Noncooperative oligopoly equilibrium in markets with hierarchical competition

Abstract

In this paper we study a non-cooperative sequential equilibrium concept, namely the Stackelberg–Nash equilibrium, in a game in which heterogeneous atomic traders interact in interrelated markets. To this end, we consider a two-stage quantity setting strategic market game with a finite number of traders. Within this framework, we define a Stackelberg–Nash equilibrium. Then, we show existence and local uniqueness of a Stackelberg–Nash equilibrium with trade. To this end, we use a differentiable approach: the vector mapping which determines the strategies of followers is a smooth local diffeomorphism, and the set of Stackelberg–Nash equilibria with trade is discrete, i.e., the interior equilibria of the game are locally unique. We also compare through examples the sequential and the simultaneous moves games. A striking difference is that exchange can take place in one subgame while autarky can hold in another subgame, in which case only leaders (followers) make trade.

Appendix

1.1 Appendix A: Proof of Proposition 1

Given $(\textbf{q}_{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F};\textbf{b} _{\epsilon }^{L},\textbf{b}_{\epsilon }^{F})\in \textbf{S}_{-i}$, the problem for follower i may be written:

$$\begin{aligned} \underset{q_{i,\epsilon }}{\max }\{\pi _{i}^{\epsilon }(q_{i,\epsilon }, \textbf{q}_{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F};\textbf{b} _{\epsilon }^{L},\textbf{b}_{\epsilon }^{F}):q_{i,\epsilon }\in {\mathcal {S}}_{i}\}\text {, }\epsilon >0\text {,} \end{aligned}$$

(A1)

where ${\mathcal {S}}_{i}=[0,\alpha _{i}]$ is a nonempty compact convex set, and $\pi _{i,\epsilon }(.)$ is a continuous function of $(\textbf{q} _{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F};\textbf{b}_{\epsilon }^{L}, \textbf{b}_{\epsilon }^{F};\epsilon )$ as $\pi _{i,\epsilon }(.)\in \mathcal {C}^{\infty }$. We show that $\pi _{i,\epsilon }(.)$ is a strictly quasi-concave function of $q_{i,\epsilon }$. Differentiating (12) with respect to $q_{i,\epsilon }$ leads to:

$$\begin{aligned} \frac{\partial \pi _{i}^{\epsilon }}{\partial q_{i,\epsilon }}{ =-} \frac{\partial u_{i}}{\partial x_{i}}{ +p}_{X}^{\epsilon }\frac{ Q_{-i,\epsilon }+\epsilon }{q_{i,\epsilon }+Q_{-i,\epsilon }+\epsilon }\frac{ \partial u_{i}}{\partial y_{i}}\text {.} \end{aligned}$$

(A2)

Differentiating (A2) with respect to $q_{i,\epsilon }$ leads to:

$$\begin{aligned} \frac{\partial ^{2}\pi _{i}^{\epsilon }}{(\partial q_{i,\epsilon })^{2}} { =-}\frac{1}{\left( \frac{\partial u_{i}}{\partial y_{i}}\right) ^{2}}\left| \bar{\mathcal {H}}_{u_{\textbf{z}_{i}}}\right| { -2p}_{X}^{\epsilon }\frac{Q_{-i,\epsilon }+\epsilon }{(Q_{\epsilon }+\epsilon )^{2}}\frac{\partial u_{i}}{\partial y_{i}}\text {,} \end{aligned}$$

(A3)

where $\left| \bar{\mathcal {H}}_{u_{\textbf{z}_{i}}}\right| =(\frac{ \partial u_{i}}{\partial y_{i}})^{2}[\frac{\partial ^{2}u_{i}}{(\partial x_{i})^{2}}-2p_{X}^{\epsilon }\frac{Q_{-i,\epsilon }+\epsilon }{Q_{\epsilon }+\epsilon }\frac{\partial ^{2}u_{i}}{\partial x_{i}\partial y_{i}}+\left( p_{X}^{\epsilon }\frac{Q_{-i,\epsilon }+\epsilon }{Q_{\epsilon }+\epsilon } \right) ^{2}\frac{\partial ^{2}u_{i}}{(\partial y_{i})^{2}}]$ is the determinant of the bordered Hessian matrix of the function $u_{i}$ at equilibrium, i.e, when $p_{X}^{\epsilon }\frac{Q_{-i,\epsilon }+\epsilon }{ Q_{\epsilon }+\epsilon }=\frac{\partial u_{i}/\partial x_{i}}{\partial u_{i}/\partial y_{i}}$. As $\left| \bar{\mathcal {H}}_{u_{\textbf{z} _{i}}}\right| >0$ (Assumption 2c), and the last term in (A3) is negative, then $\frac{\partial ^{2}{\pi }_{i}^{\epsilon }}{(\partial q_{i,\epsilon })^{2}}<0$, so ${\pi }_{i}^{\epsilon }(.)$ is strictly concave (then strictly quasi-concave) of $q_{i,\epsilon }$. But then, the solution to (A1) is unique, so the map $\phi _{i}^{\epsilon }(\textbf{q} _{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F};\textbf{b}_{\epsilon }^{L}, \textbf{b}_{\epsilon }^{F};\epsilon )$ is point-valued, i.e, a function. As the objective $\pi _{i,\epsilon }(.)$ is strictly quasi-concave (condition (d) in Arrow and Enthoven 1961 holds), and the constraint set is quasi-convex (as it is convex), the Kuhn-Tucker conditions are sufficient to identify the solution to (A1). Therefore, define the Lagrangian $\mathcal {L}_{i}^{\epsilon }:\textbf{S}\times {\mathbb {R}} _{+}^{2}\times {\mathbb {R}} _{++}\rightarrow {\mathbb {R}}$ as $\mathcal {L}_{i}^{\epsilon }(.;\epsilon ):=\pi _{i}^{\epsilon }(q_{i,\epsilon },\textbf{q}_{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F}; \textbf{b}_{\epsilon }^{L},\textbf{b}_{\epsilon }^{F};\epsilon )+\lambda _{i,\epsilon }(\alpha _{i}-q_{i,\epsilon })+\mu _{i,\epsilon }q_{i,\epsilon }$, where $\lambda _{i,\epsilon }\geqslant 0$ and $\mu _{i,\epsilon }\geqslant 0$. Then, for all $\varepsilon >0$, and given $(\textbf{q} _{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F};\textbf{b}_{\epsilon }^{L}, \textbf{b}_{\epsilon }^{F}\mathbf {)\in S}_{-i}$, $\phi _{i}^{\epsilon }( \textbf{q}_{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F};\textbf{b} _{\epsilon }^{L},\textbf{b}_{\epsilon }^{F};\epsilon )$, is the unique solution to:

$$\begin{aligned} \underset{q_{i,\epsilon }}{\max }\mathcal {L}_{i}^{\epsilon }(.;\epsilon )=u_{i}\left( \alpha _{i}-q_{i,\epsilon },\frac{B_{\epsilon }+\epsilon }{q_{i,\epsilon }+Q_{-i,\epsilon }+\epsilon }q_{i,\epsilon }\right) +\lambda _{i,\epsilon }(\alpha _{i}-q_{i,\epsilon })+\mu _{i,\epsilon }q_{i,\epsilon }\text {.} \end{aligned}$$

(A4)

For all $\epsilon >0$, by using (A2), the Kuhn-Tucker conditions may be written:

$$\begin{aligned}{} & {} \frac{\partial \mathcal {L}_{i}^{\epsilon }}{\partial q_{i}}=-\frac{\partial u_{i}}{\partial x_{i}}+p_{X}^{\epsilon }\frac{Q_{-i,\epsilon }+\epsilon }{ q_{i,\epsilon }+Q_{-i,\epsilon }+\epsilon }\frac{\partial u_{i}}{\partial y_{i}}-\lambda _{i,\epsilon }+\mu _{i,\epsilon }=0\text {,} \nonumber \\{} & {} \quad \lambda _{i,\epsilon }\geqslant 0\text {, }(\alpha _{i}-q_{i,\epsilon })\geqslant 0\text {, with }\lambda _{i,\epsilon }(\alpha _{i}-q_{i,\epsilon })=0\text {,} \nonumber \\{} & {} \quad \mu _{i,\epsilon }\geqslant 0\text {, }q_{i,\epsilon }\geqslant 0\text {, with }\mu _{i,\epsilon }q_{i,\epsilon }=0\text {.} \end{aligned}$$

(A5)

Therefore, if $q_{i,\varepsilon }>0$, then $\mu _{i,\epsilon }=0$, where $b_{i,\varepsilon }$ is the solution to:

$$\begin{aligned} { -}\frac{\partial u_{i}}{\partial x_{i}}{ +p}_{X}^{\epsilon } \frac{Q_{-i,\epsilon }+\epsilon }{q_{i,\epsilon }+Q_{-i,\epsilon }+\epsilon } \frac{\partial u_{i}}{\partial y_{i}}{ =\lambda }_{i,\epsilon }\text {,} \end{aligned}$$

(A6)

which yields $\phi _{i}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q} _{-i,\epsilon }^{F};\textbf{b}_{\epsilon }^{L},\textbf{b}_{\epsilon }^{F};\epsilon )>0$. In addition, if $\lambda _{i,\epsilon }>0$, then $q_{i,\epsilon }=\phi _{i}^{\epsilon }({.})=\alpha _{i}$, while if $\lambda _{i,\epsilon }=0$, then $\phi _{i}^{\epsilon }({.})\in (0,\alpha _{i})$. Now, if $\mu _{i,\epsilon }>0$, then $q_{i,\varepsilon }=0$, which means that $\phi _{i}^{\epsilon }({.})=0$ and $\lambda _{i,\epsilon }=0$ since $q_{i,\varepsilon }<\alpha _{i}$. Therefore, either $\phi _{i}^{\epsilon }(.)>0$ when $q_{i,\epsilon }\in (0,\alpha _{i}]$ or $\ \phi _{i}^{\epsilon }({.})=0$. Then, $\phi _{i}^{\epsilon }({.})\geqslant 0$. In each case there exists a unique solution to (A1): either $\phi _{i}^{\epsilon }({.})\in \{0,\alpha _{i}\}$ or $\phi _{i}^{\epsilon }({.})\in (0,\alpha _{i})$, $i\in T_{X}^{F}$.

Finally, we show $\phi _{i}^{\epsilon }({.})\in \mathcal {C}^{\infty }$ . The equation $\frac{\partial \mathcal {L}_{i}^{\epsilon }}{\partial q_{i}} =0$ in (A5) defines implicitly $\phi _{i}^{\epsilon }({.})$. As ${\pi }_{i}^{\epsilon }(.)\in \mathcal {C}^{\infty }$ and $\frac{ \partial ^{2}{\pi }_{i}^{\epsilon }}{(\partial q_{i,\epsilon })^{2}} \ne 0$, by the Implicit Function Theorem, $\phi _{i}^{\epsilon }({.} )\in \mathcal {C}^{\infty }$.

A similar reasoning holds for $\psi _{j}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{L},\textbf{b} _{-j,\epsilon }^{F};\epsilon )$, $j\in T_{Y}^{F}$.$\blacksquare$

1.2 Appendix B: Proof of Proposition 2

Let $(\bar{\textbf{q}};\bar{\textbf{b}})=(\bar{\textbf{q}}_{\epsilon }^{L},\bar{\textbf{q}}_{\epsilon }^{F}(\bar{\textbf{q}}_{\epsilon }^{L};\bar{\textbf{b}} _{\epsilon }^{L});\bar{\textbf{b}}_{\epsilon }^{L},\bar{\textbf{b}} _{\epsilon }^{F}(\bar{\textbf{q}}_{\epsilon }^{L};\bar{\textbf{b}}_{\epsilon }^{L}))\in \textbf{S}$. First: $\mathcal {J}_{\varvec{\phi }_{\textbf{q} _{\epsilon }^{F}}^{\epsilon }}(\bar{\textbf{q}};\bar{\textbf{b}})\in (-\mathbf {I,I)}$ , where $\textbf{I}$ is the $(n_{X}-m_{X},n_{X}-m_{X})$ unit matrix. $\mathcal {J}_{\varvec{\phi }_{\textbf{q}_{\epsilon }^{F}}^{\epsilon }}( \bar{\textbf{q}};\bar{\textbf{b}})$ has unit terms on its main diagonal. To study the partial effects of a change in the supplies of other followers, i.e., $q_{-i,\epsilon }$, $-i\ne i$, $-i\in T_{X}^{F}$, and $b_{j,\epsilon }$, $j\in T_{Y}^{F}$, consider the identity:

$$\begin{aligned} \frac{\partial \pi _{i}^{\epsilon }}{\partial q_{i,\epsilon }}(\textbf{q} _{\epsilon }^{L},\phi _{i}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q} _{{ -i},\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L},\textbf{b} _{\epsilon }^{F}(.);\epsilon ),\varvec{\phi }_{-i}^{\epsilon }(.);\textbf{b} _{\epsilon }^{L},\textbf{b}_{\epsilon }^{F}(.);\epsilon )\equiv 0\text {,} \end{aligned}$$

(B1)

where, for $i\in T_{X}^{F}$, $\phi _{i}^{\epsilon }(.)$ is the solution to (A2). Implicit partial differentiation of (B1) with respect to $q_{-i,\epsilon }$, for $-i\ne i$, leads to $\frac{\partial \phi _{i}^{\epsilon }(.)}{\partial q_{-i,\epsilon }}=-\frac{\frac{\partial ^{2}\pi _{i}^{\epsilon }}{\partial q_{i,\epsilon }\partial q_{-i,\epsilon }}}{\frac{\partial ^{2}\pi _{i}^{\epsilon }}{(\partial q_{i,\epsilon })^{2}}}$, so we deduce:

$$\begin{aligned} \frac{\partial \phi _{i}^{\epsilon }(.)}{\partial q_{-i,\epsilon }}=\frac{ p_{X}^{\epsilon }\left( \frac{q_{i,\epsilon }}{Q_{\epsilon }+\epsilon }\frac{ \partial ^{2}u_{i}}{\partial x_{i}\partial y_{i}}+\frac{q_{i,\epsilon }-(Q_{-i,\epsilon }+\epsilon )}{(Q_{\epsilon }+\epsilon )^{2}}\frac{\partial u_{i}}{\partial y_{i}}-p_{X}^{\epsilon }\frac{(Q_{-i,\epsilon }+\epsilon )q_{i,\epsilon }}{(Q_{\epsilon }+\epsilon )^{2}}\frac{\partial ^{2}u_{i}}{ (\partial y_{i})^{2}}\right) }{\frac{\partial ^{2}u_{i}}{(\partial x_{i})^{2}} -p_{X}^{\epsilon }\left( 2\frac{Q_{-i,\epsilon }+\epsilon }{Q_{\epsilon }+\epsilon }\frac{\partial ^{2}u_{i}}{\partial x_{i}\partial y_{i}}-p_{X}^{\epsilon }\left( \frac{Q_{-i,\epsilon }+\epsilon }{Q_{\epsilon }+\epsilon }\right) ^{2}\frac{ \partial ^{2}u_{i}}{(\partial y_{i})^{2}}+2\frac{Q_{-i,\epsilon }+\epsilon }{ (Q_{\epsilon }+\epsilon )^{2}}\frac{\partial u_{i}}{\partial y_{i}}\right) }\text {.} \end{aligned}$$

(B2)

Assume, without loss of generality, that, for at least one leader i or one follower $i^{\prime }$ we have $\tilde{q}_{i,\epsilon }\leqslant \frac{ \tilde{Q}_{\epsilon }}{2}$ or $\tilde{q}_{i^{\prime },\epsilon }\leqslant \frac{\tilde{Q}_{\epsilon }}{2}$ (otherwise $\tilde{q}_{i,\epsilon }+$ $\tilde{q}_{i^{\prime },\epsilon }>\tilde{Q}_{\epsilon }$). As $\left| \frac{(Q_{-i,\epsilon }+\epsilon )q_{i,\epsilon }}{(Q_{\epsilon }+\epsilon )^{2}}\right| <\left| \left( \frac{Q_{-i,\epsilon }+\epsilon }{ Q_{\epsilon }+\epsilon }\right) ^{2}\right|$, $2\frac{Q_{-i,\epsilon }+\epsilon }{Q_{\epsilon }+\epsilon }>\frac{q_{i,\epsilon }}{Q_{\epsilon }+\epsilon }$, and $\frac{q_{i,\epsilon }-(Q_{-i,\epsilon }+\epsilon )}{ (Q_{\epsilon }+\epsilon )^{2}}<2\frac{Q_{-i,\epsilon }+\epsilon }{ (Q_{\epsilon }+\epsilon )^{2}}$, then $\left| \frac{\partial \phi _{i}^{\epsilon }(.)}{\partial q_{-i,\epsilon }}\right| <1$.

Second: $\mathcal {J}_{\varvec{\phi }_{\textbf{b}_{\epsilon }^{F}}^{\epsilon }}(\bar{\textbf{q}};\bar{\textbf{b}})\in (-\mathbf {I,I)}$. Implicit partial differentiation of (B1) with respect to $b_{j,\epsilon }$, $j\in T_{Y}^{F}$ , leads to:

$$\begin{aligned} \frac{\partial \phi _{i}^{\epsilon }(.)}{\partial b_{j,\epsilon }}=\frac{ \frac{q_{i,\epsilon }}{Q_{\epsilon }+\epsilon }\frac{\partial ^{2}u_{i}}{ (\partial x_{i})^{2}}-\frac{Q_{-i,\epsilon }+\epsilon }{(Q_{\epsilon }+\epsilon )^{2}}\frac{\partial u_{i}}{\partial y_{i}}-p_{X}^{\epsilon } \frac{(Q_{-i,\epsilon }+\epsilon )q_{i,\epsilon }}{(Q_{\epsilon }+\epsilon )^{2}}\frac{\partial ^{2}u_{i}}{(\partial y_{i})^{2}}}{\frac{\partial ^{2}u_{i}}{(\partial x_{i})^{2}}-p_{X}^{\epsilon }\left( 2\frac{Q_{-i,\epsilon }+\epsilon }{Q_{\epsilon }+\epsilon }\frac{\partial ^{2}u_{i}}{\partial x_{i}\partial y_{i}}-p_{X}^{\epsilon }\left( \frac{Q_{-i,\epsilon }+\epsilon }{ Q_{\epsilon }+\epsilon }\right) ^{2}\frac{\partial ^{2}u_{i}}{(\partial y_{i})^{2}} +2\frac{Q_{-i,\epsilon }+\epsilon }{(Q_{\epsilon }+\epsilon )^{2}}\frac{ \partial u_{i}}{\partial y_{i}}\right) }\text {.} \end{aligned}$$

(B3)

A similar reasoning leads to $\left| \frac{\partial \phi _{i}^{\epsilon }(.)}{\partial b_{j,\epsilon }}\right| <1$, for $i\in T_{X}^{F}$, and $j\in T_{Y}^{F}$.$\blacksquare$

1.3 Appendix C: Proof of Lemma 1

Define the function $\Phi _{i}^{\epsilon }:\textbf{S}\times {\mathbb {R}} _{++}\rightarrow {\mathcal {S}}_{i}$, with $\Phi _{i}^{\epsilon }(\textbf{q} _{\epsilon }^{L},\textbf{q}_{\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L}, \textbf{b}_{\epsilon }^{F}(.);\epsilon ):=q_{i,\epsilon }(.)-\phi _{i}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q}_{-i,\epsilon }^{F}(.),;\textbf{b}_{\epsilon }^{L},\textbf{b}_{\epsilon }^{F}(.);\epsilon )$, $i\in T_{X}^{F}$, and the function $\Psi _{j}^{\epsilon }: \textbf{S}\times {\mathbb {R}} _{++}\rightarrow {\mathcal {S}}_{j}$, with $\Psi _{j}^{\epsilon }(\textbf{q} _{\epsilon }^{L},\textbf{q}_{\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L}, \textbf{b}_{\epsilon }^{F}(.);\epsilon ):=b_{j,\epsilon }(.)-\psi _{j}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q}_{\epsilon }^{F}(.); \textbf{b}_{\epsilon }^{L},\textbf{b}_{-j,\epsilon }^{F}(.);\epsilon \mathbf { )}$, $j\in T_{Y}^{F}$. As for each $i\in T_{X}^{F}$, $\phi _{i}^{\epsilon }(.)\in \mathcal {C}^{\infty }(.)$, then $\Phi _{i}^{\epsilon }(.)\in \mathcal {C}^{\infty }$, $i\in T_{X}^{F}$. Likewise, $\Psi _{j}^{\epsilon }(.)\in \mathcal {C}^{\infty }$, $j\in T_{Y}^{F}$. For all $\epsilon >0$, consider the system of equations for ${\varvec{\Gamma }}^{\epsilon }$:

$$\begin{aligned} \left\{ \begin{array}{c} \Phi _{i}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q}_{\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L},\textbf{b}_{\epsilon }^{F}(.);\epsilon )=0\text {, }i\in T_{X}^{F}\text {,} \\ \\ \Psi _{j}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q}_{\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L},\textbf{b}_{\epsilon }^{F}(.);\epsilon )=0\text {, }j\in T_{Y}^{F}\text {.} \end{array} \right. \end{aligned}$$

(C1)

Define the $(n_{X}-m_{X}+n_{Y}-m_{Y})$-dimensional vector function ${\varvec{\Upsilon }}^{\epsilon }:\textbf{S}\times {\mathbb {R}} _{++}\rightarrow \textbf{S}^{F}$, ${{\varvec{\Upsilon }}}^{\epsilon }(\textbf{q} _{\epsilon }^{L},\textbf{q}_{\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L}, \textbf{b}_{\epsilon }^{F}(.);\epsilon )=(\Phi _{m_{X}+1}^{\epsilon }( {.};\epsilon ),\ldots ,\Phi _{n_{X}}^{\epsilon }({.};\epsilon );\Psi _{m_{Y}+1}^{\epsilon }({.};\epsilon ),\ldots ,\Psi _{n_{Y}}^{\epsilon }({.};\epsilon ))$. Then, (C1) may be written as a $(n_{X}-m_{X}+n_{Y}-m_{Y})$-dimensional vector equation:

$$\begin{aligned} {{\varvec{\Upsilon }}}^{\epsilon }(\textbf{q}_{\epsilon }^{L},\textbf{q} _{\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L},\textbf{b}_{\epsilon }^{F}(.);\epsilon )=\textbf{0}\text {.} \end{aligned}$$

(C2)

Consider the restriction of $\textbf{S}\times {\mathbb {R}} _{++}$ to the open set $\bar{\textbf{S}}\times {\mathbb {R}} _{++}$, with $\bar{{\mathcal {S}}}_{i}\subset {\mathcal {S}}_{i}$, $i\in T_{X}$, and $\bar{{\mathcal {S}}}_{j}\subset {\mathcal {S}}_{j}$, $j\in T_{Y}$. The vector function ${{\varvec{\Upsilon }}}^{\epsilon }(\textbf{q}_{\epsilon }^{L}, \textbf{q}_{\epsilon }^{F}(.);\textbf{b}_{\epsilon }^{L},\textbf{b} _{\epsilon }^{F}(.);\epsilon )$ is $\mathcal {C}^{\infty }$ on the open set $\bar{\textbf{S}}\times {\mathbb {R}} _{++}$ as each $\Phi _{i}^{\epsilon }$ and each $\Psi _{j}^{\epsilon }$ are $\mathcal {C}^{\infty }$ functions of $(\textbf{q}_{\epsilon }^{L};\textbf{b} _{\epsilon }^{L})$ on the open set $\bar{\textbf{S}}\times {\mathbb {R}} _{++}$. Let $(\bar{\textbf{q}};\bar{\textbf{b}})=(\bar{\textbf{q}}_{\epsilon }^{L}, \bar{\textbf{q}}_{\epsilon }^{F}(\bar{\textbf{q}}_{\epsilon }^{L};\bar{\textbf{b}}_{\epsilon }^{L});\bar{\textbf{b}}_{\epsilon }^{L},\bar{\textbf{b}} _{\epsilon }^{F}(\bar{\textbf{q}}_{\epsilon }^{L};\bar{\textbf{b}}_{\epsilon }^{L}))$ be an interior point of $\textbf{S}$, where $(\bar{\textbf{q}} _{\epsilon }^{L};\bar{\textbf{b}}_{\epsilon }^{L})$ corresponds to a parameter configuration. Therefore, the following identity, which defines implicitly the strategies $\textbf{q}_{\epsilon }^{L}:={{\varvec{\sigma }}} ^{\epsilon }(.)$ and $\textbf{b}_{\epsilon }^{L}:={{\varvec{\varphi }}} ^{\epsilon }(.)$, holds in an open neighborhood of $(\bar{\textbf{q}} _{\epsilon };\bar{\textbf{b}}_{\epsilon })$:

$$\begin{aligned} {{\varvec{\Upsilon }}}^{\epsilon }(\textbf{q}_{\epsilon }^{L}, {{\varvec{\sigma }}}^{\epsilon }(\textbf{b }_{\epsilon }^{L};\textbf{q}_{\epsilon }^{L};\epsilon );\textbf{b}_{\epsilon }^{L},{{\varvec{\varphi }}}^{\epsilon }(\textbf{b}_{\epsilon }^{L};\textbf{q} _{\epsilon }^{L};\epsilon ))\equiv \textbf{0}\text {.} \end{aligned}$$

(C3)

We now show that there exists a product of open sets $\mathcal {U}\times \mathcal{V}$ in $\bar{\textbf{S}}$ and a product neighborhood $(\mathcal {U}_{L} \times \mathcal { V}_{L})$ in $\prod \nolimits _{i\in T_{X}^{L}} \bar{{\mathcal {S}}}_{i}\times \prod \nolimits _{j\in T_{Y}^{L}}\bar{{\mathcal {S}}} _{j}$, with $(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}}_{\epsilon })\subseteq$ $\mathcal {U}\times \mathcal{V}$ and $(\textbf{q}_{\epsilon }^{L};\textbf{ b}_{\epsilon }^{L})\subseteq (\mathcal {U}_{L}\times \mathcal { V}_{L})$ such that for each $(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L})$ in $(\mathcal {U}_{L}\times \mathcal { V}_{L})$, there exists (at least locally) one unique $(n_{X}-m_{X}+n_{Y}-m_{Y})$ dimensional $\mathcal {C}^{\infty }$ vector function $( {{\varvec{\sigma }}}^{\epsilon }(\textbf{q }_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon );{{\varvec{\varphi }}} ^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon ))=[{{\varvec{\Upsilon }}}^{\epsilon }\mathbf {]}^{-1}(\textbf{0})$ in some neighborhood of $(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L})$ such that $(\textbf{q}_{\epsilon }^{L}, {{\varvec{\sigma }}}^{\epsilon }({.});\textbf{b} _{\epsilon }^{L}, {\varvec{\varphi }}^{\epsilon }({.}))\in$ $\mathcal {U}\times \mathcal{V}$ and (C3) holds. Implicit partial differentiation with respect to each component of $( \bar{\textbf{q}}_{\epsilon }^{L};\bar{\textbf{b}}_{\epsilon }^{L})$ yields:

$$\begin{aligned} \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b} _{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}} _{\epsilon }).\mathcal {A}^{\epsilon }+\mathcal {B}^{\epsilon }=\textbf{0} \text {, for each }\epsilon >0\text {,} \end{aligned}$$

(C4)

where:

$$\begin{aligned} \mathcal { J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b} _{\epsilon }^{F})}^{\epsilon }}{ (}\bar{\textbf{q}}_{\epsilon }{ ;}\bar{\textbf{b}}_{\epsilon }{ )=}\left[ \begin{array}{cccccc} 1 &{} \cdots &{} \frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial q_{n_{X},\epsilon }} &{} \frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial b_{m_{Y}+1,\epsilon }} &{} \cdots &{} \frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{ \partial b_{n_{Y},\epsilon }} \\ \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \frac{\partial \Phi _{n_{X}}^{\epsilon }}{\partial q_{m_{X}+1,\epsilon }} &{} \cdots &{} 1 &{} \frac{\partial \Phi _{n_{X}}^{\epsilon }}{\partial b_{m_{Y}+1,\epsilon }} &{} \cdots &{} \frac{\partial \Phi _{n_{X}}^{\epsilon }}{ \partial b_{n_{Y},\epsilon }} \\ \frac{\partial \Psi _{m_{Y}+1}^{\epsilon }}{\partial q_{m_{X}+1,\epsilon }} &{} \cdots &{} \frac{\partial \Psi _{m_{Y}+1}^{\epsilon }}{\partial q_{n_{X},\epsilon }} &{} 1 &{} \cdots &{} \frac{\partial \Psi _{m_{Y}+1}^{\epsilon }}{ \partial b_{n_{Y},\epsilon }} \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ \frac{\partial \Psi _{n_{Y}}^{\epsilon }}{\partial q_{m_{X}+1,\epsilon }} &{} \cdots &{} \frac{\partial \Psi _{n_{Y}}^{\epsilon }}{\partial q_{n_{X,\epsilon }}} &{} \frac{\partial \Psi _{n_{Y}}^{\epsilon }}{\partial b_{m_{Y}+1,\epsilon }} &{} \cdots &{} 1 \end{array} \right] \end{aligned}$$

(C5)

is an $(n_{X}-m_{X}+n_{Y}-m_{Y},n_{X}-m_{X}+n_{Y}-m_{Y})$ matrix, and the matrices $\mathcal {A}^{\epsilon }=[\frac{\partial (q_{m_{X}+1,\epsilon },\ldots ,q_{n_{X},\epsilon },b_{m_{X}+1,\epsilon },\ldots ,b_{n_{Y},\epsilon } )}{\partial (q_{1,\epsilon },\ldots ,q_{m_{X},\epsilon },b_{1,\epsilon },\ldots ,b_{m_{Y},\epsilon })}(\bar{\textbf{q}};\bar{\textbf{b}})]$, $\mathcal {B} ^{\epsilon }=[\frac{\partial (\Phi _{m_{X}+1,\epsilon }^{\epsilon },\ldots ,\Phi _{n_{X},\epsilon }^{\epsilon },\Psi _{m_{Y}+1}^{\epsilon },\ldots ,\Psi _{n_{Y}}^{\epsilon })}{\partial (q_{1,\epsilon },\ldots ,q_{m_{X},\epsilon },b_{1,\epsilon },\ldots ,b_{m_{Y},\epsilon })}(\bar{\textbf{q}};\bar{\textbf{b}})]$ are of dimension $(n_{X}-m_{X}+n_{Y}-m_{Y},m_{X}+m_{Y})$.

The square matrix $\mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}}_{\epsilon } ;\bar{\textbf{b}}_{\epsilon })$ has unit terms on the main diagonal and each off-diagonal term is bounded below by $-1$ and above by 1, as from Proposition 2, we have $-\textbf{I}<<\mathcal {J}_{\varvec{\phi }_{(\textbf{q} _{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}<<\textbf{I}$ and $-\textbf{I}<<\mathcal {J}_{\varvec{\psi }_{(\textbf{q}_{\epsilon }^{F}; \textbf{b}_{\epsilon }^{F})}^{\epsilon }}<<\textbf{I}$. Then, $\frac{ \partial \Phi _{i}^{\epsilon }(.)}{\partial q_{-i,\epsilon }}=-\frac{ \partial \phi _{i}^{\epsilon }(.)}{\partial q_{-i,\epsilon }}\in (-1,1)$, with $-i\ne i$, and $\left| \frac{\partial \Phi _{i}^{\epsilon }(.)}{ \partial b_{j,\epsilon }}\right| =\left| -\frac{\partial \phi _{i}^{\epsilon }(.)}{\partial b_{j,\epsilon }}\right| <1$, $i\in T_{Y}^{F}$; and $\frac{\partial \Psi _{j}^{\epsilon }(.)}{\partial b_{-j,\epsilon }}=-\frac{\partial \psi _{j}^{\epsilon }(.)}{\partial b_{-j,\epsilon }}\in (-1,1)$, with $-j\ne j$, and $\left| \frac{ \partial \Psi _{j}^{\epsilon }(.)}{\partial q_{i,\epsilon }}\right| =\left| -\frac{\partial \psi _{j}^{\epsilon }(.)}{\partial q_{i,\epsilon }}\right| <1$, $j\in T_{Y}^{F}$. The signs of the off diagonal terms depend on whether the strategies of followers are complements or substitutes. But, in any case, for all $\epsilon >0$, the rows of the matrix $\mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\varepsilon }^{F}; \textbf{b}_{\varepsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}}_{\varepsilon }; \bar{\textbf{b}}_{\varepsilon })$ are linearly independent, so this matrix is of full rank, and then invertible, i.e., for all $\epsilon >0$,$\ \left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F}; \textbf{b}_{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}};\bar{\textbf{b}} )\right| \ne 0$. Then, by the Implicit Function Theorem, there exist open sets $\mathcal {U}\times \mathcal{V}$ in $\bar{\textbf{S}}$ and $\mathcal {U}_{L} \times \mathcal { V}_{L}$ in $\prod \nolimits _{i\in T_{X}^{L}}\bar{{\mathcal {S}}} _{i}\times \prod \nolimits _{j\in T_{Y}^{L}}\bar{\mathcal {azS}}_{j}$, with $( \bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}}_{\epsilon })\subseteq$ $\mathcal {U}\times \mathcal{V}$ and $(\textbf{q}_{\epsilon }^{L};\textbf{b} _{\epsilon }^{L})\subseteq (\mathcal {U}_{L}\times \mathcal { V}_{L}\mathcal {)}$ such that for each $(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L})$ in $\mathcal {U}_{L}\times \mathcal { V}_{L}$, there exists a locally unique $(n_{X}-m_{X}+n_{Y}-m_{Y})$ dimensional vector function $( \textbf{q}_{\epsilon }^{F}(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon );\textbf{b}_{\epsilon }^{F}( \textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon ))$ in some neighborhood of $(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L})$ such that $(\textbf{q}_{\epsilon }^{L},\textbf{q}_{\epsilon }^{F}(\textbf{q} _{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon );\textbf{b}_{\epsilon }^{L}, \textbf{b} _{\epsilon }^{F}(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon ))\in$ $\mathcal {U}\times \mathcal{V}$, and the identity (C3) holds. Indeed, the unique solution $( {{\varvec{\sigma }}} ^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon );{{\varvec{\varphi }}}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b} _{\epsilon }^{L};\epsilon ))$ to $(\textbf{q}_{\epsilon }^{F}(\textbf{q} _{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon );\textbf{b}_{\epsilon }^{F}(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon ))=[ {{\varvec{\Upsilon }}}^{\epsilon }\mathbf {]}^{-1}(\textbf{0})$ is defined by ${{\varvec{\sigma }}}^{\epsilon }:{\mathcal {S}}^{L}\times {\mathbb {R}} _{++}\supset$ $\mathcal {U}_{L}\times \mathcal { V}_{L}\rightarrow \prod \nolimits _{i\in T_{X}^{F}}{\mathcal {S}}_{i}$, with $\textbf{q}_{\epsilon }^{F}={{\varvec{\sigma }}}^{\epsilon }(\textbf{ q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )$, and by ${\varvec{{\varphi }}}^{\epsilon }:{\mathcal {S}}^{L}\times {\mathbb {R}} _{++}\supset$ $\mathcal {U}_{L}\times \mathcal { V}_{L}\rightarrow \mathop {\textstyle \prod }\nolimits _{j\in T_{Y}^{F}}{\mathcal {S}}_{j}$, with $\textbf{b}_{\epsilon }^{F}={{\varvec{\varphi }}}^{\epsilon }( \textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )$. For all $\epsilon >0$, each component function $\sigma _{i}^{\epsilon }({.})$ is defined as $\sigma _{i}^{\epsilon }: {\mathcal {S}}^{L}\times {\mathbb {R}} _{++}\supset \mathcal {U}_{L}\times \mathcal { V}_{L}\rightarrow {\mathcal {S}} _{i}$, with $q_{i,\epsilon }= \sigma _{i}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )$, $i\in T_{X}^{F}$. The same holds for $\varphi _{j}^{\epsilon }:{\mathcal {S}}^{L}\times {\mathbb {R}} _{++}\supset \mathcal {U}_{L}\times \mathcal { V}_{L}$, with $b_{j,\epsilon }= \varphi _{j}^{\epsilon }(\textbf{q} _{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )$ $j\in T_{Y}^{F}$. Finally, for all $\epsilon >0$, $\ \sigma _{i}^{\epsilon }(\textbf{q} _{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )\in \mathcal {C} ^{\infty }(\textbf{S},{\mathcal {S}}_{i})$, $i\in T_{X}^{F}$, and $\varphi _{j}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )\in \mathcal {C}^{\infty }(\textbf{S},{\mathcal {S}}_{j})$, $j\in T_{Y}^{F}$. $\square$

1.4 Appendix D: Proof of Proposition 3

Let $(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}}_{\epsilon })$ in $\textbf{S}$. We show $-1\leqslant \frac{\partial \sigma _{i}^{\epsilon } \mathbf {(q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )}{ \partial q_{i,\epsilon }}<1$, $i\in T_{X}^{F}$, and, $\frac{\partial \varphi _{j}^{\epsilon }(.)}{\partial q_{i,\epsilon }}\geqslant 0$, $j\in T_{Y}^{F}$ . First, we show $-1\leqslant \frac{\partial \sigma _{i}^{\epsilon }(.)}{ \partial q_{i,\epsilon }}<1$. Consider (9). From Cramer’s rule:

$$\begin{aligned} \frac{\partial q_{m_{X}+1,\epsilon }}{\partial q_{1,\epsilon }}{ =-} \frac{\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}} _{\epsilon };\bar{\textbf{b}}_{\epsilon })\right| }{\left| \mathcal {J }_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}}_{\epsilon })\right| }\text {,} \end{aligned}$$

(D1)

where $\mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{b}_{\epsilon }^{F};\textbf{q }_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}}_{\epsilon } ;\bar{\textbf{b}}_{\epsilon })$ is the $(n_{X}-m_{X}+n_{Y}-m_{Y},n_{X}-m_{X}+n_{Y}-m_{Y})$ square matrix obtained by replacing the first column in $\mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{b }_{\epsilon }^{F};\textbf{q}_{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}} _{\epsilon };\bar{\textbf{b}}_{\epsilon })$ by the first column of $\mathcal { B}^{\epsilon }$. (D1) is well-defined, as from Lemma 1, $\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}\right| \ne 0$. Let $\frac{\partial \Phi _{i}^{\epsilon }}{\partial q_{1,\epsilon }}=0$, $i\in T_{X}^{F}$, and $\frac{\partial \Psi _{j}^{\epsilon }}{\partial q_{1,\epsilon }}=0$, $j\in T_{Y}^{F}$, in $\mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F}; \textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}} _{\epsilon };\bar{\textbf{b}}_{\epsilon })$. The matrices $\mathcal {J}_{ {{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\textbf{q;b})$ and $\mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}(\textbf{q;b})$ have common terms: the off-diagonal terms of the matrix $\mathcal {B}^{\epsilon }$ coincide with the off-diagonal terms of the matrix $\mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}$ as $Q\equiv \sum \nolimits _{i\in T_{1}}q_{i}$ and $B\equiv \sum \nolimits _{j\in T_{2}}b_{j}$. If $\frac{\partial q_{m_{X}+1,\epsilon }}{\partial q_{1,\epsilon }}<-1$, then

$$\begin{aligned} \left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F}; \textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}} _{\epsilon };\bar{\textbf{b}}_{\epsilon })\right| { >}\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b} _{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}} _{\epsilon })\right| \text {.} \end{aligned}$$

(D2)

Expansion by cofactors of the both sides of (D3), and cancellation among common terms on both sides, lead to:

$$\begin{aligned} \frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial q_{1,\epsilon }} \left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F}; \textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}} _{\epsilon };\bar{\textbf{b}}_{\epsilon })\right| { >}\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b} _{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}} _{\epsilon })\right| \text {,} \end{aligned}$$

(D3)

where $\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}} _{\epsilon };\bar{\textbf{b}}_{\epsilon })\right|$ stands for the principal minor of order $(n_{X}-m_{X}+n_{Y}-m_{Y}-1).(1,1)$ of $\mathcal {J} _{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}} _{\epsilon })$. From (D3), we have $\frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial q_{1,\epsilon }}>1$. A contradiction as $\frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial q_{1,\epsilon }}<-1$. Then, we have $-\frac{\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q} _{\epsilon }^{F};\textbf{b}_{\epsilon }^{F})}^{\epsilon }}^{\prime }(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}}_{\epsilon })\right| }{\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{(\textbf{q}_{\epsilon }^{F};\textbf{b} _{\epsilon }^{F})}^{\epsilon }}(\bar{\textbf{q}}_{\epsilon };\bar{\textbf{b}} _{\epsilon })\right| }\leqslant 1$, so $\frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial q_{1,\epsilon }}\geqslant -1$. Next, if $\frac{\partial q_{m_{X}+1,\epsilon }}{\partial q_{1,\epsilon }}>1$, then $\frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial q_{1,\epsilon }}<-1$. A contradiction. Then, $\frac{\partial \Phi _{m_{X}+1}^{\epsilon }}{\partial q_{1,\epsilon }}<1$. As the same holds for all best responses, and for $i\in T_{X}^{L}$, then

$$\begin{aligned} { -}\textbf{I}\le \frac{\partial [\sigma _{m_{X}+1}^{\epsilon } \mathbf {(b}_{\epsilon }^{L};\textbf{q}_{\epsilon }^{L};\epsilon \mathbf { ),\ldots ,}\sigma _{n_{X}}^{\epsilon }{(\textbf{b}_{{\varvec{\epsilon }} }^{\textbf{L}}; \textbf{q}_{{\varvec{\epsilon }} }^{\textbf{L}};{\varvec{\epsilon }})]}}{\partial [q_{1,\epsilon },\ldots ,q_{m_{X},\epsilon }]}<<\textbf{I}\text {,} \end{aligned}$$

(D4)

where $\textbf{I}$ is the $(n_{X}-m_{X},m_{X})$ unit matrix.

Now, we show $\frac{\partial \varphi _{j}^{\epsilon }(.)}{\partial q_{i,\epsilon }}\geqslant 0$, $j\in T_{Y}^{F}$, $i\in T_{X}^{L}$. Assume $\frac{\partial \varphi _{j}^{\epsilon }(.)}{\partial q_{i,\epsilon }}<0$ (strategies are substitutes so goods are complements). Then, u(.) is not differentiable, which contradicts Assumption 2a. Then, the conclusion follows.$\blacksquare$

1.5 Appendix E: Proof of Proposition 4

Let the reduced form payoff of leader $i\in T_{X}^{L}$ be given by $\pi _{i}^{\epsilon }(q_{i,\epsilon },\textbf{q}_{-i,\epsilon }^{L},.;\textbf{b} _{\epsilon }^{L},.;\epsilon )\equiv \pi _{i}^{\epsilon }(q_{i,\epsilon }, \textbf{q}_{-i,\epsilon }^{L}, {{\varvec{\sigma }}} ^{\epsilon }(q_{i,\epsilon },\textbf{q}_{-i,\epsilon }^{L};\textbf{b} _{\epsilon }^{L};\epsilon );\textbf{b}_{\epsilon }^{L},{{\varvec{\varphi }}} ^{\epsilon }(q_{i,\epsilon },\textbf{q}_{-i,\epsilon }^{L};\textbf{b} _{\epsilon }^{L};\epsilon ))$. For all $(\textbf{q}_{-i,\epsilon }^{L}; \textbf{b}_{\epsilon }^{L})$, the problem of leader i may be written:

$$\begin{aligned} \underset{q_{i,\epsilon }}{\max }\{\pi _{i}^{\epsilon }(q_{i,\epsilon },\textbf{q}_{-i,\epsilon }^{L},.;\textbf{b} _{\epsilon }^{L},.;\epsilon ):q_{i,\epsilon }\in {\mathcal {S}}_{i}\}\text {, } \epsilon >0\text {,} \end{aligned}$$

(E1)

The set ${\mathcal {S}}_{i}=[0,\alpha _{i}]$ is nonempty, compact and convex. As $u_{i}(.)\in \mathcal {C}^{\infty }$, ${{\varvec{\sigma }}}^{\epsilon }(.)\in \mathcal {C}^{\infty }$, and ${{\varvec{\varphi }}} ^{\epsilon }(.)\in \mathcal {C}^{\infty }$, $\pi _{i}^{\epsilon }(.) \in \mathcal {C}^{\infty }$, so $\pi _{i}^{\epsilon }(.)$ is a continuous function of $(q_{i,\epsilon },\textbf{q}_{-i,\epsilon };\textbf{b}_{\epsilon };\epsilon )$. Then, there exists $q_{i,\epsilon }=\phi _{i}^{\epsilon }(\textbf{q} _{-i}^{L};\textbf{b}^{L};\epsilon )$. To prove uniqueness, we show $\pi _{i}^{\epsilon }(.)$ is a strictly quasi-concave function of $q_{i,\epsilon }$. Differentiating (10) with respect to $q_{i,\epsilon }$, and using ${{\varvec{\sigma }}}^{\epsilon }(.)$ and ${{\varvec{\varphi }}}^{\epsilon }(.)$, yield:

$$\begin{aligned} \frac{\partial \pi _{i}^{\epsilon }}{\partial q_{i,\epsilon }}{ =-} \frac{\partial u_{i}}{\partial x_{i}}{ +\chi p}_{X}^{\epsilon }\frac{ \partial u_{i}}{\partial y_{i}}\text {,} \end{aligned}$$

(E2)

where $\chi \equiv 1-(1+\nu _{i,\epsilon }^{X})\frac{q_{i,\epsilon }}{ Q_{\epsilon }+\epsilon }+\eta _{i,\epsilon }^{X}\frac{q_{i,\epsilon }}{ B_{\epsilon }+\epsilon }$, $\nu _{i,\epsilon }^{X}=\frac{\partial \sum \nolimits _{i}\sigma _{i}^{\epsilon }(.)}{\partial q_{i,\epsilon }}$, $\eta _{i,\epsilon }^{X}=\frac{\partial \sum \nolimits _{j}\varphi _{j}^{\epsilon }(.)}{\partial q_{i,\epsilon }}$, and $p_{X}^{\epsilon }\overset{\vartriangle }{=}$ $p_{X}(\textbf{q}_{\epsilon }^{L},{{\varvec{\sigma }}}^{\epsilon } (\textbf{q} _{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon );\textbf{b}_{\epsilon }^{L},{{\varvec{\varphi }}}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b} _{\epsilon }^{L};\epsilon ))$. By construction $\nu _{i,\epsilon }^{X}=\nu _{\epsilon }^{X}$, and $\eta _{i,\epsilon }^{X}=\eta _{\epsilon }^{X}$, with $\nu _{\epsilon }^{X}\in [-1,1)$ and $\eta _{\epsilon }^{X}\geqslant 0$ (from 2a.). Indeed, as $\chi \in [0,1]$, then $0\leqslant -(1+\nu _{\epsilon }^{X})\frac{q_{i,\epsilon }}{\textbf{Q}_{\epsilon }+\epsilon } +\eta _{\epsilon }^{X}\frac{q_{i,\epsilon }}{\textbf{B}_{\epsilon }+\epsilon }\leqslant 1$, which leads to $\frac{\eta _{\epsilon }^{X}}{2-\nu _{\epsilon }^{X}}\leqslant \frac{\textbf{B}_{\epsilon }+\epsilon }{\textbf{Q}_{\epsilon }+\epsilon }\leqslant \frac{\eta _{\epsilon }^{X}}{1+\nu _{\epsilon }^{X}}$. Then, $\nu _{\epsilon }^{X}\leqslant \frac{1}{2}$. And, from Proposition 3, we get $\nu _{\epsilon }^{X}\geqslant -1$ as $\frac{\partial \sigma _{i}^{\epsilon }(.)}{\partial q_{i,\epsilon }}\geqslant -1$. Next, by differentiating partially (E2) with respect to $q_{i,\epsilon }$ leads to:

$$\begin{aligned} \frac{\partial ^{2}\pi _{i}^{\epsilon }}{(\partial q_{i,\epsilon })^{2}} { =}\frac{\partial ^{2}u_{i}}{(\partial x_{i})^{2}}{ -2\chi p} _{X}^{\epsilon }\frac{\partial u_{i}}{\partial x_{i}\partial y_{i}}{ +(\chi p}_{X}^{\epsilon }{ )}^{2}\frac{\partial ^{2}u_{i}}{(\partial y_{i})^{2}}{ -\kappa }\frac{\partial u_{i}}{\partial y_{i}}\text {,} \end{aligned}$$

(E3)

where $\kappa \equiv p_{X}^{\epsilon }(1-\frac{(1+\nu _{\epsilon }^{X})q_{i,\epsilon }}{Q_{\epsilon }+\epsilon })(1-\chi )$. The first three terms on the right hand side of (E3) are equal to the negative of the determinant of the bordered Hessian matrix of $u_{i}$ at equilibrium times the positive term $\frac{1}{(\frac{\partial u_{i}}{\partial y_{i}})^{2}}$, so it is positive from (2c). For (E3) to be strictly negative, it is sufficient that $\kappa \geqslant 0$. As $\chi \leqslant 1$, and $1-\frac{ (1+\nu _{\epsilon }^{X})q_{i,\epsilon }}{Q_{\epsilon }+\epsilon }\geqslant 0$ (the reverse inequality would lead to $\chi <\eta _{i,\epsilon }^{X}\frac{ q_{i,\epsilon }}{B_{\epsilon }+\epsilon }$, with $\eta _{i,\epsilon }^{X}\geqslant 0$, a contradiction). Then, $\kappa \geqslant 0$. As $\frac{ \partial ^{2}\pi _{i}^{\epsilon }}{(\partial q_{i,\epsilon })^{2}}<0$, the solution to (E1) is unique, so the map $\phi _{i}^{\epsilon }(\textbf{q} _{-i}^{L};\textbf{b}^{L};\epsilon )$ is point-valued, a function. As $\pi _{i}^{\epsilon }(.)$ is strictly quasi-concave, and ${\mathcal {S}}_{i}$ is quasi-convex, the Kuhn-Tucker conditions are sufficient to identify the solution to (E1). Let $\mathcal {L}_{i}^{\epsilon }(.;\epsilon ):\textbf{S}^{L}\times {\mathbb {R}} _{+}^{2}\times {\mathbb {R}} _{++}\rightarrow {\mathbb {R}}$, be given by

$$\begin{aligned} \mathcal {L}_{i}^{\epsilon }(.;\epsilon ):=\pi _{i}^{\epsilon }(q_{i,\epsilon },\textbf{q}_{-i,\epsilon }^{L}, {{\varvec{\sigma }}} ^{\epsilon }(q_{i,\epsilon },.;\epsilon );\textbf{b}_{\epsilon }^{L},{{\varvec{\varphi }}}^{\epsilon }(q_{i,\epsilon },{.};\epsilon ))+\lambda _{i,\epsilon }(\alpha _{i}-q_{i,\epsilon })+\mu _{i,\epsilon }q_{i,\epsilon } \text {,} \end{aligned}$$

(E4)

where $\lambda _{i,\epsilon },\mu _{i,\epsilon }\geqslant 0$, $i\in T_{X}^{L}$. Then, $\phi _{i}^{\epsilon }(\textbf{q}_{-i,\epsilon }^{L}; \textbf{b}_{\epsilon }^{L};\epsilon )$ is the unique solution to:

$$\begin{aligned} \underset{q_{i,\epsilon }}{\max }\mathcal {L}_{i}^{\epsilon }(.;\epsilon )= { u}_{i}{ (\alpha }_{i}{ -q}_{i,\epsilon }{ ,}\frac{ \sum _{j\in T_{Y}^{L}}b_{j,\epsilon }+\sum \nolimits _{j\in T_{Y}^{F}}\varphi _{j}^{\epsilon }({.};\epsilon )+\epsilon }{\sum _{k\in T_{X}^{L}}q_{k,\epsilon }+\sum \nolimits _{k\in T_{X}^{F}}\sigma _{k}^{\epsilon }({.};\epsilon )+\epsilon }{ q} _{i,\epsilon }{ )+\lambda }_{i,\epsilon }{ (\alpha }_{i}{ -q}_{i,\epsilon }{ )+\mu }_{i,\epsilon }{ q}_{i,\epsilon }\text {. } \end{aligned}$$

(E5)

For all $\epsilon >0$, the Kuhn-Tucker conditions may be written:

$$\begin{aligned} \begin{array}{c} \frac{\partial \mathcal {L}_{i}^{\epsilon }}{\partial q_{i,\epsilon }}=-\frac{ \partial u_{i}}{\partial x_{i}}+\chi p_{X}^{\epsilon }\frac{\partial u_{i}}{ \partial y_{i}}-\lambda _{i,\epsilon }+\mu _{i,\epsilon }=0\text {,} \\ \\ \lambda _{i,\epsilon }\geqslant 0\text {, }(\alpha _{i}-b_{i,\epsilon })\geqslant 0\text {, with }\lambda _{i,\epsilon }(\alpha _{i}-b_{i,\epsilon })=0\text {,} \\ \\ \mu _{i,\epsilon }\geqslant 0\text {, }b_{i,\epsilon }\geqslant 0\text {, with }\mu _{i,\epsilon } b_{i,\epsilon }=0\text {.} \end{array} \end{aligned}$$

(E6)

If $\phi _{i}^{\epsilon }(.)>0$, then $\mu _{i,\epsilon }=0$, where $b_{i,\epsilon }$ is the solution to:

$$\begin{aligned} { -}\frac{\partial u_{i}}{\partial x_{i}}{ +\chi p} _{X}^{\epsilon }\frac{\partial u_{i}}{\partial y_{i}}{ =\mu } _{i,\epsilon }\text {.} \end{aligned}$$

(E7)

If $\lambda _{i,\epsilon }>0$, then $q_{i,\epsilon }=\phi _{i}^{\epsilon }( .)=\alpha _{i}$; if $\lambda _{i,\epsilon }=0$, then $\phi _{i}^{\epsilon }(.)\in (0,\alpha _{i})$. If $\mu _{i,\epsilon }>0$ , then $\phi _{i}^{\epsilon }(.)=0$ and $\lambda _{i,\epsilon }=0$ since $q_{i,\epsilon }<\alpha _{i}$. Then, either $\ \phi _{i}^{\epsilon }( .)>0$ when $b_{i,\epsilon }\in (0,\alpha _{i}]$ or $\ \phi _{i}^{\epsilon }(.)=0$. Then, there is a unique maximum $q_{i,\epsilon }=\phi _{i}^{\epsilon }(.)\geqslant 0$, $i\in T_{X}^{L}$.

Finally, we show $\phi _{i}^{\epsilon }({.})\in \mathcal {C}^{\infty }$ . Equation $\frac{\partial \mathcal {L}_{i}^{\epsilon }}{\partial q_{i,\epsilon }}=0$ in (E6) defines implicitly $\phi _{i}^{\epsilon }( {.})$. As ${\pi }_{i}^{\epsilon }(.)\in \mathcal {C}^{\infty }$ and $\frac{\partial ^{2}{\pi }_{i}^{\epsilon }}{(\partial q_{i,\epsilon })^{2}}\ne 0$, from the Implicit Function Theorem, $\phi _{i}^{\epsilon }({.})\in \mathcal {C}^{\infty }$.

A similar reasoning holds for $\psi _{j}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b}_{-j,\epsilon }^{L};\epsilon )$, $j\in T_{Y}^{L}$.$\blacksquare$

1.6 Appendix F: Proof of Lemma 3

To show Lemma 3, we adapt to our sequential framework one result based on the Uniform Monotonicity Lemma (see Lemma C, p. 8, in Dubey and Shubik 1978).

Lemma 6

(Uniform monotonicity). Let $c\in \left\{ X,Y\right\}$, let $u_{k}: {\mathbb {R}} _{+}^{2}\rightarrow {\mathbb {R}}$, $z_{k}\mapsto u_{k}(z_{k})$, $k=i,j$, $i\in T_{X}$, $j\in T_{Y}$, be a continuous and increasing function, and let H be a positive constant. Then, there exists a positive real number $h(u_{k}(.),c,H)\in (0,1)$ such that, for all $s_{k},z_{k}\in {\mathbb {R}} _{+}^{2}$, if $\left\| \textbf{z}_{k}\right\| \leqslant H$ and $\left\| \textbf{s}_{k}-\textbf{z}_{k}\right\|$ $\leqslant h(u_{k}(.),c,H)$, then $u_{k}(s_{k}+e^{c})>u_{k}(z_{k})$, where $\left\| .\right\|$ denotes the Euclidean norm, and $e^{c}$ denotes the vector in ${\mathbb {R}} _{+}^{2}$ whose c-th component is 1 and the other 0.

Proof

Lemma 6 is a direct consequence of Lemma C in Dubey and Shubik (1978) (see Appendix B, p. 19) as, for each k, $u_{k}(.)$ satisfies Assumptions 2a-2b.

We now show that there exist some uniform bounds on the relative price in each perturbed subgame. Let $(\tilde{\textbf{q}}_{\epsilon }^{L},\tilde{\textbf{q}}_{\epsilon }^{F}(\tilde{\textbf{q}}_{\epsilon }^{L};\tilde{\textbf{ b}}_{\epsilon }^{L});\tilde{\textbf{b}}_{\epsilon }^{L},\tilde{\textbf{b}} _{\epsilon }^{F}(\tilde{\textbf{q}}_{\epsilon }^{L};\tilde{\textbf{b}} _{\epsilon }^{L}))$ be an $\epsilon$-SNE, and let $\tilde{p}_{X}^{\epsilon } \overset{\vartriangle }{=}p_{X}^{\epsilon }(\tilde{\textbf{q}}_{\epsilon }^{L},\tilde{\textbf{q}}_{\epsilon }^{F}(\tilde{\textbf{q}}_{\epsilon }^{L}; \tilde{\textbf{b}}_{\epsilon }^{L});\tilde{\textbf{b}}_{\epsilon }^{L}, \tilde{\textbf{b}}_{\epsilon }^{F}(\tilde{\textbf{q}}_{\epsilon }^{L}; \tilde{\textbf{b}}_{\epsilon }^{L}))$ be the corresponding relative price.

1. First, $\exists \xi _{1}>0\mid \tilde{p}_{X}^{\epsilon }>\xi _{1}$. Consider one leader j and one follower $j^{\prime }$. Let

$$\begin{aligned} { H}&{ =}&\max { (\bar{\alpha },\bar{\beta })}\text {, with } { \bar{\alpha }\equiv }\mathop {\textstyle \sum }_{i\in T_{X}}{ \alpha }_{i}\text { and }{ \bar{\beta }\equiv }\mathop {\textstyle \sum }_{j\in T_{Y}}{ \beta }_{j}\text {;} \nonumber \\ { h}&{ =}&\min { (h(u}_{j}{ ,Y,H),h(u}_{j^{\prime }} { ,Y,H))}\text {;} \nonumber \\ { A}&{ =}&\frac{1}{2}\min { (\beta }_{j}{ ,\beta } _{j^{\prime }}{ )}\text {, }{ j\ne j}^{\prime }\text {.} \end{aligned}$$

(F1)

Assume, without loss of generality, that $\tilde{b}_{j,\epsilon }\leqslant \frac{\tilde{B}_{\epsilon }}{2}$ or $\tilde{b}_{j^{\prime },\epsilon }\leqslant \frac{\tilde{B}_{\epsilon }}{2}$, for at least one leader j or one follower $j^{\prime }$ (otherwise $\tilde{b}_{j,\epsilon }+$ $\tilde{b} _{j^{\prime },\epsilon }>\tilde{B}_{\epsilon }$). Consider an increase of strategic supply at each stage.

Consider first follower $j^{\prime }$. Suppose $\beta _{j^{\prime }}-\tilde{ b}_{j^{\prime },\epsilon }\geqslant A$. Then, an increase $\delta$ in follower $j^{\prime }$’s supply such that $b_{j^{\prime },\epsilon }(\delta )=\tilde{b}_{j^{\prime },\epsilon }+\delta$, with $\delta \in (0,\frac{1}{2} \min \{\epsilon ,A\}]$, has the following incremental effect on his final holding:

$$\begin{aligned} { x}_{j^{\prime },\epsilon }{ (\delta )-x}_{j^{\prime },\epsilon }= & {} \frac{\tilde{Q}_{\epsilon }+\epsilon }{\tilde{B}_{\epsilon }+\epsilon +\delta }{ (\tilde{b}}_{j^{\prime },\epsilon }{ +\delta )}-\frac{ \tilde{Q}_{\epsilon }+\epsilon }{\tilde{B}_{\epsilon }+\epsilon }{ \tilde{b}}_{j^{\prime },\epsilon } \nonumber \\= & {} { \delta }\frac{\tilde{B}_{\epsilon }+\epsilon -\tilde{b} _{j^{\prime },\epsilon }}{\tilde{B}_{\epsilon }+\epsilon +\delta }\frac{ \tilde{Q}_{\epsilon }+\epsilon }{\tilde{B}_{\epsilon }+\epsilon } \nonumber \\> & {} { \delta }\frac{\frac{\tilde{B}_{\epsilon }}{2}+\frac{\epsilon }{2}+ \frac{\delta }{2}}{\tilde{B}_{\epsilon }+\epsilon +\delta }\frac{\tilde{Q} _{\epsilon }+\epsilon }{\tilde{B}_{\epsilon }+\epsilon }=\frac{\delta }{2} \frac{1}{\tilde{p}_{X}^{\epsilon }}\text {,} \end{aligned}$$

(F2)

and

$$\begin{aligned} y_{j^{\prime },\epsilon }(\delta )-y_{j^{\prime },\epsilon }=(\beta _{j^{\prime }}-\tilde{q}_{j^{\prime },\epsilon }-\delta )-(\beta _{j^{\prime }}-\tilde{q}_{j^{\prime },\epsilon })=-\delta \text {,} \end{aligned}$$

(F3)

where the strict inequality in (F2) results from $\tilde{B}_{\epsilon }+\epsilon -\tilde{b}_{j^{\prime },\epsilon }\geqslant \frac{\tilde{B} _{\epsilon }}{2}+\epsilon >\frac{\tilde{B}_{\epsilon }}{2}+\frac{\epsilon }{2 }+\frac{\delta }{2}$ (as $\tilde{b}_{j^{\prime },\epsilon }\leqslant \frac{ \tilde{B}_{\epsilon }}{2}$ and $\delta \leqslant \frac{1}{2}\epsilon )$. Let us define

$$\begin{aligned} t=-2\tilde{p}_{X}^{\epsilon }\textbf{e}^{Y}\text {, where } \,\, \textbf{e} ^{Y}=(0,1)\text {.} \end{aligned}$$

(F4)

Then, the following vector inequality holds:

$$\begin{aligned} { \textbf{z}}_{j^{\prime },\epsilon }{ (b}_{j^{\prime },\epsilon }{ (\delta ),p}_{X}^{\epsilon }{ (\tilde{\textbf{q}}}_{\epsilon }^{L}{ ,\tilde{\textbf{q}}}_{\epsilon }^{F}{ (.);\tilde{\textbf{b }}_{\epsilon }^{L},b}_{j^{\prime },\epsilon }{ (\delta ),\tilde{\textbf{b}}}_{-j,\epsilon }^{F}{ (.)))\ge \textbf{z}}_{j^{\prime },\epsilon }{ (\tilde{b}}_{j^{\prime },\epsilon }{ ,\tilde{p}} _{X}^{\epsilon }{ )+}\frac{\delta }{2}\frac{1}{\tilde{p}_{X}^{\epsilon }}{ (\textbf{e}}^{X}{ +t)}\text {,} \end{aligned}$$

(F5)

where $\textbf{e}^{X}=(1,0)$, and where, by (F2), the inequality (F5) is strict for the first component of $\textbf{z}_{j^{\prime },\epsilon }$. We can now apply Lemma 6, with $c=X$, $\textbf{z}_{j^{\prime },\epsilon }= \textbf{z}_{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon },\tilde{p }_{X}^{\epsilon })$ and $\textbf{s}_{j^{\prime },\epsilon }=\textbf{z}_{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon }, \tilde{p}_{X}^{\epsilon })+t$. We know that $\textbf{z} _{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon },\tilde{p} _{X}^{\epsilon })\in {\mathbb {R}} _{+}^{2}$ and $\left\| \textbf{z}_{j^{\prime },\epsilon }(\tilde{b} _{j^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon }) \right\| \leqslant H$. If $\textbf{s}_{j^{\prime },\epsilon }\in {\mathbb {R}} _{+}^{2}$ and $\left\| t\right\| \leqslant h$, then, by Lemma 6, we get:

$$\begin{aligned} u_{j^{\prime }}(\textbf{z}_{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon })+\textbf{e}^{X}+t)>u_{j^{\prime }}( \textbf{z}_{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon },\tilde{p }_{X}^{\epsilon }))\text {.} \end{aligned}$$

(F6)

As $u_{j^{\prime }}$ satisfies Assumptions (2b) and (2c), and as $0<\frac{ \delta }{2}\frac{1}{\tilde{p}_{X}^{\epsilon }}<1$, then we deduce:

$$\begin{aligned} u_{j^{\prime }}\left( \textbf{z}_{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon })+\frac{\delta }{2} \frac{1}{\tilde{p}_{X}^{\epsilon }}(\textbf{e}^{X}+t)\right) >u_{j^{\prime }}( \textbf{z}_{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon },\tilde{p }_{X}^{\epsilon }))\text {.} \end{aligned}$$

(F7)

Then, as (F5) holds strictly for its first component, from (2b), we deduce:

$$\begin{aligned} u_{j^{\prime }}(\textbf{z}_{j^{\prime },\epsilon }(b_{j^{\prime },\epsilon }(\delta ),p_{X}^{\epsilon }(\tilde{\textbf{q}}_{\epsilon }^{L},\tilde{\textbf{q}}_{\epsilon }^{F}({.});\tilde{\textbf{b}}_{\epsilon }^{L},b_{j^{\prime },\epsilon }(\delta ),\tilde{\textbf{b}}_{-j,\epsilon }^{F}({.}))))>u_{j^{\prime }}(\textbf{z}_{j^{\prime },\epsilon }(\tilde{b}_{j^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon } ))\text {,} \end{aligned}$$

(F8)

a contradiction. Hence, either $\textbf{z}_{j^{\prime },\epsilon }(b_{j^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon })+t< \textbf{0}$ or $\left\| t\right\| >h$. If $\textbf{z}_{j^{\prime },\epsilon }(b_{j^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon })+t<\textbf{0}$, then, $\tilde{y}_{j^{\prime },\epsilon }-2\tilde{p }_{X}^{\epsilon }<0$. As $\tilde{y}_{j^{\prime },\epsilon }=\beta _{j^{\prime }}-\tilde{b}_{j^{\prime },\epsilon }\geqslant A$, we deduce:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }>\frac{A}{2}\text {.} \end{aligned}$$

(F9)

Suppose now we have $\left\| t\right\| >h$. Then, we deduce:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }>\frac{h}{2}\text {.} \end{aligned}$$

(F10)

Finally, assume that the inequality $\beta _{j^{\prime }}-\tilde{b} _{j^{\prime },\epsilon }\geqslant A$ does not hold, which means that $\beta _{j^{\prime }}-\tilde{b}_{j^{\prime },\epsilon }<A$. Then, we have $\tilde{b }_{j^{\prime },\epsilon }>\beta _{j}-A\geqslant A$. Then $\tilde{b} _{j^{\prime },\epsilon }>A$, so, we get:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }>\frac{A}{\bar{\alpha }}\text {.} \end{aligned}$$

(F11)

Therefore, it suffices to take for follower $j^{\prime }$:

$$\begin{aligned} \xi _{1}^{j^{\prime }}=\min \left( \frac{A}{2},\frac{h}{2},\frac{A}{\bar{ \alpha }}\right) \text {.} \end{aligned}$$

(F12)

Consider now leader j. Assume $\beta _{j}-\tilde{b}_{j,\epsilon }\geqslant A$. Let $b_{j,\epsilon }(\delta )=\tilde{b}_{j,\epsilon }+\delta$ , with $\delta \in (0,\frac{1}{2}\min (\epsilon ,A)]$. As $\tilde{p} _{X}^{\epsilon }=\frac{\sum \nolimits _{j\in T_{Y}^{L}}\tilde{b}_{j,\epsilon }+\mathop {\textstyle \sum }\nolimits _{j\in T_{Y}^{F}}\varphi _{j}^{\epsilon }(\textbf{b} _{\epsilon }^{L};\textbf{q}_{\epsilon }^{L})+\epsilon }{ \sum \nolimits _{i\in T_{X}^{L}}\tilde{q}_{i,\epsilon }+\mathop {\textstyle \sum }\nolimits _{i\in T_{X}^{F}}\sigma _{i}^{\epsilon }(\textbf{b}_{\epsilon }^{L};\textbf{q} _{\epsilon }^{L})+\epsilon }$, then we have:

$$\begin{aligned} { x}_{j,\epsilon }{ (\delta )-x}_{j,\epsilon }= & {} \frac{ \sum \nolimits _{i}\tilde{q}_{i,\epsilon }+\sum \nolimits _{i}\sigma _{i}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L}+\delta )+\epsilon }{\sum \nolimits _{j}\tilde{b}_{j,\epsilon }+\delta +\sum \nolimits _{j}\varphi _{j}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L}+\delta )+\epsilon }{ (\tilde{ b}}_{j,\epsilon }{ +\delta )}-\frac{\tilde{b}_{j,\epsilon }}{\tilde{p} _{X}^{\epsilon }} \nonumber \\= & {} { \delta }\frac{\tilde{B}_{\epsilon }+\epsilon -(1+\nu _{\epsilon }^{Y})\tilde{b}_{j,\epsilon }}{\tilde{B}_{\epsilon }+\epsilon +(1+\nu _{\epsilon }^{Y})\delta }\frac{\tilde{Q}_{\epsilon }+\epsilon }{\tilde{B} _{\epsilon }+\epsilon }+{ \delta \eta }_{\epsilon }^{Y}\frac{\tilde{b} _{j,\epsilon }+\delta }{\tilde{B}_{\epsilon }+\epsilon +(1+\nu _{\epsilon }^{Y})\delta } \nonumber \\> & {} { \delta (1-\nu }_{\epsilon }^{Y}{ )}\frac{\frac{\tilde{\textbf{B}}_{\epsilon }}{2}+\frac{\epsilon }{2}+(1+\nu _{\epsilon }^{Y})\frac{ \delta }{2}}{\tilde{B}_{\epsilon }+\epsilon +(1+\nu _{\epsilon }^{Y})\delta } \frac{1}{\tilde{p}_{X}^{\epsilon }}+{ \delta a\eta }_{\epsilon }^{Y}\nonumber \\= & {} \frac{\delta }{2}\left( \frac{1-\nu _{\epsilon }^{Y}}{\tilde{p} _{X}^{\epsilon }}+2a\eta _{\epsilon }^{Y}\right) \text {, and} \end{aligned}$$

(F13)

$$\begin{aligned} y_{j,\epsilon }(\delta )-y_{j,\epsilon }=-\delta \text {,} \end{aligned}$$

(F14)

where $a\equiv \frac{\tilde{b}_{j,\epsilon }+\delta }{\tilde{B}_{\epsilon }+\epsilon +(1+\nu _{\epsilon }^{Y})\delta }$, with $0<a\leqslant 1$, $\nu _{\epsilon }^{Y}=\frac{\partial \sum \nolimits _{j}\varphi _{j}^{\epsilon }(. )}{\partial b_{j,\epsilon }}$ and $\eta _{\epsilon }^{Y}=\frac{ \partial \sum \nolimits _{i}\sigma _{i}^{\epsilon }(.)}{\partial b_{j,\epsilon }}$, and where the strict inequality results from $\tilde{B} _{\epsilon }+\epsilon -(1+\nu _{\epsilon }^{Y})\tilde{b}_{j,\epsilon }\geqslant (1-\nu _{\epsilon }^{Y})\frac{\tilde{B}_{\epsilon }}{2}+\epsilon >(1-\nu _{\epsilon }^{Y})(\frac{\tilde{B}_{\epsilon }}{2}+\frac{\epsilon }{2} +(1+\nu _{\epsilon }^{Y})\frac{\delta }{2})$ as $\tilde{b}_{j,\epsilon }$ $\leqslant \frac{\tilde{B}_{\epsilon }}{2}$, $\delta \leqslant \frac{1}{2} \epsilon$ and $\nu _{\epsilon }^{Y}\in [-1,\frac{1}{2}]$. Let us define

$$\begin{aligned} t=-2\frac{\tilde{p}_{X}^{\epsilon }}{1-\nu _{\epsilon }^{Y}+2a\eta _{\epsilon }^{Y}\tilde{p}_{X}^{\epsilon }}\textbf{e}^{Y}\text {.} \end{aligned}$$

(F15)

Then, the following vector inequality holds:

$$\begin{aligned}{} & {} \textbf{z}_{j,\epsilon }(b_{j,\epsilon }(\delta ),p_{X}^{\epsilon }(\tilde{{\textbf{q}}}_{\epsilon }^{L}\mathbf {;\textbf{q}}_{\epsilon }^{F} {(\tilde{{\textbf{q}}}}_{\epsilon }^{L}{;}b_{j,\epsilon }(\delta ),\tilde{\textbf{b}}_{-j,\epsilon }^{L});b_{j,\epsilon }(\delta ),\tilde{\textbf{b}}_{-j,\epsilon }^{L},\textbf{b}_{\epsilon }^{F}( {\tilde{{\textbf{q}}}}_{\epsilon }^{L}\textbf{;}b_{j,\epsilon }(\delta ),\tilde{\textbf{b}}_{-j,\epsilon }^{L}))) \nonumber \\{} & {} \quad \ge \textbf{z}_{j,\epsilon }(\tilde{b}_{j,\epsilon },\tilde{p}_{X}^{\epsilon } )+\frac{\delta }{2}\frac{1-\nu _{\epsilon }^{Y}+2a\eta _{\epsilon }^{Y}\tilde{p}_{X}^{\epsilon }}{\tilde{p}_{X}^{\epsilon }}( \textbf{e}^{X}+t)\text {.} \end{aligned}$$

(F16)

Let $c=X$, $\textbf{z}_{j,\epsilon }(\tilde{b}_{j,\epsilon },\tilde{p} _{X}^{\epsilon })$ and $\textbf{s}_{j,\epsilon }=\textbf{z} _{j,\epsilon }(\tilde{b}_{j,\epsilon },\tilde{p}_{X}^{\epsilon }\mathbf { )}+t$. We know $\textbf{z}_{j,\epsilon }(\tilde{b}_{j,\epsilon }, \tilde{p}_{X}^{\epsilon })\in {\mathbb {R}} _{+}^{2}$ and $\left\| \textbf{z}_{j,\epsilon }(\tilde{b}_{j,\epsilon }, \tilde{p}_{X}^{\epsilon })\right\| \leqslant H$. Suppose $\textbf{s}_{j,\epsilon }\in {\mathbb {R}} _{+}^{2}$ and $\left\| t\right\| \leqslant h$. Then, by Lemma 6:

$$\begin{aligned} u_{j}(\textbf{z}_{j,\epsilon }(\tilde{b}_{j,\epsilon },\tilde{p} _{X}^{\epsilon })+\textbf{e}^{X}+t)>u_{j}(\textbf{z} _{j,\epsilon }(\tilde{b}_{j,\epsilon },\tilde{p}_{X}^{\epsilon }\mathbf { )})\text {.} \end{aligned}$$

(F17)

From Assumptions (2b) and (2c) and as $0<\delta (\frac{1}{2}\frac{1-\nu _{\epsilon }^{Y}}{\tilde{p}_{X}^{\epsilon }}+a\eta _{\epsilon }^{Y})<1$, we deduce:

$$\begin{aligned} u_{j}(\textbf{z}_{j,\epsilon }(\tilde{b}_{j,\epsilon },\tilde{p} _{X}^{\epsilon })+\frac{\delta }{2}\frac{1-\nu _{\epsilon }^{Y}+2a\eta _{\epsilon }^{Y}\tilde{p}_{X}^{\epsilon }}{\tilde{p} _{X}^{\epsilon }}(\textbf{e}^{X}+t))>u_{j}(\textbf{z}_{j,\epsilon }(\tilde{b} _{j,\epsilon },\tilde{p}_{X}^{\epsilon }))\text {.} \end{aligned}$$

(F18)

But then, by Assumptions (2b) and (2c), we deduce:

$$\begin{aligned}{} & {} u_{j}(z_{j,\epsilon }(b_{j,\epsilon }(\delta ),p_{X}^{\epsilon }(\tilde{q} _{\epsilon }^{L};q_{\epsilon }^{F}(\tilde{q}_{\epsilon }^{L};b_{j,\epsilon }(\delta ),\tilde{b}_{-j,\epsilon }^{L});b_{j,\epsilon }(\delta ),\tilde{b} _{-j,\epsilon }^{L},b_{\epsilon }^{F}(\tilde{q}_{\epsilon }^{L};b_{j,\epsilon }(\delta ),\tilde{b}_{-j,\epsilon }^{L}))))\nonumber \\{} & {} \quad >u_{j}(\textbf{z}_{j,\epsilon }(\tilde{b}_{j,\epsilon },\tilde{p} _{X}^{\epsilon }))\text {,} \end{aligned}$$

(F19)

a contradiction. Hence, either $\textbf{z}_{j,\epsilon }(\tilde{b} _{j,\epsilon },\tilde{p}_{X}^{\epsilon })+t<\textbf{0}$ or $\left\| t\right\| >h$. Thus, if $\textbf{z}_{j,\epsilon }(\tilde{b} _{j,\epsilon },\tilde{p}_{X}^{\epsilon })+t<\textbf{0}$, then, $\tilde{y}_{j,\epsilon }-\frac{2\tilde{p}_{X}^{\epsilon }}{1-\nu _{\epsilon }^{Y}+2a\eta _{\epsilon }^{Y}\tilde{p}_{X}^{\epsilon }}<0$. Then, we have:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }>\frac{A}{2}\left( \frac{1-\nu _{\epsilon }^{Y}}{ 1-a\eta _{\epsilon }^{Y}A}\right) \text {,} \end{aligned}$$

(F20)

as $\tilde{y}_{j,\epsilon }=\beta _{j}-\tilde{b}_{j,\epsilon }\geqslant A$, where $\frac{A}{2}\frac{1-\nu _{\epsilon }^{Y}}{1-a\eta _{\epsilon }^{Y}A}>0$ . Reason: $\frac{A}{2}\frac{1-\nu _{\epsilon }^{Y}}{1-a\eta _{\epsilon }^{Y}A}\geqslant \frac{A}{2}(1-\nu _{\epsilon }^{Y})>0$. The strict inequality holds as $\frac{A}{2}>0$ and $\nu _{\epsilon }^{Y}<1$, while the weak inequality results from $a\eta _{\epsilon }^{Y}A\geqslant 0$ since $0<a\leqslant 1$, $A>0$, and $\eta _{\epsilon }^{Y}\geqslant 0$ (remind, from (2a), that $u_{j^{\prime }}\in \mathcal {C}^{\infty }$ so $\eta _{\epsilon }^{Y}$ cannot be negative, and $\chi \in [-1,1]$ in (E2)). Next, if $\left\| t\right\| >h$, then:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }>\frac{h}{2}\left( \frac{1-\nu _{\epsilon }^{Y}}{ 1-a\eta _{\epsilon }^{Y}h}\right) \text {,} \end{aligned}$$

(F21)

where $\frac{h}{2}\frac{1-\nu _{\epsilon }^{Y}}{1-a\eta _{\epsilon }^{Y}h}>0$ . Reason: $\frac{h}{2}\frac{1-\nu _{\epsilon }^{Y}}{1-a\eta _{\epsilon }^{Y}h}\geqslant \frac{h}{2}(1-\nu _{\epsilon }^{Y})>0$. The strict inequality holds as $\frac{h}{2}\in (0,\frac{1}{2})$ and $\nu _{\epsilon }^{Y}<1$, while the weak inequality results from $a\eta _{\epsilon }^{Y}h\geqslant 0$ since $0<a\leqslant 1$, $h\in (0,1)$, and $\eta _{\epsilon }^{Y}\geqslant 0$. Finally, assume that $\beta _{j}-\tilde{b} _{j,\epsilon }\geqslant A$ does not hold, i.e. $\beta _{j}-\tilde{b} _{j,\epsilon }<A$. Then, $\tilde{b}_{j,\epsilon }>\beta _{j}-A\geqslant A$ . Then, $\tilde{b}_{j,\epsilon }>A$, so we deduce:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }>\frac{A}{\bar{\alpha }}\text {.} \end{aligned}$$

(F22)

Therefore, it suffices to take for leader j:

$$\begin{aligned} \xi _{1}^{j}=\min \left( \frac{A}{2}\left( \frac{1-\nu _{\epsilon }^{Y}}{ 1-a\eta _{\epsilon }^{Y}A}\right) ,\frac{h}{2}\left( \frac{1-\nu _{\epsilon }^{Y}}{1-a\eta _{\epsilon }^{Y}h}\right) ,\frac{A}{\bar{\alpha }}\right) \text {.} \end{aligned}$$

(F23)

Then, by taking $\xi _{1}=\min (\xi _{1}^{j},\xi _{1}^{j^{\prime }})$, where $\xi _{1}>0$, we conclude that:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }>\xi _{1}\text {. } \end{aligned}$$

(F24)

2. Second, $\exists \xi _{2}>0\mid \tilde{p}_{X}^{\epsilon }<\xi _{2}$. Consider one leader i and one follower $i^{\prime }$. Let

$$\begin{aligned} \hat{h}= & {} \min (h(u_{j},Y,H),h(u_{j^{\prime }},Y,H))\text {;} \nonumber \\ \hat{A}= & {} \frac{1}{2}\min (\alpha _{i},\alpha _{i^{\prime }})\text {, }i\ne i^{\prime }\text {.} \end{aligned}$$

(F25)

Assume $\tilde{q}_{i,\epsilon }\leqslant \frac{\tilde{Q}_{\epsilon }}{2}$ or $\tilde{q}_{i^{\prime },\epsilon }\leqslant \frac{\tilde{Q}_{\epsilon }}{2}$ . Consider follower $i^{\prime }$. Assume $\alpha _{i^{\prime }}-\tilde{q} _{i^{\prime },\epsilon }\geqslant \hat{A}$. Let $q_{i^{\prime },\epsilon }(\delta )=\tilde{q}_{i^{\prime },\epsilon }+\delta$, with $\delta \in (0, \frac{1}{2}\min \{\epsilon ,\hat{A}\}]$. Then, his final holding is such that:

$$\begin{aligned} x_{i^{\prime },\epsilon }(\delta )-x_{i^{\prime },\epsilon }=-\delta \text { and }y_{i^{\prime },\epsilon }(\delta )-y_{i^{\prime },\epsilon }>\frac{ \delta }{2}\tilde{p}_{X}^{\epsilon }\text {,} \end{aligned}$$

(F26)

as $\tilde{Q}_{\epsilon }+\epsilon -\tilde{q}_{i^{\prime },\epsilon }\geqslant \frac{\tilde{Q}_{\epsilon }}{2}+\epsilon \geqslant \frac{\tilde{Q} _{\epsilon }+\epsilon -\tilde{q}_{i^{\prime },\epsilon }}{2}+\frac{\epsilon }{2}+\frac{\delta }{2}$ (as $\delta <\epsilon )$. Let us define

$$\begin{aligned} t=-\frac{2}{\tilde{p}_{X}^{\epsilon }}\textbf{e}^{X}\text {.} \end{aligned}$$

(F27)

Then, we have the vector inequality:

$$\begin{aligned} { \textbf{z}}_{i^{\prime },\epsilon }{ (q}_{i^{\prime },\epsilon }{ (\delta ),p}_{X}^{\epsilon }{ (\tilde{\textbf{q}}}_{\epsilon }^{L}{ ,q}_{i^{\prime },\epsilon }{ (\delta ),\tilde{\textbf{q}}} _{-i^{\prime },\epsilon }^{F}{ (.);\tilde{\textbf{b}}}_{\epsilon }^{L} { ,\tilde{\textbf{b}}}_{\epsilon }^{F}{ (.)))\ge \textbf{z}} _{i^{\prime },\epsilon }{ (\tilde{q}}_{i^{\prime },\epsilon }{ , \tilde{p}}_{X}^{\epsilon }{ )+}\frac{\delta }{2}{ \tilde{p}} _{X}^{\epsilon }{ (t+\textbf{e}}^{Y}{ )}\text {.} \end{aligned}$$

(F28)

Suppose that $\textbf{r}_{i,\epsilon }\in {\mathbb {R}} _{+}^{2}$ and $\left\| t\right\| \leqslant h$. Then, by Lemma 6, with $c=Y$, we get:

$$\begin{aligned} u_{i^{\prime }}(\textbf{z}_{i^{\prime },\epsilon }(\tilde{q}_{i^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon })+t+\textbf{e}^{Y})>u_{i^{\prime }}( \textbf{z}_{i^{\prime },\epsilon }(\tilde{q}_{i^{\prime },\epsilon },\tilde{p }_{X}^{\epsilon })). \end{aligned}$$

(F29)

As Assumptions (2b) and (2c) hold for $u_{i^{\prime }}$, and as $0<\frac{ \delta }{2}\tilde{p}^{\epsilon }<1$, then:

$$\begin{aligned} u_{i^{\prime }}\left( \textbf{z}_{i^{\prime },\epsilon }(\tilde{q}_{i^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon })+\frac{\delta }{2}\tilde{p} _{X}^{\epsilon }(t+\textbf{e}^{Y})\right) >u_{i^{\prime }}(\textbf{z}_{i^{\prime },\epsilon }(\tilde{q}_{i^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon })) \text {.} \end{aligned}$$

(F30)

But then, by Assumptions (2b) and (2c), we have that:

$$\begin{aligned} u_{i^{\prime }}(\textbf{z}_{i^{\prime },\epsilon }(q_{i^{\prime },\epsilon }(\delta ),p_{X}^{\epsilon }(q_{i^{\prime },\epsilon }(\delta ), \tilde{{\textbf{q}}}_{-i^{\prime },\epsilon };\tilde{\textbf{b}}_{\epsilon } )))>u_{i^{\prime }}(\textbf{z}_{i^{\prime },\epsilon }( \tilde{q}_{i^{\prime },\epsilon },\tilde{p}_{X}^{\epsilon }))\text {,} \end{aligned}$$

(F31)

a contradiction. Then, either $\textbf{z}_{i^{\prime },\epsilon }(\tilde{q} _{i^{\prime },\epsilon },p_{X}^{\epsilon })+t<\textbf{0}$ or $\left\| t\right\| >h$. Thus, if $\textbf{z}_{i^{\prime },\epsilon }(\tilde{q} _{i^{\prime },\epsilon },p_{X}^{\epsilon })+t<\textbf{0}$, then, $\tilde{x} _{i^{\prime },\epsilon }-\frac{2}{\tilde{p}_{X}^{\epsilon }(\tilde{\textbf{q} }_{\epsilon };\tilde{\textbf{b}}_{\epsilon })}<0$. As $\tilde{x}_{i^{\prime },\epsilon }=\alpha _{i^{\prime }}-\tilde{q}_{i^{\prime },\epsilon }\geqslant \hat{A}$, we deduce:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }<\frac{2}{\hat{A}}\text {.} \end{aligned}$$

(F32)

Suppose now we have $\left\| t\right\| >h$. Then, we deduce:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }<\frac{2}{\hat{h}}\text {.} \end{aligned}$$

(F33)

Finally, assume $\alpha _{i}-\tilde{q}_{i^{\prime },\epsilon }<\hat{A}$. Then, $\tilde{q}_{i^{\prime },\epsilon }>\alpha _{i}-\hat{A}\geqslant \hat{A}$, so $\tilde{q}_{i^{\prime },\epsilon }>\hat{A}$. We deduce:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }<\frac{\bar{\beta }}{\hat{A}}\text {.} \end{aligned}$$

(F34)

Therefore, it is sufficient to take:

$$\begin{aligned} \xi _{2}^{i^{\prime }}=\max \left( \frac{2}{\hat{A}},\frac{2}{\hat{h}},\frac{ \bar{\beta }}{\hat{A}}\right) \text {.} \end{aligned}$$

(F35)

Consider now leader i. Assume that $\alpha _{i}-\tilde{q}_{i,\epsilon }\geqslant \hat{A}$. Let $q_{i,\epsilon }(\delta )=\tilde{q}_{i,\epsilon }+\delta$, with $\delta \in (0,\frac{1}{2}\min (\epsilon ,\hat{A})]$. Such an increase has the following effect on her final holding:

$$\begin{aligned}{} & {} x_{i,\epsilon }(\delta )-x_{i,\epsilon }=-\delta \text {, and} \end{aligned}$$

(F36)

$$\begin{aligned}{} & {} { y}_{i,\epsilon }{ (\delta )-y}_{i,\epsilon } =\frac{ \sum \nolimits _{j}\tilde{b}_{j,\epsilon }+\sum \nolimits _{j}\varphi _{j}^{\epsilon }(\textbf{b}_{\epsilon }^{L};\textbf{q}_{\epsilon }^{L}+\delta )+\epsilon }{\sum \nolimits _{i}\tilde{q}_{i,\epsilon }+\delta +\sum \nolimits _{i}\sigma _{i}^{\epsilon }(\textbf{b}_{\epsilon }^{L};\textbf{q}_{\epsilon }^{L}+\delta )+\epsilon }{ (\tilde{ q}}_{i,\epsilon }{ +\delta )}-{ \tilde{p}}_{X}^{\epsilon } { \tilde{q}}_{i,\epsilon } \nonumber \\= & {} { \delta }\frac{\tilde{Q}_{\epsilon }+\epsilon -(1+\nu _{\epsilon }^{X})\tilde{q}_{i,\epsilon }}{\tilde{Q}_{\epsilon }+(1+\nu _{\epsilon }^{X})\delta +\epsilon }\frac{\tilde{B}_{\epsilon }+\epsilon }{\tilde{Q} _{\epsilon }+\epsilon }+{ \delta \eta }_{\epsilon }^{X}\frac{\tilde{q} _{i,\epsilon }+\delta }{\tilde{Q}_{\epsilon }+(1+\nu _{\epsilon }^{X})\delta +\epsilon } \nonumber \\> & {} { \delta (1-\nu }_{\epsilon }^{X}{ )}\frac{\frac{\tilde{Q} _{\epsilon }}{2}+(1+\nu _{\epsilon }^{X})\frac{\delta }{2}+\frac{\epsilon }{2 }}{\tilde{Q}_{\epsilon }+(1+\nu _{\epsilon }^{X})\delta +\epsilon }{ \tilde{p}}_{X}+{ \delta d\eta }_{\epsilon }^{X} \nonumber \\= & {} \frac{\delta }{2}{ ((1-\nu }_{\epsilon }^{X}{ )\tilde{p}} _{X}^{\epsilon }{ +2d\eta }_{\epsilon }^{X}{ )}\text {,} \end{aligned}$$

(F37)

where $d\equiv \frac{\tilde{q}_{i,\epsilon }+\delta }{\tilde{Q}_{\epsilon }+(1+\nu _{\epsilon }^{X})\delta +\epsilon }$, with $0<d\leqslant 1$,$\ \nu _{\epsilon }^{X}=\frac{\partial \sum \nolimits _{i}\sigma _{i}^{\epsilon }( \textbf{q}_{\epsilon }^{L};\textbf{b}_{\epsilon }^{L};\epsilon )}{ \partial q_{i,\epsilon }}$ and $\eta _{\epsilon }^{X}=\frac{\partial \sum \nolimits _{j}\varphi _{j}^{\epsilon }(\textbf{q}_{\epsilon }^{L};\textbf{ b}_{\epsilon }^{L};\epsilon )}{\partial q_{i,\epsilon }}$ for $\delta$ sufficiently small, and where the strict inequality results from $\tilde{Q}_{\epsilon }+\epsilon -(1+\nu _{\epsilon }^{X})\tilde{q} _{i,\epsilon }\geqslant (1-\nu _{\epsilon }^{X})(\frac{\tilde{Q}_{\epsilon } }{2}+\frac{\epsilon }{2}+(1+\nu _{\epsilon }^{X})\frac{\delta }{2})$ as $\tilde{q}_{i,\epsilon }\leqslant \frac{\tilde{Q}_{\epsilon }}{2}$ always holds, and as $\delta <\epsilon$, with $\nu _{\epsilon }^{X}\in [-1,1)$. Let us define

$$\begin{aligned} t=-2\frac{1}{(1-\nu _{\epsilon }^{X})\tilde{p}_{X}^{\epsilon }+2d\eta _{\epsilon }^{X}}\textbf{e}^{X}\text {.} \end{aligned}$$

(F38)

Then, the following vector inequality holds:

$$\begin{aligned}{} & {} \textbf{z}_{i,\epsilon }(q_{i,\epsilon }(\delta ),p_{X}^{\epsilon }(q_{i,\epsilon }(\delta ),\tilde{{\textbf{q}}}_{-i,\epsilon }^{L} \mathbf {;\textbf{q}}_{\epsilon }^{F}\mathbf {(}q_{i,\epsilon }(\delta ), {\tilde{{\textbf{q}}}}_{\epsilon }^{L};\tilde{\textbf{b}} _{\epsilon }^{L});\tilde{\textbf{b}}_{\epsilon }^{L},\textbf{b} _{\epsilon }^{F}(q_{j,\epsilon }(\delta ),{\tilde{{\textbf{q}}} }_{-i,\epsilon }^{L};\tilde{\textbf{b}}_{\epsilon }^{L})))\nonumber \\{} & {} \quad \ge \textbf{z}_{i,\epsilon }(\tilde{q}_{i,\epsilon },\tilde{p}_{X}^{\epsilon } )+\frac{\delta }{2}((1-\nu _{\epsilon }^{X})\tilde{p} _{X}^{\epsilon }+2d\eta _{\epsilon }^{X})(t+\textbf{e}^{Y})\text {.} \end{aligned}$$

(F39)

Suppose that $\textbf{s}_{i,\epsilon }\in {\mathbb {R}} _{+}^{2}$ and $\left\| t\right\| \leqslant h$. Then, by Lemma 6, we deduce:

$$\begin{aligned} u_{i}(\textbf{z}_{i,\epsilon }(\tilde{q}_{i,\epsilon },\tilde{p} _{X}^{\epsilon })+t+\textbf{e}^{Y})>u_{i}(\textbf{z} _{i,\epsilon }(\tilde{q}_{i,\epsilon },\tilde{p}_{X}^{\epsilon }\mathbf { )})\text {.} \end{aligned}$$

(F40)

From (2b) and (2c) and as $0<\delta (\frac{1-\nu _{\epsilon }^{X}}{2}\tilde{p }_{X}^{\epsilon }+d\eta _{\epsilon }^{X})<1$, we deduce:

$$\begin{aligned} u_{i}(\textbf{z}_{i,\epsilon }(\tilde{q}_{i,\epsilon },\tilde{p} _{X}^{\epsilon }))+\frac{\delta }{2}((1-\nu _{\epsilon }^{X})\tilde{p}_{X}^{\epsilon }+2d\eta _{\epsilon }^{X})(t+\textbf{e} ^{Y}))>u_{i}(\textbf{z}_{i,\epsilon }(\tilde{q}_{i,\epsilon },\tilde{p} _{X}^{\epsilon }))\text {.} \end{aligned}$$

(F41)

But then, by Assumptions (2b) and (2c), we have that:

$$\begin{aligned}{} & {} u_{i}(q_{i,\epsilon }(\delta ),p_{X}^{\epsilon }(q_{i,\epsilon }(\delta ), \tilde{{\textbf{q}}}_{-i,\epsilon }^{L};\textbf{q}_{\epsilon }^{F}(q_{i,\epsilon }(\delta ),\tilde{{\textbf{q}}} _{-i,\epsilon }^{L};\tilde{\textbf{b}}_{\epsilon }^{L});\tilde{\textbf{b}}_{\epsilon }^{L},\textbf{b}_{\epsilon }^{F}(q_{j,\epsilon }(\delta ),\tilde{{\textbf{q}}}_{-i,\epsilon }^{L};\tilde{\textbf{b}} _{\epsilon }^{L})))) \nonumber \\{} & {} \quad >u_{i}(\textbf{z}_{i,\epsilon }(\tilde{q}_{i,\epsilon },\tilde{p} _{X}^{\epsilon }))\text {,} \end{aligned}$$

(F42)

a contradiction. Then, either $\textbf{z}_{i,\epsilon }(\tilde{q} _{i,\epsilon },\tilde{p}_{X}^{\epsilon })+t<\textbf{0}$ or $\left\| t\right\| >h$. Thus, if $\textbf{z}_{i,\epsilon }(\tilde{q} _{i,\epsilon },\tilde{p}_{X}^{\epsilon })+t<\textbf{0}$, then, $\tilde{x}_{i,\epsilon }-2\frac{1}{(1-\nu _{\epsilon }^{X})\tilde{p} _{X}^{\epsilon }+2d\eta _{\epsilon }^{X}}<0$. As $\tilde{x}_{i,\epsilon }=\alpha _{i}-\tilde{q}_{i,\epsilon }\geqslant \hat{A}$, we get:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }<\frac{2}{\hat{A}}\left( \frac{1-d\eta _{\epsilon }^{X}\hat{A}}{1-\nu _{\epsilon }^{X}}\right) \text {,} \end{aligned}$$

(F43)

where $\frac{2}{\hat{A}}\frac{1-d\eta _{\epsilon }^{X}\hat{A}}{1-\nu _{\epsilon }^{X}}>0$. Reason: $\frac{2}{\hat{A}}\frac{1-d\eta _{\epsilon }^{X}\hat{A}}{1-\nu _{\epsilon }^{X}}\geqslant \frac{2}{\hat{A}}\frac{d^{2}}{ 1-\nu _{\epsilon }^{X}}>0$ as $d\in (0,1]$ and $\nu _{\epsilon }^{X}<1$. The weak inequality leads to $d^{2}+d\eta _{\epsilon }^{X}\hat{A}-1\leqslant 0$, so $d\leqslant -\frac{\eta _{\epsilon }^{X}\hat{A}}{2}+\frac{\sqrt{(\eta _{\epsilon }^{X}\hat{A})^{2}+4}}{2}$, with $0\leqslant d\leqslant 1$. Then we must have $-\frac{\eta _{\epsilon }^{X}\hat{A}}{2}+\frac{\sqrt{(\eta _{\epsilon }^{X}\hat{A})^{2}+4}}{2}\leqslant 1$, which holds as $\eta _{\epsilon }^{X}\hat{A}\geqslant 0$.

Next, if $\left\| t\right\| >h$, then:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }<\frac{2}{\hat{h}}\left( \frac{1-d\eta _{\epsilon }^{X}\hat{h}}{1-\nu _{\epsilon }^{X}}\right) \text {,} \end{aligned}$$

(F44)

where $\frac{2}{\hat{h}}\frac{1-d\eta _{\epsilon }^{X}\hat{h}}{1-\nu _{\epsilon }^{X}}>0$. Reason: $\frac{2}{\hat{h}}\frac{1-d\eta _{\epsilon }^{X}\hat{h}}{1-\nu _{\epsilon }^{X}}\geqslant \frac{2}{\hat{h}}\frac{d^{2}}{ 1-\nu _{\epsilon }^{X}}>0$. The weak inequality leads to $d^{2}+d\eta _{\epsilon }^{X}\hat{h}-1\leqslant 0$, which yields $d\leqslant -\frac{\eta _{\epsilon }^{X}\hat{h}}{2}+\frac{\sqrt{(\eta _{\epsilon }^{X}\hat{h})^{2}+4} }{2}$, with $d>0$. As $d\leqslant 1$, we must have $-\frac{\eta _{\epsilon }^{X}\hat{h}}{2}+\frac{\sqrt{(\eta _{\epsilon }^{X}\hat{h})^{2}+4}}{2} \leqslant 1$, which is satisfied as $\eta _{\epsilon }^{X}\hat{h}\geqslant 0$ .

Finally, assume that the inequality $\alpha _{i}-\tilde{q}_{i,\epsilon }\geqslant \hat{A}$ does not hold, i.e., $\alpha _{i}-\tilde{q}_{i,\epsilon }<\hat{A}$. Then, we have $\tilde{q}_{i,\epsilon }>\alpha _{i}-\hat{A} \geqslant \hat{A}$. Then, we have $\tilde{q}_{i,\epsilon }>\hat{A}$, so we deduce:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }<\frac{\bar{\beta }}{\hat{A}}\text {.} \end{aligned}$$

(F45)

Therefore, it suffices to take for leader i:

$$\begin{aligned} \xi _{1}^{i}=\max \left( \frac{2}{\hat{A}}\left( \frac{1-d\eta _{\epsilon }^{X}\hat{A}}{1-\nu _{\epsilon }^{X}}\right) ,\frac{2}{\hat{h}}\left( \frac{ 1-d\eta _{\epsilon }^{X}\hat{h}}{1-\nu _{\epsilon }^{X}}\right) ,\frac{\bar{ \beta }}{\hat{A}}\right) \text {.} \end{aligned}$$

(F46)

Then, by taking $\xi _{2}=\max (\xi _{2}^{i},\xi _{2}^{i^{\prime }})$, we conclude that:

$$\begin{aligned} \tilde{p}_{X}^{\epsilon }<\xi _{2}\text {.} \end{aligned}$$

(F47)

$\square$

1.7 Appendix G: Proof of Theorem 2

To show that the Stackelberg–Nash equilibria are locally unique, i.e., that the set of critical equilibria is negligible, we use the Regular Value Theorem and Sard’s Theorem, which we restate, and we show that the set of regular values of the mapping which defines the equilibrium strategies is of full measure. Before we give one definition (see notably Debreu 1970; and Milnor 1997).

Definition 7

(Critical point, critical value, regular point, regular value) Let $\textbf{F}:\mathcal {O}{\subset {\mathbb {R}} }^{n}\rightarrow { {\mathbb {R}} }^{m}$, $\mathcal {O}$ open, be a differentiable vector function. A point $\textbf{x}\in \mathcal {O}$ is a critical point of $\textbf{F}$ at $\textbf{x}$ if $r(\mathcal {J}_{\textbf{F}}(\textbf{x}))<m$, where r is the rank of the Jacobian matrix $\mathcal {J}_{\textbf{F}}(\textbf{x})$ of $\textbf{F}$ at $\textbf{x}$. A point $\textbf{y}\in { {\mathbb {R}} }^{m}$ is a critical value of $\textbf{F}$ if there a critical point $\textbf{x}\in \mathcal {O}$ with $\textbf{y}=\textbf{F}(\textbf{x})$. A point $\textbf{x}\in \mathcal {O}$ (resp.$\ \textbf{y}\in { {\mathbb {R}} }^{m}$) is a regular point (resp. value) of $\textbf{F}$ if $\textbf{x}$ (resp.$\ \textbf{F}$) is not a critical point (resp. value).

Theorem 3

(Regular Value Theorem) Let $\textbf{F}: {\mathbb {R}} ^{n}\rightarrow {\mathbb {R}} ^{m}$ be a $C^{k}$ function. Let $\textbf{y}\in {\mathbb {R}} ^{m}$ be a regular value of $\textbf{F}$. Then, the preimage $\textbf{F}^{-1}(\textbf{y})$ is a smooth manifold of ${\mathbb {R}} ^{n}$ of dimension $n-m$.

Theorem 4

(Sard’s Theorem) Let $\textbf{F}: {\mathbb {R}} ^{n}\rightarrow {\mathbb {R}} ^{m}$ be a $C^{k}$ function, where $k\geqslant \max (n-m+1,1)$. Let C be the set of critical points of $\textbf{F}$, i.e., $C=\{\textbf{x}\in {\mathbb {R}} ^{n}\mid \left| \mathcal {J}_{\textbf{F}_{\textbf{x}}}(\textbf{x} )\right| =0\}$. Then, the set of criticial values of $\textbf{F }$ has Lebesgue measure zero in ${\mathbb {R}} ^{m}$, i.e., the set of regular values of $\textbf{F}$ is dense in ${\mathbb {R}} ^{m}$.

Consider the subgame $\Gamma _{L}$. Define the family of functions ${{\varvec{\Lambda }}}_{L}:{\mathcal {S}}^{L}\rightarrow {\mathcal {S}}^{L}$, with ${{\varvec{\Lambda }}}_{L}(\textbf{q}^{L},{{\varvec{\sigma }}}(\textbf{q}^{L};\textbf{b}^{L}); \textbf{b}^{L},{{\varvec{\varphi }}}(\textbf{q}^{L};\textbf{b}^{L}))={\times }_{i\in T_{X}^{L}}\phi _{i}{\times }_{j\in T_{Y}^{L}}\psi _{j}$ , where the smooth functions $\phi _{i}$, for $i\in T_{X}^{L}$, and $\psi _{j}$, for $j\in T_{Y}^{L}$, exist from Proposition 4 and Lemma 5, and the smooth vector valued functions ${{\varvec{\sigma }}}(\textbf{q}^{L};\textbf{b}^{L})$ and ${{\varvec{\varphi }}}(\textbf{q}^{L};\textbf{b}^{L})$ exist from Lemmas 1 and 5. To show that the set of equilibria of $\Gamma _{L}$ is discrete, we rewrite the equilibrium conditions in $\Gamma _{L}$ as follows. Define the function $\Phi _{i}:\textbf{S}^{L}\rightarrow {\mathcal {S}}_{i}$, $\Phi _{i}( \textbf{q}^{L};\textbf{b}^{L}):=q_{i}-\phi _{i}(\textbf{q} _{-i}^{L};\textbf{b}^{L})$, with $\Phi _{i}(.)\in \mathcal {C} ^{\infty }$, $i\in T_{X}^{L}$, and the function $\Psi _{j}:\textbf{S} ^{L}\rightarrow {\mathcal {S}}_{j}$, $\Psi _{j}(\textbf{q}^{L};\textbf{b} ^{L}):=b_{j}-\psi _{j}(\textbf{q}^{L};\textbf{b}^{L})$, with $\Psi _{j}(.)\in \mathcal {C}^{\infty }$, $j\in T_{Y}^{L}$. Consider the system of $m_{X}+m_{Y}$ equations:

$$\begin{aligned} \left\{ \begin{array}{c} \Phi _{i}(\textbf{q}^{L};\textbf{b}^{L})=0\text {, }i\in T_{X}^{F}\text {,} \\ \\ \Psi _{j}(\textbf{q}^{L};\textbf{b}^{L})=0\text {, }j\in T_{Y}^{F}\text {.} \end{array} \right. \end{aligned}$$

(G1)

The system (G1) may be rewritten as

$$\begin{aligned} {{\varvec{\Upsilon }}}_{L}(\textbf{q}^{L};\textbf{b}^{L})=\textbf{0}\text {,} \end{aligned}$$

(G2)

where ${{\varvec{\Upsilon }}}:\textbf{S}^{L}\rightarrow \textbf{S}^{L}$, ${{\varvec{\Upsilon }}}(\textbf{q}^{L};\textbf{b}^{L})=(\Phi _{1},\ldots ,\Phi _{m_{X}};\Psi _{1},\ldots ,\Psi _{m_{Y}})$, with ${{\varvec{\Upsilon }}}_{L}(\textbf{ q}^{L};\textbf{b}^{L})\in \mathcal {C}^{\infty }$.

First, we show the differential $d{{\varvec{\Upsilon }}}_{L}$ of the mapping ${{\varvec{\Upsilon }}}_{L}$ is surjective, which is equivalent to show that the Jacobian matrix $\mathcal {J}_{{{\varvec{\Upsilon }}}_{L}}$of ${{\varvec{\Upsilon }}} _{L}$ at $(\textbf{q}^{L};\textbf{b}^{L})$ has rank $m_{X}+m_{Y}$. This Jacobian matrix may be written:

$$\begin{aligned} \mathcal {J}_{{{\varvec{\Upsilon }}}_{L}}{ (}\textbf{q}^{L}{ ;} \textbf{b}^{L}{ )=}\left[ \begin{array}{cccc} 1 &{} -\frac{\partial \phi _{1}}{\partial q_{2}} &{} \cdots &{} -\frac{\partial \phi _{1}}{\partial b_{m_{Y}}} \\ -\frac{\partial \phi _{2}}{\partial q_{1}} &{} 1 &{} \cdots &{} -\frac{\partial \phi _{21}}{\partial b_{m_{Y}}} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ -\frac{\partial \psi _{m_{Y}}}{\partial q_{1}} &{} \ldots &{} -\frac{\partial \psi _{m_{Y}}}{\partial b_{m_{Y}-1}} &{} 1 \end{array} \right] \text {,} \end{aligned}$$

(G3)

and it has dimension $(m_{X}+m_{Y},m_{X}+m_{Y})$. To show that the matrix (G3) has rank $m_{X}+m_{Y}$, it is sufficient to show that its off-diagonal elements are such that for each $i\in T_{X}^{L}$, $\frac{\partial \phi _{i} }{\partial q_{-i}}\ne -1$, $-i\ne i$, $-i\in T_{X}^{L}$, and $\frac{ \partial \phi _{i}}{\partial b_{j}}\ne -1$, $j\in T_{Y}^{L}$ (likewise, for each $j\in T_{Y}^{L}$,$\ \frac{\partial \psi _{j}}{\partial b_{-j}}\ne -1$, $-j\ne j$, $-j\in T_{Y}^{L}$, and $\frac{\partial \psi _{j}}{\partial q_{i}}\ne -1$, $i\in T_{X}^{L}$). The result obtains by using a similar argument as the one used in Appendix B.¹³ Then, the rows of the matrix $\mathcal {J}_{{{\varvec{\Upsilon }}} _{L}}(\textbf{q}^{L};\textbf{b}^{L})$ are linearly independent, so we have $\left| \mathcal {J}_{{{\varvec{\Upsilon }}}_{L}}(\textbf{q}^{L};\textbf{b} ^{L})\right| \ne 0$. Then, the matrix $\mathcal {J}_{{{\varvec{\Upsilon }}} _{L}}(\textbf{q}^{L};\textbf{b}^{L})$ has rank $m_{X}+m_{Y}$. By the Regular Value Theorem, we deduce from the above that the set of solutions $\mathcal {N}_{L}$ is a manifold of dimension zero, i.e., a set of isolated points. Then, the set $\mathcal {N}_{L}$ of regular equilibria of ${\varvec{\Gamma }}_{L}$ is discrete. Second, we show that the set of critical equilibria is a null set. As $\#T_{X}=m_{X}<\infty$, $\#T_{Y}=m_{Y}<\infty$, and ${{\varvec{\Upsilon }}}_{L}\in \mathcal {C}^{\infty }$, then, from Sard’s Theorem, the set of critical equilibria is a null set. Then, as a complement of this set, the set of regular equilibria of ${\varvec{\Gamma }} _{L}$ is a dense set in the product space of strategy sets (it is of full measure set in ${\mathcal {S}}^{L}$). Finally, as the smooth mapping ${{\varvec{\Upsilon }}}_{L}$ is proper (${{\varvec{\Upsilon }}}_{L}^{-1}(A)$ is compact whenever $A\subset {\mathcal {S}}^{L}$ is compact) , the set of critical equilibria of ${\varvec{\Gamma }}_{L}$ is closed.¹⁴ Therefore, the set $\mathcal {N}_{L}$ of regular equilibria of ${\varvec{\Gamma }}_{L}$ is open.

Consider now the subgame ${\varvec{\Gamma }}_{F}$, and let $(\tilde{\textbf{q} }^{L};\tilde{\textbf{b}}^{L})$ be a locally unique equilibrium of $\Gamma _{L}$. Define ${\varvec{\Lambda }}_{F}:\textbf{S}\rightarrow \textbf{S}^{F}$, ${\varvec{\Lambda }}_{F}(\tilde{\textbf{q}}^{L},\textbf{q}^{F};\tilde{\textbf{b }}^{L},\textbf{b}^{F})={\times }_{i\in T_{X}^{F}}\phi _{i}{\times }_{j\in T_{Y}^{F}}\psi _{j}$, where $\textbf{q}^{F}\in {{\varvec{\sigma }}}(\textbf{q}^{L};\textbf{b}^{L})$ and $\textbf{b}^{F}\in {{\varvec{\varphi }}}( \textbf{q}^{L};\textbf{b}^{L})$. Then, as from Lemma 1, i.e., there exist unique $\mathcal {C}^{\infty }$ functions $b_{i,\epsilon }= \sigma _{i}^{\epsilon }(q_{\epsilon }^{L};b_{\epsilon }^{L};\epsilon )$, for $i\in T_{X}^{F}$, and $q_{j,\epsilon }= \varphi _{j}^{\epsilon }(q^{L};b^{L};\epsilon )$, for $j\in T_{Y}^{F}$, and from Lemma 5, $(\textbf{q}^{F};\textbf{b}^{F}):=({{\varvec{\sigma }}}(\textbf{q}^{L};\textbf{b}^{L});{{\varvec{\varphi }}}(\textbf{q}^{L}; \textbf{b}^{L}))$ is a well-defined strategy profile of ${\varvec{\Gamma }} _{F}$, we deduce that the set of equilibria in ${\varvec{\Gamma }}_{F}$, i.e., the set of fixed points $(\tilde{\textbf{q}}^{F};\tilde{\textbf{b}}^{F})$ of ${\varvec{\Lambda }}_{F}(.)$, with ${\varvec{\Lambda }}_{F}(\tilde{\textbf{q}} ^{L},\textbf{q}^{F};\tilde{\textbf{b}}^{L},\textbf{b}^{F})={\times } _{i\in T_{X}^{F}}\phi _{i}{\times }_{j\in T_{Y}^{F}}\psi _{j}$, is discrete. $\square$