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

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.

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

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

(16)

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,

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

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

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

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

(17)

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. $\triangle$