## Abstract

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.

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## Acknowledgements

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.

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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.

## APPENDIX

### APPENDIX

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

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 \((\bar{p},\bar{x})\).

### Proof

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,

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

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

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

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

### 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

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

### Proof

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

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

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

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

one has

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

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

This yields the formula

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

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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

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### Keywords

- Generalized equation
- Equilibrium
- Cost of Change

### Mathematics Subject Classification

- 90C33
- 91B52
- 49J40
- 90C31