On computation of optimal strategies in oligopolistic markets respecting the cost of change


The paper deals with a class of parameterized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability, under appropriate conditions, is established. This theory is then applied to evolution of an oligopolistic market in which the firms adapt their production strategies to changing input costs, while each change of the production is associated with some “costs of change”. We examine both the Cournot-Nash equilibria as well as the two-level case, when one firm decides to take over the role of the Leader (Stackelberg equilibrium). The impact of costs of change is illustrated by academic examples.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. Allgower EL, Georg K (1997) Numerical path following. In: Ciarlet PG, Lions JL (eds) Techniques of scientific computing, Part 2, Handbook of Numerical Analysis, vol 5. North-Holland, Amsterdam, pp 3–207

    Google Scholar 

  2. Aubin J-P (1998) Optima and Equilibria. Springer, Berlin

    Google Scholar 

  3. Basilico N, Coniglio S, Gatti N, Marchesi A (2020) Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games. EURO J Comput Optim 8:3–31

    MathSciNet  Article  Google Scholar 

  4. Brent RP (1973) Algorithms for minimization without derivatives. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  5. Dempe S. Bilevel optimization: theory, algorithms and applications. Preprint 2018-11, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg

  6. Dontchev AL, Rockafellar RT (2014) Implicit functions and solution mappings. Springer, Heidelberg

    Google Scholar 

  7. Facchinei F, Pang J-S (2003) Finite-dimensional variational inequalities and complementarity problems. Springer, Berlin

    Google Scholar 

  8. Flåm SD (2020) Games and cost of change. Ann Oper Res. https://doi.org/10.1007/s10479-020-03585-w

    Article  Google Scholar 

  9. Frost M, Kružík M, Valdman J (2019) Interfacial polyconvex energy-enhanced evolutionary model for shape memory alloys. Math Mech Solids 24:2619–2635

    MathSciNet  Article  Google Scholar 

  10. Gfrerer H, Outrata JV (2016) On Lipschitzian properties of implicit multifunctions. SIAM J Optim 26:2160–2189

    MathSciNet  Article  Google Scholar 

  11. Gfrerer H, Outrata JV (2019) On a semismooth\(^{*}\) Newton method for solving generalized equations. arXiv:1904.09167

  12. Kanzow Ch, Schwartz A (2018) Spieltheorie. Springer, Cham

    Google Scholar 

  13. Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic Press, New York

    Google Scholar 

  14. Mielke A, Roubíček T (2015) Rate-Independent systems - theory and applications. Springer, New York

    Google Scholar 

  15. Murphy MH, Sherali AD, Soyster AL (1982) A mathematical programming approach for determining oligopolistic market equilibria. Math Prog 24:92–106

    Article  Google Scholar 

  16. Outrata JV, Kočvara M, Zowe J (1998) Nonsmooth approach to optimization problems with equilibrium constraints. Kluwer, Dordrecht

    Google Scholar 

  17. Poliquin RA, Rockafellar RT (1998) Tilt stability of a local minimum. SIAM J Optim 8:287–299

    MathSciNet  Article  Google Scholar 

  18. Robinson SM (1976) An implicit function theorem for generalized variational inequalities. Technical Summary Report 1672, Mathematics Research Center, University of Wisconsin–Madison

  19. Rockafellar RT, Wets RJ-B (1998) Variational Analysis. Springer, Berlin

    Google Scholar 

  20. Zhu DL, Marcotte P (1996) Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J Optim 6:714–726

    MathSciNet  Article  Google Scholar 

Download references


The authors are deeply indebted to both Reviewers and the Associated Editor for their careful reading and numerous important suggestions. The also benefited from valuable advices of T. Roubíček.

Author information



Corresponding author

Correspondence to Jiří V. Outrata.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of J. V. Outrata was supported by the Czech Science Foundation (GA ČR), through the Grant 17-08182S and by the Australian Research Council, Project DP160100854F. The research of J. Valdman was supported by the Czech Science Foundation (GA ČR), through the Grant 17-04301S.



In some models of practical importance function q is piecewise linear-quadratic. Then the assumption of positive definiteness of \(\nabla _{x}F(\bar{p},\bar{x})\) in Proposition 1 can be somewhat relaxed.

Proposition 2

Assume that \(\tilde{q}\) is convex, piecewise linear-quadratic and the mapping \(\varXi : \mathbb {R}^{s} \rightrightarrows \mathbb {R}^{s}\) defined by

$$\begin{aligned} \varXi (w):=\left\{ k \in \mathbb {R}^{s} | w \in \nabla _{x}F(\bar{p},\bar{x}) k + \partial \varphi (k)\right\} \end{aligned}$$

with \(\varphi (k):=\frac{1}{2} d^{2}q (\bar{x}| -F(\bar{p},\bar{x}))(k)\) is single-valued on \(\mathbb {R}^{s}\). Then S has a single-valued and Lipschitzian localization around \((\bar{p},\bar{x})\).


By virtue of (Dontchev and Rockafellar 2014, Theorem 3G.4) it suffices to show that the single-valuedness of \(\varXi \) implies the existence of a single-valued and Lipschitzian localization of \(\varSigma \) (defined in (6)) around \((0,\bar{x})\). Clearly,

$$\begin{aligned} \mathrm {gph}\,\varSigma = \left\{ (w,x) \left| \left[ \begin{aligned}&\qquad x - \bar{x}\\&w - \nabla _{x}F(\bar{p},\bar{x})(x-\bar{x}) \end{aligned}\right] \in \mathrm {gph}\,\partial \tilde{q} -\left[ \begin{aligned}&\qquad \bar{x}\\&- F(\bar{p},\bar{x}) \end{aligned}\right] \right. \right\} \end{aligned}$$

so that \(\varSigma \) is a polyhedral multifunction due to our assumptions imposed on \( \tilde{q}\), cf. (Rockafellar and Wets 1998, Theorem 12.30). It follows from Robinson (1976) (see also (Outrata et al. 1998, Cor.2.5)) that due to the polyhedrality of \(\varSigma \), it suffices to ensure the single-valuedness of \(\varSigma (\cdot ) \cap \mathcal {V}\) on \(\mathcal {U}\), where \(\mathcal {U}\) is a convex neighborhood of \(0 \in \mathbb {R}^{s}\) and \(\mathcal {V}\) is a neighborhood of \(\bar{x}\). Let us select these neighborhoods in such a way that

$$\begin{aligned} \mathrm {gph}\,\partial \tilde{q} - \left[ \begin{aligned}&\qquad \bar{x}\\&- F(\bar{p},\bar{x}) \end{aligned}\right] = T_{\mathrm {gph}\,\partial \tilde{q}} (\bar{x}, - F(\bar{p},\bar{x})), \end{aligned}$$

which is possible due to the polyhedrality of \(\partial \tilde{q}\). Then one has

$$\begin{aligned}&\mathrm {gph}\,\varSigma \cap (\mathcal {U} \times \mathcal {V})=\{(w,\bar{x} +k)\in \mathcal {U} \\&\quad \times \mathcal {V} | w \in \nabla _{x}F(\bar{p},\bar{x})k+D\partial \tilde{q}(\bar{x},- F(\bar{p},\bar{x}))(k)\}. \end{aligned}$$

Under the posed assumptions for any \(k \in \mathbb {R}^{n}\)

$$\begin{aligned} D\partial \tilde{q}(\bar{x},- F(\bar{p},\bar{x}))(k) = \partial \varphi (k), \end{aligned}$$

cf. (Rockafellar and Wets 1998, Theorem 13.40), so that \(\mathrm {gph}\,\varSigma \cap (\mathcal {U} \times \mathcal {V})=\{(w,\bar{x} +k)\in \mathcal {U} \times \mathcal {V} | (w,k)\in \mathrm {gph}\,\varXi \}\). Since \(D \partial \tilde{q}(\bar{x},- F(\bar{p},\bar{x}))(\cdot )\) is positively homogeneous, \(\partial \varphi (\cdot )\) is positively homogeneous as well and so the single-valuedness of \(\varSigma (\cdot )\cap \mathcal {V}\) on \(\mathcal {U}\) amounts exactly to the single-valuedness of \(\varXi \) on \(\mathbb {R}^{s}\). \(\square \)

On the basis of (Rockafellar and Wets 1998, Proposition 13.9) the single-valuedness of \(\varXi \) can be ensured via the notion of copositivity. Recall that an [\(s \times s\)] matrix H is strictly copositive with respect to a cone \(\mathcal {K} \subset \mathbb {R}^s\) provided

$$\begin{aligned} \langle d, H d \rangle > 0 \quad \text{ for } \text{ all } d \in {\mathcal {K}}, d \not =0. \end{aligned}$$
Fig. 3

The set in equilibria \(S(p_1, p_2)\) (left) and the graphical derivative \(DS(\bar{p},\bar{x})(h_1,h_2)\) of Example 1

Proposition 3

Assume that \(\tilde{q}\) is convex, piecewise linear-quadratic and \( \tilde{q}^{\prime \prime }(\bar{x};\cdot )\) is convex. Further suppose that \(\nabla _{x}F(\bar{p},\bar{x})\) is strictly copositive with respect to \(K-K\), where

$$\begin{aligned} K:= \{k | \tilde{q}^{\prime }(\bar{x}; k)=\langle -F(\bar{p},\bar{x}), k\rangle \}. \end{aligned}$$

Then S has a single-valued and Lipschitzian localization around \((\bar{p},\bar{x})\).


By virtue of (Rockafellar and Wets 1998, Proposition 13.9) the second subderivative \(d^{2}\tilde{q}(\bar{x}| -F(\bar{p},\bar{x}))(\cdot )\) is proper convex and piecewise linear-quadratic and one has

$$\begin{aligned} \partial \varphi (k)= \partial \frac{1}{2}d^{2}\tilde{q}(\bar{x}| -F(\bar{p},\bar{x}))(k)= \partial \frac{1}{2} \tilde{q}^{\prime \prime }(\bar{x};k)+N_{K}(k). \end{aligned}$$

It remains to show that mapping (16) is single-valued. Clearly, the GE in (16) can be written down in the form

$$\begin{aligned} 0 \in \varPsi (k) - w +N_K(k), \end{aligned}$$

where the multifunction \(\varPsi (k):= \nabla _x F(\bar{p},\bar{x}) k + \partial \frac{1}{2} \tilde{q}''(\bar{x};k).\) As explained in (Outrata et al. 1998, Theorem 4.6), under the posed assumptions there is a positive real \(\alpha \) such that

$$\begin{aligned} \langle d, \nabla _x F(\bar{p},\bar{x}) d \rangle \ge \alpha || d||^2 \quad \text{ for } \text{ all } d \in K-K. \end{aligned}$$

It follows that for all \(k_1, k_2 \in K, \xi _1 \in \partial \frac{1}{2} \tilde{q}''(\bar{x};k_1), \xi _2 \in \partial \frac{1}{2} \tilde{q}''(\bar{x};k_2)\) and

$$\begin{aligned} \eta _1= \nabla _x F(\bar{p},\bar{x}) k_1 + \xi _1 - w, \quad \eta _2= \nabla _x F(\bar{p},\bar{x}) k_2 + \xi _2 - w, \end{aligned}$$

one has

$$\begin{aligned}&\langle \eta _1 - \eta _2, k_1 - k_2 \rangle \\&\quad = \langle k_1 - k_2, \nabla _x F(\bar{p},\bar{x})(k_1-k_2) \rangle + \langle \xi _1 - \xi _2, k_1 - k_2 \rangle \ge \alpha ||k_1 - k_2||^2. \end{aligned}$$

We conclude that \(\Phi \) is strongly monotone on K and, consequently, \(\varXi \) is single-valued by virtue of (Rockafellar and Wets 1998, Proposition 12.54). \(\square \)

Example 1

Put \(m=2, s=1\) and consider the GE (4), where

$$\begin{aligned} F(p,x)=p_{1}+p_{2} x, \quad \tilde{q}(x)= |x| + \delta _{A}(x), \quad A=[0,1] \end{aligned}$$

and the reference pair \((\bar{p},\bar{x})=((-1,1),0)\). Since \(\nabla _x F(\bar{p},\bar{x})=1\), Proposition 1 applies and we may conclude that the respective mapping S has indeed the single-valued and Lipschitzian localization around \( (\bar{p},\bar{x})\).

To compute \(DS (\bar{p},\bar{x})\), we may employ formula (7), where \(\partial \varphi \) is computed according to (17). One has \(K(\bar{x},\bar{v})=\mathbb {R}_{+}, \tilde{q}^{\prime \prime } (\bar{x},w)=0\) for any \(w\in \mathbb {R}_{+}\) and so we obtain that

$$\begin{aligned} \partial \varphi (k) = N_{\mathbb {R}_{+}}(k). \end{aligned}$$

This yields the formula

$$\begin{aligned} DS (\bar{p},\bar{x})(h)=\{k \in \mathbb {R}| 0 \in h_{1} + k + N_{\mathbb {R}_{+}}(k)\} \end{aligned}$$

valid for all \(h\in \mathbb {R}^2\). Both mappings S and \(DS (\bar{p},\bar{x})\) are depicted in Fig.3. \(\triangle \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Outrata, J.V., Valdman, J. On computation of optimal strategies in oligopolistic markets respecting the cost of change. Math Meth Oper Res 92, 489–509 (2020). https://doi.org/10.1007/s00186-020-00721-x

Download citation


  • Generalized equation
  • Equilibrium
  • Cost of Change

Mathematics Subject Classification

  • 90C33
  • 91B52
  • 49J40
  • 90C31