## 4 Application to Parametric Network Games
### Basics of Network Games
Definition 4: A Nash equilibrium is a vector \(x^{*}\in A\) such that:  
\[u_{i}(x^{*}_{i},x^{*}_{-i})\geq u_{i}(x_{i},x^{*}_{-i}),\qquad\forall\;i\in\{ 1,\ldots,n\},\;\forall\;x_{i}\in A_{i}. \tag{23}\]  
The interested reader can find a comprehensive theoretical treatment of Game Theory in the book [28], along with various applications. We now adapt the above concepts to our parametric network problem and, for simplicity, still denote the set of players by \(\{1,2,\ldots,n\}\) instead of \(\{v_{1},v_{2},\ldots,v_{n}\}\). Thus, consider an interval \([0,T]\), and for each \(t\in[0,T]\), denote with \(A_{i}(t)\subset\mathbb{R}\) the parametric strategy space of player \(i\), while \(A(t)=A_{1}(t)\times\cdots\times A_{n}(t)\) is called the parametric space of action profiles. Each player \(i\) is endowed with a payoff function \(u_{i}:[0,T]\times A(t)\to\mathbb{R}\) that he/she wishes to maximize. The notation \(u_{i}(t,x,G)\) is often utilized when one wants to emphasize the dependence on the graph structure. We wish to find a Nash equilibrium of the parametric game; that is, for each \(t\in[0,T]\), we seek an element \(x^{*}(t)\in A(t)\) such that for each \(i\in\{1,\ldots,n\}\):  
\[u_{i}(t,x^{*}_{i}(t),x^{*}_{-i}(t))\geq u_{i}(t,x_{i},x^{*}_{-i}(t)),\qquad \forall\;x_{i}\in A_{i}(t). \tag{24}\]A characteristic of network games is that the vector \(x_{-i}\) is only made up of components \(x_{i}\) such that \(j\in N_{i}(g)\); that is, \(j\) is a neighbor of \(i\). For modeling purposes, and also for simplifying the analysis, it is common to make further assumptions on how variations in the actions of a player's neighbors affect her marginal utility. We assume that the type of interaction does not change in \([0,T]\) and, in the case where \(u_{i}(t,\cdot)\in C^{2}(\mathbb{R}^{n})\), we obtain the following definitions. Definition 5: We say that the network game has the property of strategic substitutes if for each player \(i\) the following condition holds:  
\[\frac{\partial^{2}u_{i}(t,x_{i},x_{-i})}{\partial x_{j}\partial x_{i}}<0, \qquad\forall(i,j):\,g_{ij}=1,\,\,\forall\,\,t\in[0,T],\,\,\forall\,\,x\in A(t).\]  
Definition 6: We say that the network game has the property of strategic complements if for each player \(i\) the following condition holds:  
\[\frac{\partial^{2}u_{i}(t,x_{i},x_{-i})}{\partial x_{j}\partial x_{i}}>0, \qquad\forall(i,j):\,g_{ij}=1,\,\,\forall\,\,t\in[0,T],\,\,\forall\,\,x\in A(t).\]  
We will use the variational inequality approach, pioneered by [23] (in the non-parametric case), to solve Nash equilibrium problems. If, for each \(t\in[0,T]\), the \(u_{i}\) are continuously differentiable functions on \(\mathbb{R}^{n}\), and \(u_{i}(t,\cdot,x_{-i})\) are concave, the Nash equilibrium problem is equivalent to the variational inequality \(VI(F,A(t))\):  
For each \(t\in[0,T]\), find \(x^{*}(t)\in A(t)\) such that  
\[\langle F(t,x^{*}(t)),x-x^{*}(t)\rangle\geq 0,\qquad\forall\,\,x\in A(t), \tag{25}\]  
where  
\[[F(t,x)]^{\top}:=-\bigg{(}\frac{\partial u_{1}}{\partial x_{1}}(t,x),\ldots, \frac{\partial u_{n}}{\partial a_{n}}(t,x)\bigg{)} \tag{26}\]  
is also called the _pseudo-gradient_ of the game, according to the terminology introduced by [PERSON] [29]. In the following subsection, we introduce the parametric linear-quadratic utility functions, which will be used both for the Nash equilibrium problem and the generalized Nash equilibrium problem. This simple functional form as been extensively used in the literature (see, e.g, [20]) because, in the case on non-negative strategy space, it allows solutions in closed form. We consider instead the case where the strategy space of each player has a (parameter-dependent) upper bound.