On Walras-Cournot-Nash equilibria and their computation

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1 On Walras-Cournot-Nash equilibria and their computation Jiří V. Outrata, Michal Červinka, Michal Outrata Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic Faculty of Social Sciences, Charles University in Prague, Czech Republic Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic December 15, 2014 Abstract This paper considers a hierarchical model of Walras-Cournot-Nash (WCN) equilibrium where Cournot-Nash concept is used to model equilibrium on an oligopolistic market with non-cooperative players/producers who share a certain amount of socalled rare resource needed for their production and the Walras equilibrium specifies the price of such rare resource. We provide stationarity conditions and existence result for WCN equilibria. Using the primal gap function we reformulate the problem to Mathematical Program with Equilibrium Constraints. Under mild assumptions we apply the Implicit programming approach to derive a suitable numerical procedure. We illustrate performance of our procedure on an academic example and comment on alternative numerical approaches to the problem. Key Words: Walras-Cournot-Nash equilibrium, Mathematical Program with Equilibrium Constraints, stationarity conditions, existence, Implicit programming approach, exact penalty. 1 Introduction and preliminaries In the literature one finds several types of hierarchical games which include, among other things, mathematical programs with equilibrium constraints (MPECs) and equilibrium problems with equilibrium constraints (EPECs). Recently, S.D.Flåm introduced another model of a similar type and called it Nash-Walras equilibrium [3]. It has been motivated by markets where the players/firms behave non-cooperatively and some of their inputs are limited but transferable. These so-called rare resources are controlled by some national or international authority that typically provides each agent with some initial endowment of these resources. Examples of such rare resources include fish quotas or rights to water usage. In the same way one can also handle production allowances or pollution permits. Since these rare resources are transferable, after the initial allocation they may be bought or sold in a market. This eventually leads to a Walras equilibrium specifying the equilibrium price of a unit of rare resources. This price is either nil, in case when the available amount of rare resources exceeds the interests of the market, or it is nonnegative provided that the demand amounts exactly to the available quantity. In either case, the initial endowments can be reallocated, which leads to a joint improvement. In [3] the author 1

2 divides the process of finding a Nash-Walras equilibrium into two phases; in the first one, the agents compute a Nash equilibrium corresponding to their initial endowments. In the second phase, the agents approach a Nash-Walras equilibrium step by step by bilateral exchanges of their shares of rare resources so that the overall amount of them remains unchanged. In this way the author attempts to model real processes leading to an equilibrium price of rare resources. In contrast to this approach, in this paper we look at this problem from a slightly different perspective. We understand this game as a bi-level one; on the upper level, there is a fictitious player whose objective is to determine the price of rare resources in such a way that a Walras equilibrium will be attained, provided that the agents, being the lower level players, apply their optimal noncooperative (Nash) strategies. This fictitious player could be the aforementioned national or international authority, trying to investigate for instance the influence of their initial allocation of the rare resources on the whole market. In this paper, we suggest a procedure for computing a Nash-Walras equilibrium in one step, without any phases and evolutionary processes. Since our agents are producers and behave according to the Cournot-Nash concept, we prefer to use the terminology Walras-Cournot-Nash (WCN) equilibrium throughout the whole sequel. The plan of the paper is as follows: In Section 2 we formulate the problem, collect the standing assumptions and analyze some elementary properties. In Section 3 the existence of a WCN equilibrium is proved. The rest of the paper is devoted to several numerical approaches enabling us to compute WCN equilibria by existing software. The first two of them are based on a reformulation to an MPEC by means of the standard primal gap function used in complementarity problems. The resulting MPEC can be solved by a variant of the Implicit programming approach (ImP) from [11], [14] which permits also state constraints. This is conducted in Section 4. Another possibility is to apply a regularization approach, developed in [6] for mathematical programs with complementarity constraints (MPCCs). This technique, working even under less restrictive assumptions, is examined in Section 5. Our notation is basically standard. For a closed cone K with vertex at 0, K 0 denote its negative polar and for a set A, dist A (x) stands for the distance of x to A. Given a multifunction F [R n R m ], GrF := {(x, y) R n R m y F (x)} is the graph of F and B denotes the unit ball. We conclude the introductory section with the definitions of some basic notions from modern variational analysis which will be extensively used in this paper. Consider a closed set A R n and x A. We define the contingent (Bouligand) cone to A at x as the cone T A ( x) := Lim sup τ 0 A x τ = {h R n h i h, λ i 0 such that x + λ i h i A} and the regular (Fréchet) normal cone to A at x as N A ( x) := (T A ( x)) 0. Moreover, the limiting (Mordukhovich) normal cone to A at x is defined by N A ( x) := Lim sup A = {x R n x x N(x) A x k x, x k x such that x k N A (x k ) for all k}. 2

3 We say that A is (normally) regular at x provided N A ( x) = N A ( x). Convex sets are regular at all points. Now consider a closed-graph multifunction Φ[R n R m ] and a point ( x, ȳ) Gr Φ. The multifunction D Φ( x, ȳ)[r m R n ] defined by D Φ( x, ȳ)(y ) := {x R n (x, y ) N GrΦ ( x, ȳ)} is the regular (Fréchet) coderivative of Φ at ( x, ȳ). The multifunction D Φ( x, ȳ)[r m R n ] defined by D Φ( x, ȳ)(y ) := {x R n (x, y ) N GrΦ ( x, ȳ)} is the limiting (Mordukhovich) coderivative of Φ at ( x, ȳ). Let f[r n R] be lsc and x dom f. The set ˆ f( x) := {v R n (v, 1) N epif ( x, f( x))} is called the regular (Fréchet) subdifferential of f at x and its elements are termed the regular (Fréchet) subgradients of f at x. Analogously, the set f( x) := {v R n (v, 1) N epif ( x, f( x))} is called the limiting (Mordukhovich) subdifferential of f at x and its elements are termed the limiting (Mordukhovich) subgradients of f at x. We say that Φ[R n R m ] is calm at ( x, ȳ), provided neighborhoods U of x, V of ȳ and a real L 0 such that Φ(x) V Φ( x) + L x x B x U. In Section 4 we will employ the following result. Proposition 1. Let F [R n R m ] be Lipschitz around x and f[r m R] be Lipschitz around F ( x). Then for the composition g = f F one has that g( x) D F ( x, ȳ)(ξ) ξ f(ȳ) with ȳ = F ( x). If f is continuously differentiable, around ȳ then g( x) = D F ( x, ȳ)( f(ȳ)). The first statement follows from [15, Theorem 10.49], the second one from [8, Theorem 1.110(ii)]. An interested reader may find a full account of properties of the above notions e.g. in the monographs [15] and [8]. 3

4 2 Problem formulation Consider an oligopolistic market with m producers, who produce a homogeneous commodity. As mentioned in the introduction, for this production each of them needs a certain amount of a rare resource dependent on the used technology. It follows that each producer optimizes his profit by using two strategies: His production and the amount of rare resource which he intends to sell or purchase. Consequently, the ith producer solves the optimization problem minimize subject to c i (y i ) + πx i p(t )y i (y i, x i ) (A i R) B i, (1) where y i is the production, x i is the amount of purchased (sold) rare resource, c i [R + R + ] specifies the production costs, π is the price of the rare resource, T = m y i signifies the overall amount of the produced commodity in the market and A i = [a i, b i ] specifies the production bounds. The function p[int R + R + ] assigns each amount T the price at which (price-taking) consumers are willing to demand. It is usually called the inverse demand curve. The relationship between y i and the needed amount of rare resource is reflected via the set B i = {(y i, x i ) q i (y i ) x i + e i }, where e i is the initial endowment with the rare resource and q i [R + R + ] is a (technological) function assigning each production value the corresponding (needed) amount of rare resource. Observe that in problem (1) the variables y j, j i, and π play a role of parameters. Throughout the whole paper we will impose the following assumptions: A1: All functions c i can be extended to open intervals containing the sets A i. These extensions are convex and twice continuously differentiable. A2: p is strictly convex and twice continuously differentiable on int R +. A3: αp(α) is a concave function of α. A4: For all i one has 0 a i < b i and there is an index i 0 such that a i0 > 0. A5: All functions q i fulfill q i (0) = 0 and can be extended to open intervals containing the sets A i. These extensions are convex, increasing and twice continuously differentiable. A6: For all i one has q i (a i ) < e i. A7: π 0. The assumptions A1 - A3 are not too restrictive and arise in a similar form in most works about oligopolistic markets, cf. [9], [11]. They ensure in particular that the objective in (1) is convex for all i. Assumption A4 ensures that p(t ) is well-defined. Assumptions A5 A5, A6 are related to the rare resource and play an important role both in the existence 4

5 proof in the next section as well as in the first numerical approach discussed in Section 4. The economic interpretation of A6 says that the initial endowment with the rare resource enables each firm to produce more than its lower production bound. Finally, A7 is natural. Denote by J i, i = 1, 2,..., m, the objective in (1) and by Ξ the overall available amount of the rare resource so that m Ξ e i. (2) Further, to simplify the notation, y = (y 1, y 2,..., y m ) and x = (x 1, x 2,..., x m ) stand for the vectors of cumulative strategies y i, x i of all producers. To introduce the WCN equilibrium, we define first the Cournot-Nash equilibrium generated by problems (1). Definition 1. The strategy pair (ȳ, x) is a Cournot-Nash equilibrium in the considered market for a given π 0 provided for all i one has J i (π, ȳ, x i ) = min J i (π, ȳ i, ȳ 2,..., ȳ i 1, y i, ȳ i+1,..., ȳ m, x i ) (3) (y i,x i ) (A i R) B i Remark 1. If the constraint (2) is neglected and all endowments e i vanish (so that we do not have to do with a rare resource), then we may put x i = q(y i ), the production costs amount to c i (y i ) + πq i (y i ) and the constraint set in (1) can be simplified to y i A i. Via Definition 1 we obtain then the classical notion of Cournot (or Cournot-Nash) equilibrium from 1838, cf. [9]. From this reason we use the terminology Cournot-Nash equilibrium also in our slightly more complex case reflecting the above described mechanism of trading with the rare resource. Remark 2. In [3] the author assumes that the production cost functions c i also depend on x i. Definition 2. (Flåm) The triple ( π, ȳ, x) is a Walras-Cournot-Nash (WCN) equilibrium in the considered market provided that (i) (ȳ, x) is a Cournot-Nash equilibrium for π = π, and (ii) one has π 0, Ξ m (e i + x i ) 0, π (Ξ m (e i + x i )) = 0. Clearly, the conditions in (ii) characterize a Walras equilibrium with respect to the rare resource which determines a price π under which the (secondary) market with the rare resource is cleared. From the point of view of the producers the computation of π is a dynamical process starting after the initial allocation has been conducted. From the point of view of the authority, controlling the rare resource, however, the whole problem can be solved in one step. The results provide the authority with information about the influence of the initial allocation on the WCN equilibrium. The Cournot-Nash equilibrium from Definition 1 can easily be characterized via standard stationarity/optimality conditions. For the readers convenience we repeat this result here, because we will make a strong use of it in the next section. 5

6 Proposition 2. Given a price π 0, under the posed assumptions, a pair (ȳ, x) is a Cournot-Nash equilibrium in the sense of Definition 1 if and only if it fulfills the relations c 1 (y 1 ) ȳ 1 p(t ) p(t ) q 1 (y 1 ) m π.. + X N Ai (y i ) (4) c m (y m ) y m p(t ) p(t ) q m (y m ) π (q i (y i ) x i e i ) = 0, (5) q i (y i ) e i + x i, i = 1, 2,..., m. Proof. The constraints in (3) satisfy the linear independence constraint qualification (LICQ) due to A4. Moreover, the standing assumptions ensure that the functions J i are jointly convex in (x i, y i ). The Cournot-Nash equilibria are henceforth characterized by the standard first-order optimality conditions for the single optimization problems (3). Putting them together, we obtain the generalized equation (GE) c 1 (y 1 ) ȳ 1 p(t ) p(t ) λ 1 q 1 (y 1 ) π λ 1 0. c m (y m ) ȳ m p(t ) p(t ) π + m X N A i R(y i, x i ) +. λ m q m (y m ) λ m, (6) where λ 1,..., λ m are nonnegative Lagrange multipliers associated with the inequalities defining the sets B i. They must fulfill the complementarity conditions λ i (q i (y i ) x i e i ) = 0 for all i = 1, 2,..., m. (7) Since N R (x i ) = {0} for all i, we immediately conclude that π = λ 1 = λ 2 =... = λ m. (8) In this way we arrive at the simplified (but equivalent) conditions (4), (5) in which only the partial derivatives yi J i arise. 3 Existence of WCN equilibria As announced in the introduction, the considered oligopolistic market can be viewed as a hierarchical bilevel game with the producers acting on the lower level and one additional player on the upper level, whose strategy is π and whose aim is to satisfy the conditions in Definition 2(ii). Let us associate with the ith producer, instead of (1), a different problem, namely minimize subject to c i (y i ) + π(q i (y i ) e i ) p(t )y i y i A i (9) 6

7 in variable y i only. It corresponds to replacing the inequality q(y i ) x i + e i by equality so that variable x i can be completely eliminated. If we replace functions J i in Definition 1 by the objectives from (9), we obtain a different non-cooperative equilibrium characterized by the GE 0 c 1 (y 1 ) + π q 1 (y 1 ) y 1 p(t ) p(t ). c m (y m ) + π q m (y m ) y m p(t ) p(t ) + m X N A i (y i ). (10) Lemma 1. Let ȳ satisfy condition (10). Then the pair (ȳ, x) with x i = q i (ȳ i ) e i for all i is a Cournot-Nash equilibrium in the sense of Definition 1. Conversely, for each solution (ȳ, x) of system (4), (5) the component ȳ fulfills GE (10) whenever π > 0. The proof follows immediately from the comparison of GE (10) with conditions (4), (5). Denote by S[R + R m ] the mapping which assigns each π 0 the set of solutions to GE (10). The statement of Lemma 1 can then be written down as follows: (i) For any π 0 one has the implication y S(π), x i = q i (y i ) e i for all i (y, x) fulfills conditions (4), (5). (ii) For π > 0 the above implication becomes equivalence. Lemma 2. The mapping S is single-valued and locally Lipschitz over R +. Proof. Let H denote the single-valued part of GE (10). We claim that at any pair ( π, ȳ) m R + X A i the partial Jacobian y H( π, ȳ) is positive definite. To prove it, consider the symmetrized partial Jacobian of H with respect to y, i.e., 2 q 1 (y 1 ) 1 2 ( yh(π, y) + y H(π, y) T ) + π... (11) 2 q m (y m ) with H(π, y) = c 1 (y 1 ) y 1 p(t ) p(t ). c m (y m ) y m p(t ) p(t ) Under the posed assumptions the first term in (11) is positive definite at ( π, ȳ) as proved in [11, Lemma 12.2]. Moreover, the second one is positive semidefinite by virtue of A5 and so the claim follows. From this property we deduce by virtue of [11, Theorem 5.8] that GE (10) is strongly regular (in the sense of Robinson) at ( π, ȳ) and so S admits a single-valued and Lipschitz localization around ( π, ȳ). Since the choice of ( π, ȳ) was arbitrary, we conclude that. 7

8 m (i) H(π, ) is strictly monotone over X A i ([10, Theorem 5.4.3]) so that S is singlevalued over R + ([11, Theorem 4.4(i)]); (ii) S admits a single-valued and Lipschitz localization around each pair (π, y) with π 0 and y = S(π). A combination of these two facts implies the statement of the lemma. In the last auxiliary statement of this section we will show that in each WCN equilibrium the price π admits an upper bound dependent solely on the problem data. Lemma 3. There is a positive real L such that in all WCN equilibria one has π < L. Proof. Assume that π > 0 is so large that { ( m ) ( m ) } c i (a i ) a i p a i p a i + π i (a i ) > 0. (12) min,...,m By virtue of (12) it follows that the stationarity condition (4) can be fulfilled only in the case when y i = a i for all i. Indeed, since the functions J i are convex in variables (y i, x i ), their partial derivatives with respect to y i are nondecreasing, and so for y i a i the quantities c i (y i ) y i p(t ) p(t ) + π q i (y i ) are positive as well. It follows that y i = a i for all i in order to bring the normal cones to A i into play. By (7) and (8) the respective values of x i are given by x i = q i (a i ) e i and thus, thanks to assumption A6, all of them are negative. This implies, however, that the corresponding excess demand m (e i + x i ) Ξ is negative as well, which contradicts the complementarity condition of the Walras equilibrium. As L we can thus choose any positive real satisfying inequality (12) with π replaced by L. On the basis of Lemmas 1-3 we are now able to state our main existence result. Theorem 1. Under the posed assumptions there is a WCN equilibrium. Proof. Define the mapping Q[R m R] by Q(y) := n q i (y i ). By virtue of Lemma 1 it suffices to show the existence of π 0 such that which amounts to the GE Ξ Q S(π) 0, π (Ξ Q S(π)) = 0, Thanks to Lemma 3, GE (13) can be replaced by the GE 0 Ξ Q S(π) + N R+ (π). (13) 0 Ξ Q S(π) + N [0,L] (π) with a bounded constraint set. Since Q S is single-valued and continuous over [0, L] by virtue of Lemma 2, GE (14) possesses a solution π as a consequence of the Brouwer Fixed Point Theorem. It follows that ( π, ȳ, x) with ȳ = S( π) and x i = q i (ȳ i ) e i is a WCN equilibrium. 8

9 4 Computation of WCN equilibria via ImP. There are several efficient methods for the numerical solution of MPECs and MPCCs. The representation of the WCN equilibrium as a bilevel game (which has been used in the preceding section in our existence result) enables us to employ the so-called primal gap function ([2, formula ]) and arrive eventually at the following MPEC: minimize subject to π (Ξ Q(y)) y = S(π) π 0 Ξ Q(y) 0, where π is the control variable and y is the state variable. In (14) we have to do with the equilibrium constraint generated by GE (10), a simple control constraint π 0 and a (not so simple) state constraint Ξ Q(y) 0. By virtue of Lemma 1 any solution ( π, ȳ) of (14) generates a WCN equilibrium with x i = q i (ȳ i ) e i provided the optimal value of the objective in (14) is zero. To the numerical solution of (14) we apply a variant of the Implicit programming approach (ImP) which is able to deal with state constraints ([14]). The terminology ImP has been coined in [4] and this approach has been deeply investigated e.g. in the monographs [7] and [11]. Given a general MPEC minimize J (α, z) subject to z S(α) α ω, where α is the control variable, z is the state variable and S is (at least in the neighborhood of an optimal control) locally single-valued and Lipschitz, the main idea consists in the replacement of (15) by the optimization problem minimize subject to Θ(α) α ω, where Θ( ) := J (, s( )) and s stands for the appropriate single-valued localization of S around the solution. In our case, however, we have to do also with a state constraint which complicates the above reformulation. In [14] the authors have suggested to respect state constraints in this framework via an exact penalty and this technique will now be applied in the case of MPEC (14). Theorem 2. [14, Proposition 3] Let ( π, ȳ) be a solution of MPEC (7) and assume that the perturbation mapping M[R R + ] defined by M(d) = {π R + Ξ Q S(π) d} is calm at (0, π). Then there is a positive real R such that π is a solution of the (augmented or penalized) program minimize π (Ξ Q S(π)) + R[(Q S(π) Ξ) + ] subject to π 0 9 (14) (15) (16) (17)

10 in variable π only. The verification of the above required calmness property is, however, a nontrivial task. We employ to this purpose a result from [5] which is based on the following notion. Definition 3. Let f[r n R] be lsc and x domf. The set > f( x) := Lim sup ˆ f(x) x x f(x) f( x) f(x) f( x) is called the limiting outer subdifferential of f at x. This notion does not directly arise in the calmness criterion below, but plays a crucial role in its proof. Theorem 3. Assume that ȳ = S( π) and Then M is calm at (0, π). 0 D S( π, ȳ) q 1 (ȳ 1 ). q m (ȳ m ) (18) Proof. It is well-known that the calmness of M at (0, π) is equivalent with the local error bound property of the function { (Q S(π) Ξ)+ if π 0 β(π) := otherwise at π. This means that for Λ := {π R β(π) = 0} there is a real c > 0 such that dist Λ (π) cβ(π) for all π 0 close to π. By virtue of [5, Theorem 2.1] the latter property is ensured by the subdifferential condition 0 > β( π) and so it remains to show that > β( π) is included in the set on the right-hand side of (18). Since β( π) = 0 and β(π) β( π) for π R + by Lemma 2, one has that > β( π) := Lim sup ˆ β(π) = Lim sup β(π). π R + π β(π)>0 π R + π β(π)>0 Assume now that β(π) > 0. Then β is continuously differentiable near π and according to Proposition 2 β(π) = D S(π, y)( q(y)) with y = S(π). Consider now an arbitrary sequence {π i } R + converging to π. Since y i = S(π i ) converges to ȳ = S( π) and Q(y i ) converges to Q(ȳ) = ( q 1 (ȳ 1 ),..., Q m (ȳ m )) T, it follows that Lim sup D S(π i, y i )( Q(y i )) D S( π, ȳ)( Q(ȳ)) i by the definition of the limiting coderivative and we are done. 10

11 The above criterion is workable only in the case when we are able to compute D S( π, z). Proposition 3. Let ȳ = S( π). Under the posed assumptions for any y R m one has D N A1 (ȳ 1 H 1 ( π, ȳ))(u 1 ) D S( π, ȳ)(y ) Q(ȳ), u 0 y + ( y H( π, ȳ)) T u +. D N Am (ȳ m H m ( π, ȳ))(u m ). (19) Proof. It suffices to apply [8, Corollary 4.48] and to take into account that the respective qualification condition is fulfilled due to the Lemma 2 and [13, Proposition 3.2]. The coderivative D N m (ȳ, H( π, ȳ)) can be expressed in the form arising in (19) thanks X A i to [15, Proposition 6.41]. Since all the sets A i are non-degenerate intervals, the coderivatives D N Ai can easily be computed, cf. [1, Section 4] and [12, ]. When ȳ i (a i, b i ) for all i, then the calmness of M at (0, π) is ensured by the implication q 1 (ȳ 1 ) 0 =. + ( y H( π, ȳ)) T b Q(ȳ), b 0. (20) q m (ȳ m ) However, this condition holds true by virtue of A5 and the positive definiteness of y H( π, ȳ). It follows that our calmness condition could be violated only in the case when one of the producers attains his production limits. Since the objective in (17) is locally Lipschitz, this optimization problem can be solved by a bundle method of nonsmooth optimization. These methods typically require the user to be able to compute at each π 0 the corresponding value of the objective and one arbitrary element of its Clarke or limiting subdifferential. As this element we will use a vector ξ computed in the following two steps: 1 Given a pair (ˆπ, ŷ) with ŷ = S(ˆπ) and a penalty parameter R > 0, solve the adjoint equation system 0 y + ( y H(ˆπ, ŷ)) T u + w (21) (w i, u i ) L i, i = 1, 2,..., m in variables u, w, where L i are linear subspaces of the cones N grnai (ŷ i, H(ˆπ, ŷ)) and q 1 (ŷ 1 ) { y 0 if Ξ Q(ŷ) 0 = ( ˆπ + v). with v = R otherwise; q 2 (ŷ 2 ) 2 Put ξ := Ξ Q(ŷ) + Q(ŷ), u. 11

12 Table 1: Parameter specification Firm 1 Firm 2 Firm 3 Firm 4 Firm 5 q i c i K i β i Via system (21) we compute a solution of the GE arising at the right-hand side of (19). Indeed, the cones N gph NAi are of a very simple nature ([1, Section 4], [12, Section 2]) and always contain suitable linear subspaces L i. In this way the computation of ξ is simplified a lot. ξ is a correct subgradient whenever the sets GrA i are regular at (ŷ i, H i (ˆπ, ŷ)) for all i ([15, Theorem 6.14]). The computational experience with the application of ImP to MPECs shows that during the iteration process we typically always meet this type of regularity. But even an occasional occurrence of a point where the subgradient information could be false does not usually cause the bundle method to collapse. The numerical results, obtained via the above described technique, are presented in the next section. 5 Numerical results In this section, to test the performance of the proposed variant of ImP we modify correspondingly the oligopolistic market example from [9], see also [11, Section 12.1] and [14, Example 3]. Analogously to these references, to obtain numerical results, we applied the Bundle-Trustregion code BT described in [18]. In order to solve the generalized equation (10), we have used an idea of Smith [19] and solved the resulting convex optimization problems via the sequential quadratic programming code NLPQL [16]. Consider an example of five producers/firms supplying a quantity y i R +, i = 1,..., 5 of some homogeneous product on the market with the inverse demand function p(t ) = γ T 1 γ, where γ is a positive parameter termed demand elasticity. Let the production cost functions be in the form β i c i (y i ) = c i y i + K i 1 + β i 1 β i (y i ) 1+β i β i, where c i, K i and β i, i = 1,..., 5, are positive parameters. Let the technological functions q i be linear in the form q i (y i ) = q i y i, i = 1,..., 5. Table 1 specifies values of parameters q i, c i, K i and β i, i = 1,..., 5. Further, let the demand elasticity γ = 1.3, initial endowments with rare resource e i = 25 for each Firm i, i = 1,..., 5, and consider production bounds A i = [0, 30], i = 1,..., 4 and A 5 = [1, 30]. Each production cost function is convex and twice continuously differentiable on some open set containing the feasible set of strategies of a corresponding player. The inverse 12

13 demand curve is twice continuously differentiable on int R +, strictly decreasing, and convex. Observe that the so-called industry revenue curve T p(t ) = γ 1 γ T γ is concave on int R + for γ 1. Thus, all assumptions A1 - A7 are satisfied. For the numerical tests, we have first considered a situation where there are no additional units of rare resource available on the market (case A). In addition, we will consider the following modifications of this example: additional 10 units of rare resource is available (case B); Firm 1 has better production technology such that c 1 = 5 (case C); upper bounds on production are increased to 35 and initial endowments with rare resource are increased to 45 (case D) upper bounds on production are lowered to 23 (case E); Producer 1 has worse technological function such that q 1 = 4 (case F); Producer 1 has worse technological function such that q 1 = 4 and upper bounds on production are lowered to 23 (case G). All results have been computed in Fortran 77 with the accuracy ɛ = in BT and ɛ = in the sequential quadratic programming code NLPQL and the starting point has been set to 50 for variable π and upper production bounds for variables y i, i = 1,..., 5. 13

14 Table 2: Production, profits and purchased rare resources Firm 1 Firm 2 Firm 3 Firm 4 Firm 5 case A π = production profit purchased rare resource case B π = production profit purchased rare resource case C π = production profit purchased rare resource case D π = 0 production profit purchased rare resource case E π = production profit purchased rare resource case F π = production profit purchased rare resource case G π = production profit purchased rare resource

15 In Table 2 we present the productions, profits and purchased rare resource of each producer along with price of the rare resource and value of the objective of (14) in each of the above specified variants of the example. Note that case D corresponds to a situation when the available amount of rare resources exceeds the interests of the market and thus this price vanishes. The firms 1 and 2 would like to sell a certain amount of rare resource but since the remaining firms are not willing to buy this whole amount for any positive price, the resulting π = 0. Cases E, F and G illuminate the effects of attaining the lower or upper production bounds. We conclude this section with a remark on application of an alternative solution method for MPEC (14). In this method, the original MPEC is replaced by a sequence of NLPs solved by standard techniques that depend on a certain parameter t such that for t 0 solutions of these relaxed NLPs approach the solution (or at least a stationary point of certain type) of the original problem. In the light of recent results from [6], we have applied a MATLAB implementation of the method by Scholtes [17] to the same examples as above. Since the obtained results differ from those reported above only up to chosen computational accuracy, we do not report them in a separate table. Acknowledgement The authors would like to express their gratitude to Alexandra Schwartz for providing her MATLAB codes implementing a family of relaxation methods for solving MPECs. This work was partially supported by the Grant Agency of the Czech Republic under Grant P402/12/1309 and by the Grant Agency of the Charles University by Grant SFG References [1] A.L. Dontchev, R.T. Rockafellar: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optimization 7(1996), [2] F. Facchinei, J. Pang: Finite-Dimensional Variational Inequalaities and Complementarity Problems. Springer, New York, [3] S.D. Flåm: Noncooperative games, coupling constraints and partial efficiency. Preprint, [4] P.T. Harker, J.-S. Pang: On the existence of optimal solutions to mathematical program with equilibrium constraints. Oper. Res. Letters 7(1988), [5] A.D. Ioffe, J.V. Outrata: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16(2008), [6] Ch. Kanzow, A. Schwartz: The price of inexactness: convergence properties of relaxation methods for mathematical programs with equilibrium constraints revisited, to appear in Math. Oper. Res

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