## 4 Application to Parametric Network Games
### The Linear-Quadratic Network Generalized Nash Equilibrium Problem  
We start this subsection with a recall of generalized Nash equilibrium problems (GNEPs) with shared constraints and their variational solutions. More details can be found in the already-mentioned paper by [PERSON], and in the more recent papers [31; 32; 33; 34; 35; 36; 37; 38]. We directly introduce the parametric case, and all the following definitions and properties hold for any fixed \(t\in[0,T]\). In GNEPs, the strategy set of each player is dependent on the strategies of their rivals. In this context, we examine a simplified scenario where the variable \(x\) is constrained to be non-negative (\(x\geq 0\)). Moreover, we consider a function \(g:[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\), which characterizes the collective constraints shared among the players. For each \(t\), and for each move \(x_{-i}\) of the other players, the strategy set of player \(i\) is then given by:  
\[K_{i}(t,x_{-i})=\{x_{i}\in\mathbb{R}_{+}:\ g(t,x)=g(t,x_{i},x_{-i})\leq 0\}.\]  
**Definition 7**. : _The GNEP is the problem of finding, for each \(t\in[0,T]\), \(x^{*}(t)\in\mathbb{R}^{n}\) such that, for any \(i\in\{1,\ldots,n\}\), \(x^{*}_{i}(t)\in K_{i}(t,x^{*}_{-i}(t))\) and_  
\[u_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t))\geq u_{i}(x_{i},x^{*}_{-i}(t)),\qquad\forall \ t\in[0,T],\ \forall\ x_{i}\in K_{i}(t,x^{*}_{-i}(t)). \tag{31}\]  
We assume that that, for each fixed \(t\in[0,T],\ x_{i}\in K_{i}(t,x^{*}_{-i})\), the functions \(u_{i}(t,\cdot,x_{-i})\) are concave and continuously differentiable, and the components of \(g(t,\cdot)\) are convex and continuously differentiable. As a consequence, a necessary and sufficient condition for \(x^{*}_{i}(t)\in K_{i}(t,x^{*}_{-i})\) to satisfy (31) is  
\[-\frac{\partial u_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t))}{\partial x_{i}}(x_{i}-x^ {*}_{i}(t))\geq 0,\qquad\forall t\in[0,T],\ \forall\ x_{i}\in K_{i}(t,x^{*}_{-i}(t)). \tag{32}\]  
Thus, if we define \(F(t,x)\) as in (26), and  
\[K(t,x)=K_{1}(t,x_{-1})\times\cdots\times K_{n}(t,x_{-n}),\]  
it follows that \(x^{*}(t)\) is a GNE if and only if, \(\forall t\in[0,T]\), \(x^{*}(t)\in K(t,x^{*}(t))\) and  
\[\langle F(t,x^{*}(t)),x-x^{*}(t)\rangle\geq 0,\qquad\forall\ x\in K(t,x^{*}(t)). \tag{33}\]  
The above problem, where the feasible set also depends on the solution, is called a quasi-variational inequality (with parameter \(t\)), and its solution is as difficult as the original GNEP. Let us now remark that even if the strict monotonicity assumption on \(F\) is satisfied for each \(t\), the above problem (equivalent to the original GNEP) has infinite solutions. We can select a specific solution by studying its Lagrange multipliers. Thus, assume that, for each \(t\in[0,T]\), \(x^{*}(t)\) is a solution of GNEP. Hence, for each \(i\), \(x^{*}_{i}(t)\) solves the maximization problem  
\[\max_{x_{i}}\{u_{i}(t,x_{i},x^{*}_{-i}(t)):\ g(t,x_{i},x^{*}_{-i}(t))\leq 0, \ x_{i}\geq 0\}.\]  
Under some standard constraint qualification, we can then write the KKT conditions for each maximization problem. We then introduce the Lagrange multiplier \(\lambda^{i}(t)\in\mathbb{R}^{m}\) associated with the constraint \(g(t,x_{i},x^{*}_{-i})\leq 0\) and the multiplier \(\mu_{i}(t)\in\mathbb{R}\) associated with the nonnegativity constraint \(x_{i}\geq 0\).