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Two-tier equilibria for supply chains with a single manufacturer and multiple asymmetric retailers
Operations Research Letters 41 (2013) 540–544
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Two-tier equilibria for supply chains with a single manufacturer and multiple asymmetric retailers George J. Kyparisis ∗ , Christos Koulamas Department of Decision Sciences and Information Systems, College of Business Administration, Florida International University, Miami, FL 33199, USA
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Article history: Received 31 May 2012 Received in revised form 11 July 2013 Accepted 12 July 2013 Available online 23 July 2013 Keywords: Supply chains Stackelberg leader–follower Nash–Cournot equilibria Oligopoly
abstract We derive sufficient conditions for the existence and uniqueness of the Stackelberg–Nash–Cournot equilibria for a supply chain problem with a single manufacturer and multiple asymmetric retailers and characterize the first and second order derivatives of the total equilibrium quantities. The Stackelberg manufacturer is assumed to supply a homogeneous product to all retailers with the retail price determined by a general nonlinear inverse demand function. We provide several extensions of our previous results [G.J. Kyparisis, C. Koulamas, A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers, Operations Research Letters 39 (2011) 471–474] obtained for a similar supply chain with symmetric retailers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction We consider a generalization of the two-tier supply chain problem studied earlier in Kyparisis and Koulamas [6] (to be called KK in the sequel) in which the single Stackelberg manufacturer is assumed to supply a homogeneous product to multiple downstream symmetric retailers. The common retail price p(Q ) is determined by a general nonlinear inverse demand function, where Q is the total quantity delivered to the market. The single manufacturer acting as a Stackelberg leader selects the profit maximizing wholesale price w by taking into account the product quantities qi to be chosen by the retailers. Then, in response, the symmetric retailers (acting as Stackelberg followers) engage in quantity Cournot competition and maximize their individual profits by taking the wholesale price w as given and selecting their individual product quantities qi (w), i = 1, . . . , n. In this paper, we generalize the above problem by considering asymmetric retailers with nonidentical variable cost functions and derive sufficient conditions for the existence/uniqueness of twotier equilibria. We also characterize the first and second order derivatives of the total equilibrium quantities. The symmetric case studied in KK was simpler to analyze because all the individual quantities were identical, that is qi (w) = q(w), i = 1, . . . , n. The rest of the paper is organized as follows. Our two-tier model is formulated in Section 2. The existence/uniqueness results and the characterization of the first and second order derivatives of the
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total equilibrium quantities for the two-tier model are presented in Section 3. 2. The two-tier supply chain model with a manufacturer and asymmetric retailers We consider a two-tier supply chain consisting of a single monopolistic manufacturer at the upstream tier supplying a homogeneous product to n multiple oligopolistic asymmetric retailers at the downstream tier. The manufacturer sells the product to the retailers at a wholesale price w and has a fixed marginal unit cost c. We assume that w ≥ c ≥ 0 so that the manufacturer is able to earn a nonnegative profit. The retailers, in turn, sell the product to the market at a retail price p(Q ) determined by a general nonlinear inverse demand function p(Q ), where Q ≥ 0 is the total quantity of the product supplied to the market. It is assumed that the reservation retail price p(0) exceeds the marginal unit cost c, that is p(0) > c. The retailers also incur asymmetric selling costs expressed by nonidentical variable cost functions vi (qi ), where qi ≥ 0 denotes the quantity supplied to the market by retailer i. It is assumed that no selling costs are incurred in the case of zero sales, that is vi (0) = 0, i = 1, . . . , n. We consider the following game theoretic scenario. The manufacturer acts as a Stackelberg leader and, in the first stage of the game, selects the wholesale price w to maximize her profit knowing in advance the quantities qi ≥ 0 that the retailers will select given w . In the second stage of the game, the retailers engage in quantity competition within a Cournot oligopoly framework, that is, they simultaneously maximize their individual profits by taking w as given and selecting their individual quantities qi . The gross
G.J. Kyparisis, C. Koulamas / Operations Research Letters 41 (2013) 540–544
profit of retailer i is
Πi (q1 , . . . , qn , w) = Πi (qi , Q−i , w) = qi [p(qi + Q−i ) − w] − vi (qi ), n where Q = j=1 qj and Q−i = Q − qi is the total quantity supplied
by all retailers other than retailer i. The gross profit of the manufacturer is
Π0 (w) = Q (w)(w − c ), where qi (w) n is the quantity supplied by retailer i at price w and Q (w) = j=1 qj (w). It is implicitly assumed that the total quantity produced by the manufacturer, Q (w), equals the total nquantity supplied to the market by the retailers, that is Q (w) = j=1 qj (w). For a fixed w ≥ c, the quantities [q1 (w), . . . , qn (w)] constitute a single-tier Nash–Cournot equilibrium if qi (w) ≥ 0, i = 1, . . . , n, solves the profit maximization problem of retailer i max qi [p(qi + Q−i (w)) − w] − vi (qi ),
(1)
q i ≥0
where Q−i (w) = Q (w) − qi (w). The quantities [w ∗ , q∗1 , . . . , q∗n ] constitute a two-tier Stackelberg–Nash–Cournot equilibrium if the quantity w ∗ ≥ c solves the manufacturer’s profit maximization problem max Q (w)(w − c ), w≥c
(2)
and q∗i = qi (w ∗ ) ≥ 0, i = 1, . . . , n, where [q1 (w), . . . , qn (w)] constitute a single-tier Nash–Cournot equilibrium. A similar model of a single-tier Nash–Cournot equilibrium was considered by Murphy et al. [7] who characterized its properties using equivalent mathematical programming formulations. Murphy et al. [7], however, did not include the wholesale price w in the retailer’s profit function formulation and did not study the twolevel Stackelberg–Nash–Cournot equilibrium which is the main focus of this paper. Subsequently, Sherali et al. [10] extended the single-tier equilibrium model of Murphy et al. [7] to a two-level Stackelberg–Nash–Cournot equilibrium model in which one of the retailers acts as a Stackelberg leader. In contrast, the manufacturer acts as a Stackelberg leader in our model. In the sequel, we denote the first order and second order derivatives of a function g (x) with respect to x as g ′ (x) and g ′′ (x) respectively. 3. Existence and uniqueness of two-tier Stackelberg–Nash– Cournot equilibria We first prove the existence of Stackelberg–Nash–Cournot equilibria for the two-tier supply chain with a single manufacturer and multiple asymmetric retailers. (All proofs are relegated to the Appendix.) The results presented here rely on the existence results for the single-level Nash–Cournot equilibria with multiple asymmetric retailers. A comprehensive review of known existence results for the single-level Nash–Cournot equilibria with multiple symmetric and asymmetric retailers along new general existence results were presented in Amir [1] utilizing the theory of supermodular games. A general existence result of Amir [1] (Theorem 3.1) for the singlelevel Nash–Cournot equilibria with multiple asymmetric retailers utilized the assumption that p(Q ) is log-concave (that is, that ln p(Q ) is concave). In order to extend the existence results of Amir [1], Ewerhart [3] introduced the concept of an (α, β)-biconcave function which is a generalization of a log-concave function. For α ̸= 0, β ̸= 0, a function f defined on (0, ∞) is called (α, β)-biconcave if for all x1 > 0, x2 > 0 such that f (x1 ) > 0, f (x2 ) > 0, f
1 β β 1 [λx1 + (1 − λ)x2 ] β ≥ [λf (x1 )α + (1 − λ)f (x2 )α ] α
(3)
holds for all λ ∈ [0, 1]. For α = 0, the expression on the right-hand side of (3) becomes f (x1 )λ f (x2 )1−λ and for β = 0, the expression on the left-hand side of (3) becomes f (xλ1 x12−λ ). Note that in the special
541
case where α = 0, (α, 1 − α)-biconcavity of p(Q ) in assumption (A4) is equivalent to log-concavity of p(Q ) (that is, concavity of ln p(Q )). Ewerhart [3] used the concept of an (α, β)-biconcave function to prove a very general result on the existence of a Nash–Cournot equilibrium [q∗1 , . . . , q∗n ] with multiple asymmetric retailers under assumptions (A1)–(A4) stated next in Theorem 1. In Theorem 1, we utilize this result of Ewerhart [3] to prove the existence of Stackelberg–Nash–Cournot equilibria for the two-tier supply chain with a single manufacturer and multiple asymmetric retailers. Theorem 1. Suppose that (A1) p : R1+ → R1+ is continuous and nonincreasing, where R1+ = [0, ∞), (A2) vi : R1+ → R1+ , i = 1, . . . , n, is continuous and nondecreasing, (A3) there is Q0 > 0 such that for all Q > Q0 , Qp(Q ) < Qc + min1≤i≤n vi (Q ), (A4) p(Q ) is (α, 1 − α)-biconcave for some α ∈ [0, 1]. Then, a Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ] exists for the two-tier supply chain. Theorem 1 extends the corresponding result in KK to asymmetric retailers by considering nonidentical variable cost functions vi (qi ). In addition, for the symmetric case, Theorem 1 modifies the result in KK by replacing the differentiability assumptions on p(Q ) and v(Q ) and the condition −p′ (Q ) + v ′′ (q) > 0 with the assumption (A4) stating that p(Q ) is (α, 1 − α)-biconcave for some α ∈ [0, 1]. Condition (A1) states that p(Q ) is downward sloping. Assumption (A2) implies that the total cost function of retailer i, Ci (qi ) = wqi +vi (qi ) (defined for a given w ≥ c), is nondecreasing in qi . Condition (A3) (together with condition (A1)) assures that the profit function in (1) becomes negative for sufficiently large quantities qi for any fixed w ≥ c which in turn restricts the domain in the maximization problem (1). Theorem 1 established the existence of the two-tier Stackelberg–Nash–Cournot equilibria [w ∗ , q∗1 , . . . , q∗n ] under assumptions which do not exclude the possibility of nonunique Nash–Cournot equilibria. In the next theorem, we impose the alternative assumptions (B1)–(B4) used by Quartieri [9] to prove the existence of a unique single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] for multiple asymmetric retailers. We utilize the result of Quartieri [9] to show that, if the assumptions (B1)–(B4) hold, then there exists a Stackelberg–Nash–Cournot equilibrium for the two-tier supply chain with multiple asymmetric retailers. Theorem 2. Suppose that (B1) p : R1+ → R1+ is once continuously differentiable and nonincreasing, (B2) vi : R1+ → R1+ , i = 1, . . . , n, is nondecreasing and convex, (B3) there is Q0 > 0 such that p(Q ) > 0 for Q ∈ [0, Q0 ) and p(Q ) = 0 for Q ∈ [Q0 , ∞), (B4) Qp(Q ) is strictly concave in Q on (0, Q0 ), Then, a Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ] exists for the two-tier supply chain such that [q∗1 , . . . , q∗n ] = [q1 (w ∗ ), . . . , qn (w ∗ )], where [q1 (w), . . . , qn (w)] is a unique single-tier Nash–Cournot equilibrium for all w ∈ [c , p(0)]. We next comment on the assumptions (B1)–(B4) used to prove Theorem 2. Observe that Quartieri [9] assumed in his model of Cournot oligopoly that the functions vi : R1+ → R1+ , are convex and that the quantities qi (w) are restricted to bounded sets of productive capacities Ki of the form Ki = [0, Bi ], Bi > 0, i = 1, . . . , n. We show in the Appendix that this restriction on qi (w) holds without any loss of generality for the single-tier equilibrium problem in (1) with Bi = Q0 , i = 1, . . . , n. Assumptions (B1) and (B2) in Theorem 2 are similar to assumptions (A1) and (A2) in Theorem 1 except that in (B2) the functions vi , i = 1, . . . , n, (and consequently the total cost functions Ci (qi ) = w qi + vi (qi )) are assumed convex.
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Assumption (B3) requires that the function p(Q ) is equal to zero for all large enough quantities Q which is economically justifiable. Assumption (B4) imposes a concavity condition on p(Q ) which is not directly comparable with the condition of (α, 1 − α)-biconcavity of p(Q ) stated in assumption (A4) in Theorem 1. An alternative set of assumptions needed to prove the existence of an asymmetric unique single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] was given earlier in Gaudet and Salant [4]. Let us define the functions Fi (q1 , . . . , qn ) = p(Q ) + qi p′ (Q ) − vi′ (qi ),
i = 1, . . . , n,
(4)
for (q1 , . . . , qn ) ∈ (0, Q0 )n . Next, we extend the proof of Theorem 2 in KK to the case of asymmetric retailers considered in this paper as follows. We first extend Lemmas 3 and 4 in KK. The next lemma shows that the unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] is continuous in w and that is satisfies a system of nonlinear equations which can be solved numerically. Lemma 3. Assume that vi : R1+ → R1+ , i = 1, . . . , n are once continuously differentiable and the conditions (B1)–(B4) of Theorem 2 hold. Suppose that, in addition, (B5) p(0) > min1≤i≤n vi′ (0), (B6) for all w ∈ (c , p(0)− min1≤i≤n vi′ (0)), qi (w) > 0, i = 1, . . . , n, Then, (1) for all w ∈ [c , p(0)], there exists a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] such that qi (w), i = 1, . . . , n, is a continuous function on (c , p(0)) and qi (w) ∈ (0, Q0 ) for w ∈ (c , p(0) − min vi′ (0)); 1≤i≤n
qi (w) = 0 for w ∈ [p(0) − min vi (0), p(0)]. ′
1≤i≤n
(5)
(2) For w ∈ (c , p(0) − min1≤i≤n vi′ (0)), [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n solves the system of nonlinear equations Fi (q1 , . . . , qn ) = p(Q ) + qi p′ (Q ) − vi′ (qi )
= w,
i = 1, . . . , n.
(6)
The next lemma shows that the unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] which solves the system of nonlinear equations (6) is such that the first and second order derivatives of n Q (w) = j=1 qj (w) are well-defined and can be computed numerically. Lemma 4. Assume that p : R1+ → R1+ and vi : R1+ → R1+ , i = 1, . . . , n are thrice continuously differentiable and the conditions (B1)–(B6) of Lemma 3 hold. Suppose that, in addition, (B7) the n × n Jacobian matrix [JF ] of the functions Fi (q1 , . . . , qn ), i = 1, . . . , n, in (4) with respect to (q1 , . . . , qn ) is invertible for all (q1 , . . . , qn ) ∈ (0, Q0 )n , Then, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), there exists a unique asymmetric Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n which solves the system of nonlinear equations n (6) such that the first and second order derivatives of Q (w) = j=1 qj (w) are welldefined and given by Q ′ (w) =
n
aij ,
(B8) for all (q1 , . . . , qn ) ∈ (0, Q0 )n , i,j=1 aij (q1 , . . . , qn ) < 0, where aij (q1 , . . . , qn ), i, j = 1, . . . , n, denotes the ijth element of the inverse Jacobian matrix [JF ]−1 , (B9) for all (q1 , . . . , qn ) ∈ (0, Q0 )n ,
n
n n n ∂ aij ∂ aij a1l + · · · + anl ≤ 0. ∂ q1 l=1 ∂ qn l=1 i ,j = 1 Then, a unique Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 ,
. . . , q∗n ] exists for the two-tier supply chain.
Theorem 3 extends the corresponding result in KK to asymmetric retailers by considering nonidentical variable cost functions vi (qi ). Moreover, for the symmetric case, Theorem 3 differs from the corresponding result in KK because the assumption that p(Q ) is positive for all Q ≥ 0 is replaced by the assumption (B3) which requires that p(Q ) = 0 for Q ≥ Q0 . The latter assumption applies to situations in which p(Q ) becomes zero for some Q0 as is the case with the linear inverse demand function. In general, Theorem 3 requires several additional assumptions not needed in the case of symmetric retailers including the convexity and strict concavity assumptions on vi (qi ) and Qp(Q ) respectively and several conditions on the Jacobian matrix [JF ] of the functions Fi . The assumptions (B5)–(B9) used to prove Lemmas 3 and 4 and Theorem 3 are needed to prove the existence of a unique twotier Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ]. Assumption (B5) ensures that the reservation price, p(0), exceeds the minimum value of the marginal variable costs at zero, vi′ (0), for all retailers. Assumption (B6) states a condition that needs to be verified for the single-tier asymmetric equilibrium problem in (1) which can be done in many particular instances of the problem. This is a nondegeneracy-type condition which states that the quantity qi supplied to the market by each retailer i is positive for the range of w values specified in (B6). From the economic standpoint, this assumption assures that all retailers receive some supplies from the manufacturer. It can be shown that assumption (B6) is automatically satisfied in the special case of n symmetric retailers and hence we conjecture that it will also be satisfied when the variable cost functions vi (q) are sufficiently similar. Finally, assumptions (B7)–(B9) are needed to prove the differentiability properties of the asymmetric single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)], which is a unique solution of the system of nonlinear equations Fi (q1 , . . . , qn ) = w , i = 1, . . . , n, and the strict concavity property of the manufacturer’s profit function Q (w)(w − c ). Acknowledgment We would like to thank the anonymous referee for his/her insightful comments which helped us improve an earlier version of this paper.
(7) Appendix
i,j=1
Q ′′ (w) =
Theorem 3. Assume that p : R1+ → R1+ and vi : R1+ → R1+ , i = 1, . . . , n are thrice continuously differentiable and the conditions (B1)–(B7) of Lemma 4 hold. Suppose that, in addition,
n n n ∂ aij ∂ aij a1l + · · · + anl , ∂ q1 l=1 ∂ qn l=1 i ,j = 1
(8)
∂a
where the quantities aij (q1 , . . . , qn ), ∂ qij (q1 , . . . , qn ), i, j, k = 1, . . . , k n, are evaluated at [q1 (w), . . . , qn (w)]. Based on Theorem 2 and Lemmas 3 and 4, we state the next result which proves the existence of a unique Stackelberg–Nash– Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ] for the two-tier supply chain with a single manufacturer and multiple asymmetric retailers.
Lemma 1. Suppose that the assumptions (A1)–(A3) of Theorem 1 hold. Then, (1) for all w ≥ c, the quantities [q1 (w), . . . , qn (w)] are a single-tier Nash–Cournot equilibrium if and only if qi (w) ∈ [0, Q0 ], i = 1, . . . , n, solves the profit maximization problem of retailer i max qi [p(qi + Q−i (w)) − w] − vi (qi ),
(9)
0≤qi ≤Q0
(2) the quantities [w ∗ , q∗1 , . . . , q∗n ] constitute a two-tier Stackelberg– Nash–Cournot equilibrium if and only if w ∗ ∈ [c , p(0)] solves
G.J. Kyparisis, C. Koulamas / Operations Research Letters 41 (2013) 540–544
the manufacturer’s profit maximization problem max Q (w)(w − c ),
(10)
c ≤w≤p(0)
and q∗i = qi (w ∗ ) ∈ [0, Q0 ], i = 1, . . . , n, solves the retailer’s problem (9), where, for w ∈ [c , p(0)], qi (w) ∈ [0, Q0 ]. Lemma 1 above is an adaptation of Lemma 1 in KK for the case of asymmetric retailers; its proof is analogous to the proof of Lemma 1 in KK and is therefore omitted. Similarly as in KK, we define the following notions. For w ∈ [c , p(0)], the quantities [q1 (w), . . . , qn (w)] constitute a restricted single-tier Nash–Cournot equilibrium if qi (w) ∈ [0, Q0 ], i = 1, . . . , n, solves the maximization problem of retailer i max qi [p(qi + Q−i (w)) − w] − v(qi ).
0≤qi ≤Q0
The quantities [w ∗ , q∗1 , . . . , q∗n ] constitute a restricted two-tier Stackelberg–Nash–Cournot equilibrium if w ∗ ∈ [0, p(0)] solves the manufacturer’s maximization problem (10) and q∗i = qi (w ∗ ) ∈ [0, Q0 ], i = 1, . . . , n, where [q1 (w), . . . , qn (w)] are a restricted single-tier Nash–Cournot equilibrium. The restricted two-tier Stackelberg–Nash–Cournot equilibrium problem defined above can be equivalently stated as a mathematical programming problem with equilibrium constraints (Harker and Pang [5], Pang [8]) defined as MPEC
max f (x, y)
subject to x ∈ X , y ∈ Y (x),
where X = [c , p(0)] ⊂ Rl , f = Q (w − c ) = (
(11) j=1 qj )(w − c ) :
n
Rl × Rm → R1 , l = 1, m = n, x = w ∈ X , y = [q1 , . . . , qn ] ∈ Y (x) = Y (w), Y (w) = {[q1 (w), . . . , qn (w)] : qi (w) solves max qi [p(qi + Q−i (w)) − w] − v(qi )}.
0≤qi ≤Q0
Lemma 2. Suppose that the assumptions (A1)–(A4) of Theorem 1 hold. Then, the point-to-set solution map Y is nonempty-valued on [c , p(0)] and closed on [c , p(0)]. Lemma 2 above is an adaptation of Lemma 2 in KK for the case of asymmetric retailers; its proof is analogous to the proof of Lemma 2 in KK except that it utilizes the result of Ewerhart [3] who proved that, under the conditions (A1)–(A4), there exists at least one Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] for all w ≥ c, instead of the result of Amir and Lambson [2]. Proof of Theorem 1. By Lemma 2, the solution map Y is nonempty-valued and closed on [0, p(0)]. By the definition of Y (w), for all w ∈ [0, p(0)], Y (w) ⊂ [0, Q0 ]n . Thus, the MPEC problem den fined in (11) with X = [0, p(0)] ⊂ Rl and f = Q w = ( j=1 qj )w :
Rl × Rm → R1 , where l = 1, m = n, x = w ∈ X , y = [q1 , . . . , qn ] ∈ Y (x) = Y (w), has a compact feasible set. Since f (x, y) = f (w, q1 , . . . , qn ) = Q w is also continuous in (w, q1 , . . . , qn ), this implies the existence of a restricted Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗ , . . . , q∗ ]. By part (2) of Lemma 1, the quantities [w ∗ , q∗ , . . . , q∗ ] are a Stackelberg–Nash–Cournot equilibrium if and only if they are a restricted Stackelberg–Nash–Cournot equilibrium which completes the proof. ∥
543
then there exists a unique restricted single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)], and, consequently, Y is singlevalued and closed on [c , p(0)]. Similar to the proof of Lemma 3 in KK, this implies that qi (w), i = 1, . . . , n, are continuous functions on (c , p(0)). Moreover, by the equivalence of the restricted and the unrestricted single-tier Nash–Cournot equilibria (shown in Lemma 1), for all w ∈ [c , p(0)] there exists a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] such that qi (w) ∈ [0, Q0 ], i = 1, . . . , n. Let w ∈ (c , p(0) − min1≤i≤n vi′ (0)) be fixed, where p(0) − min1≤i≤n vi′ (0) > 0 by assumptions (B5). Suppose that qi (w) = Q0 for some i and let qi ∈ [0, Q0 ). In view of (B3), Q0 p(Q0 + Q−i (w)) = 0 ≤ qi p(qi + Q−i (w)). Also, since w > 0, Q0 w > qi w and since vi (q) is nondecreasing, vi (Q0 ) ≥ vi (qi ). Together, these inequalities imply that, for qi ∈ [0, Q0 ),
Πi (qi , Q−i (w), w) − Πi (qi (w), Q−i (w), w) qi [p(qi + Q−i (w)) − w] − vi (qi )
(12)
− Q0 [p(Q0 + Q−i (w)) − w] + vi (Q0 ) > 0.
This contradicts the assumption that qi (w) = Q0 solves (1) and therefore qi (w) ∈ [0, Q0 ) for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)). By assumption (B6), we also have that qi (w) > 0 for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)) and thus qi (w) ∈ (0, Q0 ), i = 1, . . . , n, for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Let w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)] be fixed. For all w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)], the quantities [q1 (w), . . . , qn (w)] = [0, . . . , 0] are the unique Nash–Cournot equilibrium if and only if, for each i = 1, . . . , n,
Πi (qi (w), Q−i (w), w) − Πi (qi , Q−i (w), w) = 0[p(0) − w] − vi (0) − qi [p(qi ) − w] + vi (qi ) ≥ 0 for all qi ∈ (0, Q0 ]. Since vi (0) = 0, i = 1, . . . , n, the last inequality v (q ) is equivalent to the inequality w ≥ p(qi ) − i q i for all qi ∈ (0, Q0 ]. i This, in turn, is equivalent to the inequality
w ≥ sup
qi ∈(0,Q0 ]
p(qi ) −
vi (qi ) qi
,
i = 1, . . . , n.
(13)
Since, by (B2), vi (qi ), i = 1, . . . , n, is convex for qi > 0 it is easy v (q ) to see that i q i is nondecreasing for qi > 0. Together with the i fact that, by (B1), p(q) is nonincreasing for q ≥ 0, this implies that vi (qi ) p(qi ) − q is nonincreasing for qi > 0 and i
sup
qi ∈(0,Q0 ]
p(qi ) −
vi (qi ) qi
= lim
qi →0+
p(qi ) −
= p(0) − vi′ (0),
vi (qi )
qi i = 1, . . . , n.
(14)
Proof of Theorem 2. Quartieri [9] proved that if the conditions (B1)–(B4) hold, then, for w ≥ c, there exists a unique restricted single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)]. Since the conditions (B1)–(B4) also imply the conditions (A1)–(A3), we conclude that Lemmas 1 and 2 also hold under the conditions (B1)–(B4). The proof of Theorem 2 then utilizes Lemmas 1 and 2 and the MPEC problem defined in the proof of Theorem 1 and is analogous to the proof of Theorem 1 in KK. ∥
In view of inequality (14), inequality (13) is equivalent to w ≥ p(0) − vi′ (0), i = 1, . . . , n, which is equivalent to w ≥ max1≤i≤n [p(0) − vi′ (0)] = p(0) − min1≤i≤n vi′ (0). In view of the earlier remarks, this proves that the quantities [q1 (w), . . . , qn (w)] = [0, . . . , 0] are the unique Nash–Cournot equilibrium for all w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)]. (2) It follows from the equivalence of the restricted and the unrestricted single-tier Nash–Cournot equilibria and from part (1) of Lemma 3 that, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] exists and is such that qi (w) ∈ (0, Q0 ) solves problem (9). For any fixed w ≥ c, consider the objective function Πi (qi , Q−i (w), w), i = 1, . . . , n, in (9). Because qi (w) ∈ (0, Q0 ) maximizes Πi (qi , Q−i (w), w) on [0, Q0 ], for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), the first order necessary conditions
Proof of Lemma 3. (1) Since Lemma 2 holds as shown in the proof of Theorem 2, Y is nonempty-valued on [c , p(0)] and closed on [c , p(0)]. Quartieri [9] proved that if the conditions (B1)–(B4) hold,
(15)
∂ Πi (qi , Q−i (w), w) = p(qi + Q−i (w)) − w ∂ qi + qi p′ (qi + Q−i (w)) − vi′ (qi ) = 0
544
G.J. Kyparisis, C. Koulamas / Operations Research Letters 41 (2013) 540–544
hold at qi (w), i = 1, . . . , n. The result follows in view of the fact that conditions (15) hold at qi (w) if and only if [q1 (w), . . . , qn (w)] solves the system of Eq. (6). ∥ Proof of Lemma 4. By part (2) of Lemma 3, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), there exists a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n which solves the system of nonlinear equations (6). By our assumptions, the functions Fi (q1 , . . . , qn ), i = 1, . . . , n, are twice continuously differentiable. Thus, by differentiating the system of equations (6) evaluated at [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n with respect to w , we obtain the following system of equations
∂ F1 ∂ q1 ... ∂ Fn ∂ q1
∂ F1 ′ q (w) 1 ∂ qn 1 ... ... = ... , ∂ Fn q′n (w) 1 ∂ qn
... ... ...
(16)
where the leftmost matrix is evaluated at [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n . Denote the three matrices in (16) as [JF (w)], [Dq(w)] and [1] respectively, where [JF (w)] denotes the Jacobian matrix of the vector of functions Fi (q1 , . . . , qn ), i = 1, . . . , n, evaluated at [q1 (w), . . . , qn (w)], [Dq(w)] denotes the vector of the first order derivatives of functions qi (w), i = 1, . . . , n, and [1] denotes the vector of 1s. Thus, (16) can be written compactly as [JF (w)] [Dq(w)] = [1]. Let [JF (w)]−1 denote the inverse of [JF (w)] evaluated at [q1 (w), . . . , qn (w)] which exists by assumptions (B7). We can then express [JF (w)]−1 as follows a11 (q1 , . . . , qn )
... an1 (q1 , . . . , qn )
[JF (w)]−1 =
... ... ...
a1n (q1 , . . . , qn ) ... , ann (q1 , . . . , qn )
(17)
(18)
Consider the expression Q (w) = j=1 qj (w) for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). In view of (16), the first order derivative of Q (w), denoted by Q ′ (w), can be calculated as follows
n
Q ′ (w) =
n
q′j (w) = [1]T [Dq(w)],
(19)
j =1
where AT denotes the transpose of matrix A. By combining (17)–(19), we obtain Q ′ (w) = [1]T [JF (w)]−1 [1]
=
n
aij (q1 (w), . . . , qn (w)).
(20)
i,j=1
In view of (20), the second order derivative of Q (w), denoted by Q ′′ (w), can be calculated as follows n d aij (q1 (w), . . . , qn (w)) d w i ,j = 1 n ∂ aij ′ ∂ aij ′ = q1 (w) + · · · + qn (w) ∂ q1 ∂ qn i,j=1
Q ′′ (w) =
∂a
n n n ∂ aij ∂ aij a1l + · · · + anl , Q (w) = ∂ q1 l=1 ∂ qn l=1 i,j=1 ′′
(22)
∂a
where the quantities aij , ∂ qij i, j, k = 1, . . . , n, are evaluated at k [q1 (w), . . . , qn (w)]. ∥ Proof of Theorem 3. Theorem 3 implies that a Stackelberg–Nash– Cournot equilibrium [w ∗ , q∗1 , . . . , q∗2 ] exists. Moreover, Lemma 3 implies that a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)], which satisfies (5), exists for all w ∈ (c , p(0)]. The equivalence of the restricted and the unrestricted two-tier Stackelberg– Nash–Cournot equilibria (which follows from Lemma 1) implies that the profit function Π0 (w) = Q (w)(w − c ) in (10) is maximized at w ∗ ∈ (c , p(0)]. Observe that Π0 (c ) = 0, Π0 (w) = 0 for w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)] and Π0 (w) = Q (w)(w − c ) > 0 for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Therefore, the problem (10) is maximized at w ∗ ∈ (c , p(0) − min1≤i≤n vi′ (0)). By Lemma 4, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), Q ′ (w) and Q ′′ (w) are well defined. Thus, the first and second order derivatives of the profit function Π0 (w) = Q (w)(w − c ) are given by Π0′ (w) = Q ′ (w)w + Q (w) − Q ′ (w)c and Π0′′ (w) = Q ′′ (w)w + 2Q ′ (w) − Q ′′ (w)c, respectively. It follows that Π0′′ (w) < 0 if Q ′′ (w) ≤ 0 and Q ′ (w) < 0 since w ≥ c. In view of (7), the condition Q ′ (w) < 0 is equivalent to the condition n
aij (q1 (w), . . . , qn (w)) < 0.
(23)
i,j=1
where the quantities aij (q1 , . . . , qn ), i, j = 1, . . . , n, are evaluated at [q1 (w), . . . , qn (w)] and are well defined for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Thus, the system (16) implies that
[Dq(w)] = [JF (w)]−1 [1].
k = 1, . . . , n. This fact combined with (21), implies that
(21)
where the quantities ∂ qij , i, j, k = 1, . . . , n, are evaluated at k [q1 (w), . . . , qn (w)]. In view of (17) and [Dq(w)] = [JF (w)]−1 (18), n ′ [1] is the vector of quantities qk (w) = l=1 akl (q1 (w), . . . , qn (w)),
In view of (8), the condition Q ′′ (w) ≤ 0 is equivalent to the condition
n n n ∂ aij ∂ aij a1l + · · · + anl ≤ 0. ∂ q1 l=1 ∂ qn l=1 i,j=1
(24)
In view of the assumptions (B8)–(B9) of Theorem 3, the inequalities in (23) and (24) hold which implies that Q ′′ (w) ≤ 0 and Q ′ (w) < 0 and thus that Π0′′ (w) < 0 for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Therefore, Π0 (w) is strictly concave for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). This implies that w ∗ is the unique solution of problem (10), [w ∗ , q∗1 , . . . , q∗n ] is the unique restricted Stackelberg–Nash– Cournot equilibrium. In view of the equivalence of the restricted and the unrestricted two-tier Stackelberg–Nash–Cournot equilibria, this completes the proof. ∥ References [1] R. Amir, Cournot oligopoly and the theory of supermodular games, Games and Economic Behavior 15 (1996) 132–148. [2] R. Amir, V.E. Lambson, On the effects of entry in cournot markets, Review of Economic Studies 67 (2000) 235–254. [3] C. Ewerhart, Cournot oligopoly and concavo-concave demand. Department of Economics, University of Zurich, Working Paper No. 16, 2011. [4] G. Gaudet, S.W. Salant, Uniqueness of cournot equilibrium: new results from old methods, Review of Economic Studies 58 (1991) 399–404. [5] P.T. Harker, J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Mathematical Programming 48 (1990) 161–220. [6] G.J. Kyparisis, C. Koulamas, A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers, Operations Research Letters 39 (2011) 471–474. [7] F.H. Murphy, H.D. Sherali, A.L. Soyster, A mathematical programming approach for determining oligopolistic market equilibrium, Mathematical Programming 24 (1982) 92–106. [8] J.S. Pang, Partially B-regular optimization and equilibrium problems, Mathematics of Operations Research 32 (2007) 687–699. [9] F. Quartieri, Necessary and sufficient conditions for the existence of a unique Cournot equilibrium, Working Paper, 2008. http://www.siepi.luiss.it/cms/ media/2/20080715-Quartieri-Paper.pdf. [10] H.D. Sherali, A.L. Soyster, F.H. Murphy, Stackelberg–Nash–Cournot equilibria: characterizations and computations, Operations Research 31 (1993) 253–276.
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Two-tier equilibria for supply chains with a single manufacturer and multiple asymmetric retailers George J. Kyparisis ∗ , Christos Koulamas Department of Decision Sciences and Information Systems, College of Business Administration, Florida International University, Miami, FL 33199, USA
article
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Article history: Received 31 May 2012 Received in revised form 11 July 2013 Accepted 12 July 2013 Available online 23 July 2013 Keywords: Supply chains Stackelberg leader–follower Nash–Cournot equilibria Oligopoly
abstract We derive sufficient conditions for the existence and uniqueness of the Stackelberg–Nash–Cournot equilibria for a supply chain problem with a single manufacturer and multiple asymmetric retailers and characterize the first and second order derivatives of the total equilibrium quantities. The Stackelberg manufacturer is assumed to supply a homogeneous product to all retailers with the retail price determined by a general nonlinear inverse demand function. We provide several extensions of our previous results [G.J. Kyparisis, C. Koulamas, A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers, Operations Research Letters 39 (2011) 471–474] obtained for a similar supply chain with symmetric retailers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction We consider a generalization of the two-tier supply chain problem studied earlier in Kyparisis and Koulamas [6] (to be called KK in the sequel) in which the single Stackelberg manufacturer is assumed to supply a homogeneous product to multiple downstream symmetric retailers. The common retail price p(Q ) is determined by a general nonlinear inverse demand function, where Q is the total quantity delivered to the market. The single manufacturer acting as a Stackelberg leader selects the profit maximizing wholesale price w by taking into account the product quantities qi to be chosen by the retailers. Then, in response, the symmetric retailers (acting as Stackelberg followers) engage in quantity Cournot competition and maximize their individual profits by taking the wholesale price w as given and selecting their individual product quantities qi (w), i = 1, . . . , n. In this paper, we generalize the above problem by considering asymmetric retailers with nonidentical variable cost functions and derive sufficient conditions for the existence/uniqueness of twotier equilibria. We also characterize the first and second order derivatives of the total equilibrium quantities. The symmetric case studied in KK was simpler to analyze because all the individual quantities were identical, that is qi (w) = q(w), i = 1, . . . , n. The rest of the paper is organized as follows. Our two-tier model is formulated in Section 2. The existence/uniqueness results and the characterization of the first and second order derivatives of the
∗
Corresponding author. Tel.: +1 305 348 3403; fax: +1 305 348 4126. E-mail address: [email protected] (G.J. Kyparisis).
0167-6377/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.orl.2013.07.003
total equilibrium quantities for the two-tier model are presented in Section 3. 2. The two-tier supply chain model with a manufacturer and asymmetric retailers We consider a two-tier supply chain consisting of a single monopolistic manufacturer at the upstream tier supplying a homogeneous product to n multiple oligopolistic asymmetric retailers at the downstream tier. The manufacturer sells the product to the retailers at a wholesale price w and has a fixed marginal unit cost c. We assume that w ≥ c ≥ 0 so that the manufacturer is able to earn a nonnegative profit. The retailers, in turn, sell the product to the market at a retail price p(Q ) determined by a general nonlinear inverse demand function p(Q ), where Q ≥ 0 is the total quantity of the product supplied to the market. It is assumed that the reservation retail price p(0) exceeds the marginal unit cost c, that is p(0) > c. The retailers also incur asymmetric selling costs expressed by nonidentical variable cost functions vi (qi ), where qi ≥ 0 denotes the quantity supplied to the market by retailer i. It is assumed that no selling costs are incurred in the case of zero sales, that is vi (0) = 0, i = 1, . . . , n. We consider the following game theoretic scenario. The manufacturer acts as a Stackelberg leader and, in the first stage of the game, selects the wholesale price w to maximize her profit knowing in advance the quantities qi ≥ 0 that the retailers will select given w . In the second stage of the game, the retailers engage in quantity competition within a Cournot oligopoly framework, that is, they simultaneously maximize their individual profits by taking w as given and selecting their individual quantities qi . The gross
G.J. Kyparisis, C. Koulamas / Operations Research Letters 41 (2013) 540–544
profit of retailer i is
Πi (q1 , . . . , qn , w) = Πi (qi , Q−i , w) = qi [p(qi + Q−i ) − w] − vi (qi ), n where Q = j=1 qj and Q−i = Q − qi is the total quantity supplied
by all retailers other than retailer i. The gross profit of the manufacturer is
Π0 (w) = Q (w)(w − c ), where qi (w) n is the quantity supplied by retailer i at price w and Q (w) = j=1 qj (w). It is implicitly assumed that the total quantity produced by the manufacturer, Q (w), equals the total nquantity supplied to the market by the retailers, that is Q (w) = j=1 qj (w). For a fixed w ≥ c, the quantities [q1 (w), . . . , qn (w)] constitute a single-tier Nash–Cournot equilibrium if qi (w) ≥ 0, i = 1, . . . , n, solves the profit maximization problem of retailer i max qi [p(qi + Q−i (w)) − w] − vi (qi ),
(1)
q i ≥0
where Q−i (w) = Q (w) − qi (w). The quantities [w ∗ , q∗1 , . . . , q∗n ] constitute a two-tier Stackelberg–Nash–Cournot equilibrium if the quantity w ∗ ≥ c solves the manufacturer’s profit maximization problem max Q (w)(w − c ), w≥c
(2)
and q∗i = qi (w ∗ ) ≥ 0, i = 1, . . . , n, where [q1 (w), . . . , qn (w)] constitute a single-tier Nash–Cournot equilibrium. A similar model of a single-tier Nash–Cournot equilibrium was considered by Murphy et al. [7] who characterized its properties using equivalent mathematical programming formulations. Murphy et al. [7], however, did not include the wholesale price w in the retailer’s profit function formulation and did not study the twolevel Stackelberg–Nash–Cournot equilibrium which is the main focus of this paper. Subsequently, Sherali et al. [10] extended the single-tier equilibrium model of Murphy et al. [7] to a two-level Stackelberg–Nash–Cournot equilibrium model in which one of the retailers acts as a Stackelberg leader. In contrast, the manufacturer acts as a Stackelberg leader in our model. In the sequel, we denote the first order and second order derivatives of a function g (x) with respect to x as g ′ (x) and g ′′ (x) respectively. 3. Existence and uniqueness of two-tier Stackelberg–Nash– Cournot equilibria We first prove the existence of Stackelberg–Nash–Cournot equilibria for the two-tier supply chain with a single manufacturer and multiple asymmetric retailers. (All proofs are relegated to the Appendix.) The results presented here rely on the existence results for the single-level Nash–Cournot equilibria with multiple asymmetric retailers. A comprehensive review of known existence results for the single-level Nash–Cournot equilibria with multiple symmetric and asymmetric retailers along new general existence results were presented in Amir [1] utilizing the theory of supermodular games. A general existence result of Amir [1] (Theorem 3.1) for the singlelevel Nash–Cournot equilibria with multiple asymmetric retailers utilized the assumption that p(Q ) is log-concave (that is, that ln p(Q ) is concave). In order to extend the existence results of Amir [1], Ewerhart [3] introduced the concept of an (α, β)-biconcave function which is a generalization of a log-concave function. For α ̸= 0, β ̸= 0, a function f defined on (0, ∞) is called (α, β)-biconcave if for all x1 > 0, x2 > 0 such that f (x1 ) > 0, f (x2 ) > 0, f
1 β β 1 [λx1 + (1 − λ)x2 ] β ≥ [λf (x1 )α + (1 − λ)f (x2 )α ] α
(3)
holds for all λ ∈ [0, 1]. For α = 0, the expression on the right-hand side of (3) becomes f (x1 )λ f (x2 )1−λ and for β = 0, the expression on the left-hand side of (3) becomes f (xλ1 x12−λ ). Note that in the special
541
case where α = 0, (α, 1 − α)-biconcavity of p(Q ) in assumption (A4) is equivalent to log-concavity of p(Q ) (that is, concavity of ln p(Q )). Ewerhart [3] used the concept of an (α, β)-biconcave function to prove a very general result on the existence of a Nash–Cournot equilibrium [q∗1 , . . . , q∗n ] with multiple asymmetric retailers under assumptions (A1)–(A4) stated next in Theorem 1. In Theorem 1, we utilize this result of Ewerhart [3] to prove the existence of Stackelberg–Nash–Cournot equilibria for the two-tier supply chain with a single manufacturer and multiple asymmetric retailers. Theorem 1. Suppose that (A1) p : R1+ → R1+ is continuous and nonincreasing, where R1+ = [0, ∞), (A2) vi : R1+ → R1+ , i = 1, . . . , n, is continuous and nondecreasing, (A3) there is Q0 > 0 such that for all Q > Q0 , Qp(Q ) < Qc + min1≤i≤n vi (Q ), (A4) p(Q ) is (α, 1 − α)-biconcave for some α ∈ [0, 1]. Then, a Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ] exists for the two-tier supply chain. Theorem 1 extends the corresponding result in KK to asymmetric retailers by considering nonidentical variable cost functions vi (qi ). In addition, for the symmetric case, Theorem 1 modifies the result in KK by replacing the differentiability assumptions on p(Q ) and v(Q ) and the condition −p′ (Q ) + v ′′ (q) > 0 with the assumption (A4) stating that p(Q ) is (α, 1 − α)-biconcave for some α ∈ [0, 1]. Condition (A1) states that p(Q ) is downward sloping. Assumption (A2) implies that the total cost function of retailer i, Ci (qi ) = wqi +vi (qi ) (defined for a given w ≥ c), is nondecreasing in qi . Condition (A3) (together with condition (A1)) assures that the profit function in (1) becomes negative for sufficiently large quantities qi for any fixed w ≥ c which in turn restricts the domain in the maximization problem (1). Theorem 1 established the existence of the two-tier Stackelberg–Nash–Cournot equilibria [w ∗ , q∗1 , . . . , q∗n ] under assumptions which do not exclude the possibility of nonunique Nash–Cournot equilibria. In the next theorem, we impose the alternative assumptions (B1)–(B4) used by Quartieri [9] to prove the existence of a unique single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] for multiple asymmetric retailers. We utilize the result of Quartieri [9] to show that, if the assumptions (B1)–(B4) hold, then there exists a Stackelberg–Nash–Cournot equilibrium for the two-tier supply chain with multiple asymmetric retailers. Theorem 2. Suppose that (B1) p : R1+ → R1+ is once continuously differentiable and nonincreasing, (B2) vi : R1+ → R1+ , i = 1, . . . , n, is nondecreasing and convex, (B3) there is Q0 > 0 such that p(Q ) > 0 for Q ∈ [0, Q0 ) and p(Q ) = 0 for Q ∈ [Q0 , ∞), (B4) Qp(Q ) is strictly concave in Q on (0, Q0 ), Then, a Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ] exists for the two-tier supply chain such that [q∗1 , . . . , q∗n ] = [q1 (w ∗ ), . . . , qn (w ∗ )], where [q1 (w), . . . , qn (w)] is a unique single-tier Nash–Cournot equilibrium for all w ∈ [c , p(0)]. We next comment on the assumptions (B1)–(B4) used to prove Theorem 2. Observe that Quartieri [9] assumed in his model of Cournot oligopoly that the functions vi : R1+ → R1+ , are convex and that the quantities qi (w) are restricted to bounded sets of productive capacities Ki of the form Ki = [0, Bi ], Bi > 0, i = 1, . . . , n. We show in the Appendix that this restriction on qi (w) holds without any loss of generality for the single-tier equilibrium problem in (1) with Bi = Q0 , i = 1, . . . , n. Assumptions (B1) and (B2) in Theorem 2 are similar to assumptions (A1) and (A2) in Theorem 1 except that in (B2) the functions vi , i = 1, . . . , n, (and consequently the total cost functions Ci (qi ) = w qi + vi (qi )) are assumed convex.
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Assumption (B3) requires that the function p(Q ) is equal to zero for all large enough quantities Q which is economically justifiable. Assumption (B4) imposes a concavity condition on p(Q ) which is not directly comparable with the condition of (α, 1 − α)-biconcavity of p(Q ) stated in assumption (A4) in Theorem 1. An alternative set of assumptions needed to prove the existence of an asymmetric unique single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] was given earlier in Gaudet and Salant [4]. Let us define the functions Fi (q1 , . . . , qn ) = p(Q ) + qi p′ (Q ) − vi′ (qi ),
i = 1, . . . , n,
(4)
for (q1 , . . . , qn ) ∈ (0, Q0 )n . Next, we extend the proof of Theorem 2 in KK to the case of asymmetric retailers considered in this paper as follows. We first extend Lemmas 3 and 4 in KK. The next lemma shows that the unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] is continuous in w and that is satisfies a system of nonlinear equations which can be solved numerically. Lemma 3. Assume that vi : R1+ → R1+ , i = 1, . . . , n are once continuously differentiable and the conditions (B1)–(B4) of Theorem 2 hold. Suppose that, in addition, (B5) p(0) > min1≤i≤n vi′ (0), (B6) for all w ∈ (c , p(0)− min1≤i≤n vi′ (0)), qi (w) > 0, i = 1, . . . , n, Then, (1) for all w ∈ [c , p(0)], there exists a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] such that qi (w), i = 1, . . . , n, is a continuous function on (c , p(0)) and qi (w) ∈ (0, Q0 ) for w ∈ (c , p(0) − min vi′ (0)); 1≤i≤n
qi (w) = 0 for w ∈ [p(0) − min vi (0), p(0)]. ′
1≤i≤n
(5)
(2) For w ∈ (c , p(0) − min1≤i≤n vi′ (0)), [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n solves the system of nonlinear equations Fi (q1 , . . . , qn ) = p(Q ) + qi p′ (Q ) − vi′ (qi )
= w,
i = 1, . . . , n.
(6)
The next lemma shows that the unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] which solves the system of nonlinear equations (6) is such that the first and second order derivatives of n Q (w) = j=1 qj (w) are well-defined and can be computed numerically. Lemma 4. Assume that p : R1+ → R1+ and vi : R1+ → R1+ , i = 1, . . . , n are thrice continuously differentiable and the conditions (B1)–(B6) of Lemma 3 hold. Suppose that, in addition, (B7) the n × n Jacobian matrix [JF ] of the functions Fi (q1 , . . . , qn ), i = 1, . . . , n, in (4) with respect to (q1 , . . . , qn ) is invertible for all (q1 , . . . , qn ) ∈ (0, Q0 )n , Then, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), there exists a unique asymmetric Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n which solves the system of nonlinear equations n (6) such that the first and second order derivatives of Q (w) = j=1 qj (w) are welldefined and given by Q ′ (w) =
n
aij ,
(B8) for all (q1 , . . . , qn ) ∈ (0, Q0 )n , i,j=1 aij (q1 , . . . , qn ) < 0, where aij (q1 , . . . , qn ), i, j = 1, . . . , n, denotes the ijth element of the inverse Jacobian matrix [JF ]−1 , (B9) for all (q1 , . . . , qn ) ∈ (0, Q0 )n ,
n
n n n ∂ aij ∂ aij a1l + · · · + anl ≤ 0. ∂ q1 l=1 ∂ qn l=1 i ,j = 1 Then, a unique Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 ,
. . . , q∗n ] exists for the two-tier supply chain.
Theorem 3 extends the corresponding result in KK to asymmetric retailers by considering nonidentical variable cost functions vi (qi ). Moreover, for the symmetric case, Theorem 3 differs from the corresponding result in KK because the assumption that p(Q ) is positive for all Q ≥ 0 is replaced by the assumption (B3) which requires that p(Q ) = 0 for Q ≥ Q0 . The latter assumption applies to situations in which p(Q ) becomes zero for some Q0 as is the case with the linear inverse demand function. In general, Theorem 3 requires several additional assumptions not needed in the case of symmetric retailers including the convexity and strict concavity assumptions on vi (qi ) and Qp(Q ) respectively and several conditions on the Jacobian matrix [JF ] of the functions Fi . The assumptions (B5)–(B9) used to prove Lemmas 3 and 4 and Theorem 3 are needed to prove the existence of a unique twotier Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ]. Assumption (B5) ensures that the reservation price, p(0), exceeds the minimum value of the marginal variable costs at zero, vi′ (0), for all retailers. Assumption (B6) states a condition that needs to be verified for the single-tier asymmetric equilibrium problem in (1) which can be done in many particular instances of the problem. This is a nondegeneracy-type condition which states that the quantity qi supplied to the market by each retailer i is positive for the range of w values specified in (B6). From the economic standpoint, this assumption assures that all retailers receive some supplies from the manufacturer. It can be shown that assumption (B6) is automatically satisfied in the special case of n symmetric retailers and hence we conjecture that it will also be satisfied when the variable cost functions vi (q) are sufficiently similar. Finally, assumptions (B7)–(B9) are needed to prove the differentiability properties of the asymmetric single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)], which is a unique solution of the system of nonlinear equations Fi (q1 , . . . , qn ) = w , i = 1, . . . , n, and the strict concavity property of the manufacturer’s profit function Q (w)(w − c ). Acknowledgment We would like to thank the anonymous referee for his/her insightful comments which helped us improve an earlier version of this paper.
(7) Appendix
i,j=1
Q ′′ (w) =
Theorem 3. Assume that p : R1+ → R1+ and vi : R1+ → R1+ , i = 1, . . . , n are thrice continuously differentiable and the conditions (B1)–(B7) of Lemma 4 hold. Suppose that, in addition,
n n n ∂ aij ∂ aij a1l + · · · + anl , ∂ q1 l=1 ∂ qn l=1 i ,j = 1
(8)
∂a
where the quantities aij (q1 , . . . , qn ), ∂ qij (q1 , . . . , qn ), i, j, k = 1, . . . , k n, are evaluated at [q1 (w), . . . , qn (w)]. Based on Theorem 2 and Lemmas 3 and 4, we state the next result which proves the existence of a unique Stackelberg–Nash– Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ] for the two-tier supply chain with a single manufacturer and multiple asymmetric retailers.
Lemma 1. Suppose that the assumptions (A1)–(A3) of Theorem 1 hold. Then, (1) for all w ≥ c, the quantities [q1 (w), . . . , qn (w)] are a single-tier Nash–Cournot equilibrium if and only if qi (w) ∈ [0, Q0 ], i = 1, . . . , n, solves the profit maximization problem of retailer i max qi [p(qi + Q−i (w)) − w] − vi (qi ),
(9)
0≤qi ≤Q0
(2) the quantities [w ∗ , q∗1 , . . . , q∗n ] constitute a two-tier Stackelberg– Nash–Cournot equilibrium if and only if w ∗ ∈ [c , p(0)] solves
G.J. Kyparisis, C. Koulamas / Operations Research Letters 41 (2013) 540–544
the manufacturer’s profit maximization problem max Q (w)(w − c ),
(10)
c ≤w≤p(0)
and q∗i = qi (w ∗ ) ∈ [0, Q0 ], i = 1, . . . , n, solves the retailer’s problem (9), where, for w ∈ [c , p(0)], qi (w) ∈ [0, Q0 ]. Lemma 1 above is an adaptation of Lemma 1 in KK for the case of asymmetric retailers; its proof is analogous to the proof of Lemma 1 in KK and is therefore omitted. Similarly as in KK, we define the following notions. For w ∈ [c , p(0)], the quantities [q1 (w), . . . , qn (w)] constitute a restricted single-tier Nash–Cournot equilibrium if qi (w) ∈ [0, Q0 ], i = 1, . . . , n, solves the maximization problem of retailer i max qi [p(qi + Q−i (w)) − w] − v(qi ).
0≤qi ≤Q0
The quantities [w ∗ , q∗1 , . . . , q∗n ] constitute a restricted two-tier Stackelberg–Nash–Cournot equilibrium if w ∗ ∈ [0, p(0)] solves the manufacturer’s maximization problem (10) and q∗i = qi (w ∗ ) ∈ [0, Q0 ], i = 1, . . . , n, where [q1 (w), . . . , qn (w)] are a restricted single-tier Nash–Cournot equilibrium. The restricted two-tier Stackelberg–Nash–Cournot equilibrium problem defined above can be equivalently stated as a mathematical programming problem with equilibrium constraints (Harker and Pang [5], Pang [8]) defined as MPEC
max f (x, y)
subject to x ∈ X , y ∈ Y (x),
where X = [c , p(0)] ⊂ Rl , f = Q (w − c ) = (
(11) j=1 qj )(w − c ) :
n
Rl × Rm → R1 , l = 1, m = n, x = w ∈ X , y = [q1 , . . . , qn ] ∈ Y (x) = Y (w), Y (w) = {[q1 (w), . . . , qn (w)] : qi (w) solves max qi [p(qi + Q−i (w)) − w] − v(qi )}.
0≤qi ≤Q0
Lemma 2. Suppose that the assumptions (A1)–(A4) of Theorem 1 hold. Then, the point-to-set solution map Y is nonempty-valued on [c , p(0)] and closed on [c , p(0)]. Lemma 2 above is an adaptation of Lemma 2 in KK for the case of asymmetric retailers; its proof is analogous to the proof of Lemma 2 in KK except that it utilizes the result of Ewerhart [3] who proved that, under the conditions (A1)–(A4), there exists at least one Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] for all w ≥ c, instead of the result of Amir and Lambson [2]. Proof of Theorem 1. By Lemma 2, the solution map Y is nonempty-valued and closed on [0, p(0)]. By the definition of Y (w), for all w ∈ [0, p(0)], Y (w) ⊂ [0, Q0 ]n . Thus, the MPEC problem den fined in (11) with X = [0, p(0)] ⊂ Rl and f = Q w = ( j=1 qj )w :
Rl × Rm → R1 , where l = 1, m = n, x = w ∈ X , y = [q1 , . . . , qn ] ∈ Y (x) = Y (w), has a compact feasible set. Since f (x, y) = f (w, q1 , . . . , qn ) = Q w is also continuous in (w, q1 , . . . , qn ), this implies the existence of a restricted Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗ , . . . , q∗ ]. By part (2) of Lemma 1, the quantities [w ∗ , q∗ , . . . , q∗ ] are a Stackelberg–Nash–Cournot equilibrium if and only if they are a restricted Stackelberg–Nash–Cournot equilibrium which completes the proof. ∥
543
then there exists a unique restricted single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)], and, consequently, Y is singlevalued and closed on [c , p(0)]. Similar to the proof of Lemma 3 in KK, this implies that qi (w), i = 1, . . . , n, are continuous functions on (c , p(0)). Moreover, by the equivalence of the restricted and the unrestricted single-tier Nash–Cournot equilibria (shown in Lemma 1), for all w ∈ [c , p(0)] there exists a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] such that qi (w) ∈ [0, Q0 ], i = 1, . . . , n. Let w ∈ (c , p(0) − min1≤i≤n vi′ (0)) be fixed, where p(0) − min1≤i≤n vi′ (0) > 0 by assumptions (B5). Suppose that qi (w) = Q0 for some i and let qi ∈ [0, Q0 ). In view of (B3), Q0 p(Q0 + Q−i (w)) = 0 ≤ qi p(qi + Q−i (w)). Also, since w > 0, Q0 w > qi w and since vi (q) is nondecreasing, vi (Q0 ) ≥ vi (qi ). Together, these inequalities imply that, for qi ∈ [0, Q0 ),
Πi (qi , Q−i (w), w) − Πi (qi (w), Q−i (w), w) qi [p(qi + Q−i (w)) − w] − vi (qi )
(12)
− Q0 [p(Q0 + Q−i (w)) − w] + vi (Q0 ) > 0.
This contradicts the assumption that qi (w) = Q0 solves (1) and therefore qi (w) ∈ [0, Q0 ) for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)). By assumption (B6), we also have that qi (w) > 0 for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)) and thus qi (w) ∈ (0, Q0 ), i = 1, . . . , n, for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Let w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)] be fixed. For all w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)], the quantities [q1 (w), . . . , qn (w)] = [0, . . . , 0] are the unique Nash–Cournot equilibrium if and only if, for each i = 1, . . . , n,
Πi (qi (w), Q−i (w), w) − Πi (qi , Q−i (w), w) = 0[p(0) − w] − vi (0) − qi [p(qi ) − w] + vi (qi ) ≥ 0 for all qi ∈ (0, Q0 ]. Since vi (0) = 0, i = 1, . . . , n, the last inequality v (q ) is equivalent to the inequality w ≥ p(qi ) − i q i for all qi ∈ (0, Q0 ]. i This, in turn, is equivalent to the inequality
w ≥ sup
qi ∈(0,Q0 ]
p(qi ) −
vi (qi ) qi
,
i = 1, . . . , n.
(13)
Since, by (B2), vi (qi ), i = 1, . . . , n, is convex for qi > 0 it is easy v (q ) to see that i q i is nondecreasing for qi > 0. Together with the i fact that, by (B1), p(q) is nonincreasing for q ≥ 0, this implies that vi (qi ) p(qi ) − q is nonincreasing for qi > 0 and i
sup
qi ∈(0,Q0 ]
p(qi ) −
vi (qi ) qi
= lim
qi →0+
p(qi ) −
= p(0) − vi′ (0),
vi (qi )
qi i = 1, . . . , n.
(14)
Proof of Theorem 2. Quartieri [9] proved that if the conditions (B1)–(B4) hold, then, for w ≥ c, there exists a unique restricted single-tier Nash–Cournot equilibrium [q1 (w), . . . , qn (w)]. Since the conditions (B1)–(B4) also imply the conditions (A1)–(A3), we conclude that Lemmas 1 and 2 also hold under the conditions (B1)–(B4). The proof of Theorem 2 then utilizes Lemmas 1 and 2 and the MPEC problem defined in the proof of Theorem 1 and is analogous to the proof of Theorem 1 in KK. ∥
In view of inequality (14), inequality (13) is equivalent to w ≥ p(0) − vi′ (0), i = 1, . . . , n, which is equivalent to w ≥ max1≤i≤n [p(0) − vi′ (0)] = p(0) − min1≤i≤n vi′ (0). In view of the earlier remarks, this proves that the quantities [q1 (w), . . . , qn (w)] = [0, . . . , 0] are the unique Nash–Cournot equilibrium for all w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)]. (2) It follows from the equivalence of the restricted and the unrestricted single-tier Nash–Cournot equilibria and from part (1) of Lemma 3 that, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] exists and is such that qi (w) ∈ (0, Q0 ) solves problem (9). For any fixed w ≥ c, consider the objective function Πi (qi , Q−i (w), w), i = 1, . . . , n, in (9). Because qi (w) ∈ (0, Q0 ) maximizes Πi (qi , Q−i (w), w) on [0, Q0 ], for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), the first order necessary conditions
Proof of Lemma 3. (1) Since Lemma 2 holds as shown in the proof of Theorem 2, Y is nonempty-valued on [c , p(0)] and closed on [c , p(0)]. Quartieri [9] proved that if the conditions (B1)–(B4) hold,
(15)
∂ Πi (qi , Q−i (w), w) = p(qi + Q−i (w)) − w ∂ qi + qi p′ (qi + Q−i (w)) − vi′ (qi ) = 0
544
G.J. Kyparisis, C. Koulamas / Operations Research Letters 41 (2013) 540–544
hold at qi (w), i = 1, . . . , n. The result follows in view of the fact that conditions (15) hold at qi (w) if and only if [q1 (w), . . . , qn (w)] solves the system of Eq. (6). ∥ Proof of Lemma 4. By part (2) of Lemma 3, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), there exists a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n which solves the system of nonlinear equations (6). By our assumptions, the functions Fi (q1 , . . . , qn ), i = 1, . . . , n, are twice continuously differentiable. Thus, by differentiating the system of equations (6) evaluated at [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n with respect to w , we obtain the following system of equations
∂ F1 ∂ q1 ... ∂ Fn ∂ q1
∂ F1 ′ q (w) 1 ∂ qn 1 ... ... = ... , ∂ Fn q′n (w) 1 ∂ qn
... ... ...
(16)
where the leftmost matrix is evaluated at [q1 (w), . . . , qn (w)] ∈ (0, Q0 )n . Denote the three matrices in (16) as [JF (w)], [Dq(w)] and [1] respectively, where [JF (w)] denotes the Jacobian matrix of the vector of functions Fi (q1 , . . . , qn ), i = 1, . . . , n, evaluated at [q1 (w), . . . , qn (w)], [Dq(w)] denotes the vector of the first order derivatives of functions qi (w), i = 1, . . . , n, and [1] denotes the vector of 1s. Thus, (16) can be written compactly as [JF (w)] [Dq(w)] = [1]. Let [JF (w)]−1 denote the inverse of [JF (w)] evaluated at [q1 (w), . . . , qn (w)] which exists by assumptions (B7). We can then express [JF (w)]−1 as follows a11 (q1 , . . . , qn )
... an1 (q1 , . . . , qn )
[JF (w)]−1 =
... ... ...
a1n (q1 , . . . , qn ) ... , ann (q1 , . . . , qn )
(17)
(18)
Consider the expression Q (w) = j=1 qj (w) for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). In view of (16), the first order derivative of Q (w), denoted by Q ′ (w), can be calculated as follows
n
Q ′ (w) =
n
q′j (w) = [1]T [Dq(w)],
(19)
j =1
where AT denotes the transpose of matrix A. By combining (17)–(19), we obtain Q ′ (w) = [1]T [JF (w)]−1 [1]
=
n
aij (q1 (w), . . . , qn (w)).
(20)
i,j=1
In view of (20), the second order derivative of Q (w), denoted by Q ′′ (w), can be calculated as follows n d aij (q1 (w), . . . , qn (w)) d w i ,j = 1 n ∂ aij ′ ∂ aij ′ = q1 (w) + · · · + qn (w) ∂ q1 ∂ qn i,j=1
Q ′′ (w) =
∂a
n n n ∂ aij ∂ aij a1l + · · · + anl , Q (w) = ∂ q1 l=1 ∂ qn l=1 i,j=1 ′′
(22)
∂a
where the quantities aij , ∂ qij i, j, k = 1, . . . , n, are evaluated at k [q1 (w), . . . , qn (w)]. ∥ Proof of Theorem 3. Theorem 3 implies that a Stackelberg–Nash– Cournot equilibrium [w ∗ , q∗1 , . . . , q∗2 ] exists. Moreover, Lemma 3 implies that a unique Nash–Cournot equilibrium [q1 (w), . . . , qn (w)], which satisfies (5), exists for all w ∈ (c , p(0)]. The equivalence of the restricted and the unrestricted two-tier Stackelberg– Nash–Cournot equilibria (which follows from Lemma 1) implies that the profit function Π0 (w) = Q (w)(w − c ) in (10) is maximized at w ∗ ∈ (c , p(0)]. Observe that Π0 (c ) = 0, Π0 (w) = 0 for w ∈ [p(0) − min1≤i≤n vi′ (0), p(0)] and Π0 (w) = Q (w)(w − c ) > 0 for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Therefore, the problem (10) is maximized at w ∗ ∈ (c , p(0) − min1≤i≤n vi′ (0)). By Lemma 4, for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)), Q ′ (w) and Q ′′ (w) are well defined. Thus, the first and second order derivatives of the profit function Π0 (w) = Q (w)(w − c ) are given by Π0′ (w) = Q ′ (w)w + Q (w) − Q ′ (w)c and Π0′′ (w) = Q ′′ (w)w + 2Q ′ (w) − Q ′′ (w)c, respectively. It follows that Π0′′ (w) < 0 if Q ′′ (w) ≤ 0 and Q ′ (w) < 0 since w ≥ c. In view of (7), the condition Q ′ (w) < 0 is equivalent to the condition n
aij (q1 (w), . . . , qn (w)) < 0.
(23)
i,j=1
where the quantities aij (q1 , . . . , qn ), i, j = 1, . . . , n, are evaluated at [q1 (w), . . . , qn (w)] and are well defined for all w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Thus, the system (16) implies that
[Dq(w)] = [JF (w)]−1 [1].
k = 1, . . . , n. This fact combined with (21), implies that
(21)
where the quantities ∂ qij , i, j, k = 1, . . . , n, are evaluated at k [q1 (w), . . . , qn (w)]. In view of (17) and [Dq(w)] = [JF (w)]−1 (18), n ′ [1] is the vector of quantities qk (w) = l=1 akl (q1 (w), . . . , qn (w)),
In view of (8), the condition Q ′′ (w) ≤ 0 is equivalent to the condition
n n n ∂ aij ∂ aij a1l + · · · + anl ≤ 0. ∂ q1 l=1 ∂ qn l=1 i,j=1
(24)
In view of the assumptions (B8)–(B9) of Theorem 3, the inequalities in (23) and (24) hold which implies that Q ′′ (w) ≤ 0 and Q ′ (w) < 0 and thus that Π0′′ (w) < 0 for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). Therefore, Π0 (w) is strictly concave for w ∈ (c , p(0) − min1≤i≤n vi′ (0)). This implies that w ∗ is the unique solution of problem (10), [w ∗ , q∗1 , . . . , q∗n ] is the unique restricted Stackelberg–Nash– Cournot equilibrium. In view of the equivalence of the restricted and the unrestricted two-tier Stackelberg–Nash–Cournot equilibria, this completes the proof. ∥ References [1] R. Amir, Cournot oligopoly and the theory of supermodular games, Games and Economic Behavior 15 (1996) 132–148. [2] R. Amir, V.E. Lambson, On the effects of entry in cournot markets, Review of Economic Studies 67 (2000) 235–254. [3] C. Ewerhart, Cournot oligopoly and concavo-concave demand. Department of Economics, University of Zurich, Working Paper No. 16, 2011. [4] G. Gaudet, S.W. Salant, Uniqueness of cournot equilibrium: new results from old methods, Review of Economic Studies 58 (1991) 399–404. [5] P.T. Harker, J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Mathematical Programming 48 (1990) 161–220. [6] G.J. Kyparisis, C. Koulamas, A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers, Operations Research Letters 39 (2011) 471–474. [7] F.H. Murphy, H.D. Sherali, A.L. Soyster, A mathematical programming approach for determining oligopolistic market equilibrium, Mathematical Programming 24 (1982) 92–106. [8] J.S. Pang, Partially B-regular optimization and equilibrium problems, Mathematics of Operations Research 32 (2007) 687–699. [9] F. Quartieri, Necessary and sufficient conditions for the existence of a unique Cournot equilibrium, Working Paper, 2008. http://www.siepi.luiss.it/cms/ media/2/20080715-Quartieri-Paper.pdf. [10] H.D. Sherali, A.L. Soyster, F.H. Murphy, Stackelberg–Nash–Cournot equilibria: characterizations and computations, Operations Research 31 (1993) 253–276.