## 4 Application to Parametric Network Games
### The Linear-Quadratic Network Generalized Nash Equilibrium Problem
The Lagrangian function for each player \(i\) reads as:  
\[L_{i}(t,x_{i},x^{*}_{-i}(t),\lambda^{i}(t),\mu_{i}(t))=u_{i}(t,x_{i},x^{*}_{- i}(t))-\langle g(t,x_{i},x^{*}_{-i}(t)),\lambda^{i}(t)\rangle+\mu_{i}(t)x_{i}\]  
and the KKT conditions for all players are given by:  
\[\
abla_{x_{i}}L_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t),\lambda^{i*}(t),\mu^{*}_{i}(t))=0, i=1,\ldots,n, \tag{34}\] \[\lambda^{i*}_{i}(t)g_{j}(t,x^{*}(t))=0,\ \lambda^{i*}_{i}(t)\geq 0, g_{j}(t,x^{*}(t))\leq 0, i=1,\ldots,n, j=1,\ldots,m\] (35) \[\mu^{*}_{i}(t)x^{*}_{i}(t)=0,\ \mu^{*}_{i}(t)\geq 0, x^{*}_{i}(t)\geq 0, i=1,\ldots,n. \tag{36}\]  
Conversely, under the assumptions made, if, for any fixed \(t\), \((x^{*}(t),\lambda(t),\mu^{*}(t))\), where \(\lambda^{*}(t)=(\lambda^{1*}(t),\ldots,\lambda^{n*}(t))\) and \(\mu^{*}(t)=(\mu^{*}_{1}(t),\ldots,\mu^{*}_{n}(t))\), satisfy the KKT system (34)-(36), then \(x^{*}(t)\) is a GNE. We are now in position to classify equilibria. **Definition 8**. : _Let \(x^{*}(t)\) be a GNE which, together with the Lagrange multipliers \(\lambda^{*}(t)=(\lambda^{1*}(t),\ldots,\lambda^{n*}(t))\) and \(\mu^{*}(t)=(\mu^{*}_{1}(t),\ldots,\mu^{*}_{n}(t))\), satisfies the KKT system of all players. We call \(x^{*}(t)\) a normalized equilibrium if there exists a vector \(r(t)\in\mathbb{R}^{n}_{++}\) and a vector \(\bar{\lambda}(t)\in\mathbb{R}^{m}_{+}\) such that_  
\[u_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t))\geq u_{i}(x_{i},x^{*}_{-i}(t)),\qquad\forall \ t\in[0,T],\ \forall\ x_{i}\in K_{i}(t,x^{*}_{-i}(t)). \tag{31}\]0which means that, for a normalized equilibrium, the multipliers of the constraints shared by all players are proportional to a common multiplier. In the special case where, for some \(t\), \(r_{i}(t)=1\) for any \(i\), i.e., the multipliers coincide for each player, \(x^{*}(t)\) is called variational equilibrium (VE). [PERSON] (1954) proved that if the feasible set, which in our case is:_  
\[u_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t))\geq u_{i}(x_{i},x^{*}_{-i}(t)),\qquad\forall \ t\in[0,T],\ \forall\ x_{i}\in K_{i}(t,x^{*}_{-i}(t)). \tag{31}\]1  
_is compact and convex, then there exists a normalized equilibrium for each \(r(t)\in\mathbb{R}_{++}^{n}\)._  
Now, let us define, for each \(r(t)\in\mathbb{R}_{++}^{n}\), the vector function \(F^{r(t)}:\mathbb{R}^{n}\to\mathbb{R}^{n}\) as follows:  
\[u_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t))\geq u_{i}(x_{i},x^{*}_{-i}(t)),\qquad\forall \ t\in[0,T],\ \forall\ x_{i}\in K_{i}(t,x^{*}_{-i}(t)). \tag{31}\]2  
The variational inequality approach for finding the normalized equilibria of the GNEP is expressed by the following theorem, which can be viewed as a special case of Proposition 3.2 in ([PERSON] and [PERSON], 2016) in a parametric setting. **Theorem 5**. :
1. _Suppose that_ \(x^{*}(t)\) _is a solution of_ \(VI(F^{r(t)},K(t))\)_, where_ \(r(t)\in\mathbb{R}_{++}^{n}\)_, a constraint qualification holds at_ \(x^{*}(t)\) _and_ \((\bar{\lambda}(t),\bar{\mu}(t))\in\mathbb{R}^{m}\times\mathbb{R}^{n}\) _are the multipliers associated with_ \(x^{*}(t)\)_. Then,_ \(x^{*}(t)\) _is a normalized equilibrium such that the multipliers_ \((\lambda^{i*}(t),\mu^{*}_{i}(t))\) _of each player_ \(i\) _satisfy the following conditions:_ \[u_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t))\geq u_{i}(x_{i},x^{*}_{-i}(t)),\qquad\forall \ t\in[0,T],\ \forall\ x_{i}\in K_{i}(t,x^{*}_{-i}(t)). \tag{31}\]3
2.