a discussion of this literature, see Giraud (2003), p. 359 and p. 365.

It would also be worth comparing the existence proof provided by[START_REF] Bloch | Trade fragmentation and coordination in strategic market games[END_REF] with that given by[START_REF] Dickson | The strategic Marshallian cross[END_REF], based on another approach to bilateral oligopoly.

. Moreover, suppose that

} and let b γ (τ ) = b n (τ ) + γe 2 . Then, we have

2 and γ < n . Then, we obtain

Let us define

Then, we have the vector inequality

Let

.

Thus, we have that p2 n p1 n > α.

Suppose that b n 12 (ρ)

.

Thus, by the same argument used in the previous case, we have that

Let η = min{α, β}. Therefore, we can conclude that

for each n = 1, 2, . . .. Consider the sequence {p kn }. From (3), we obtain that p2 kn > η p1 kn ,

for each n = 1, 2, . . .. Then, we obtain that p2 > η p1 , as the sequence {p kn } converges to p. But then, we have that p2 > 0 as η > 0 and p1 > 0. Hence, having considered all possible cases, we can conclude that p 0.

We now follow the argument used by [START_REF] Busetto | Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities[END_REF] to prove their Theorem 2. In the next part of the proof, we apply a generalization of the Fatou lemma in several dimensions provided by [START_REF] Artstein | A note on Fatou's lemma in several dimensions[END_REF]. By Lemma 1, there is an -Cournot-Nash equilibrium b n , for each n = 1, 2, . . .. The fact that the sequence { B n } belongs to the compact set {(b ij ) ∈ R 4 + : b ij ≤ T w i (t) dµ, i, j = 1, 2} and the sequence {p n }, where p n = p n ( b n ), belongs to the unit simplex ∆, for each n = 1, 2, . . ., implies that there is a subsequence { B kn } of the sequence { B n } which converges to an element