Consider the sets $Y_1,Y_2\subset\mathbb{L}^2(\Omega,\mathbb{F}_{s},\mathbb{P})$. I am looking for some $y_1$ measurable with respect to $Y_1$ and $y_2$ measurable with respect to $Y_2$ s.t. the pair $(y^*_1,y^*_2)$ constitutes a Nash Equilibrium. In particular, the pair $(y^*_1,y^*_2)\in \mathcal{Y}= \Pi_{i=1}^2Y_i$ is the cartesian product of the strategic choices of evry agent.
I present a setup similar to Vayanos and Wang of $2011$ model, where the set of strategic actions of every agent is in the intercept of their demand functions. Consider that two agents, indexed by $i=\{1,2\}$, have the following utility functions $$g_1(y_1)=-d_1(y_1)p(y_1,y_2)+\mathbb{E}[d_1(y_1)\bar{x}+\bar{z}_1|Y_1]-\gamma_1\mathbb{V}ar[d_1(y_1)\bar{x}+\bar{z}_1|Y_1]$$ and $$g_2(y_2)=-d_2(y_2)p(y_1,y_2)+\mathbb{E}[d_2(y_2)\bar{x}+\bar{z}_2|Y_2]-\gamma_2\mathbb{V}ar[d_2(y_2)\bar{x}+\bar{z}_2|Y_2]$$
where $(\bar{x},\bar{z}_i)$ are i.d and $Y_i$ measurable and also it holds that $\bar{x}\sim N(x,\sigma_{x}^2)$, $\bar{z}_i\sim N(0,\sigma_{z_i}^2)$, $\gamma_i\in (0,+\infty)$, $\mathbb{C}ov(\bar{x},\bar{z_i})\neq 0$, $\mathbb{C}ov(\bar{z_1},\bar{z_2})\neq 0$ and $g_i:Y_i\subset\mathbb{L}^2\rightarrow \mathbb{R}$ where $i\in\{1,2\}$. For the demand functions hold that $$d_i(y_i)=\frac{y_i-p(y_1,y_2)}{2\gamma_i\mathbb{V}ar[\bar{x}|Y_i]}$$ where $y_i$ is measurable with respect to $Y_i$ and $$p(y_1,y_2)=w_1y_1+w_2y_2+w_3y$$ s.t $w_1+w_2+w_3=1$
Every agent needs to solve the following problem $$y^*_i=\max_{y_i}g_i(y_i;y_{-i})$$ where $y_{-i}$ is the strategic choice of agent $-i$ (in our case the agents are two so equilevantely $y^*_1=\max_{y_1}g_i(y_1;y_{2})$) So, I will obtain two solutions for $y_1$ and $y_2$ and by solving the system of them, I result to the pair of $(y_1^*,y_2^*)$
$\textbf{The solutions of the agents' problems}$
By solving the maximization problem of agent-$1$, we obtain: \begin{equation}y_1(y_2)=\frac{1}{1+\omega_1}P_1+\frac{\omega_1\omega_2}{(1-\omega_1)(1+\omega_1)}y_2+\frac{\omega_1\omega_3}{(1-\omega_1)(1+\omega_1)}y\end{equation} where $P_1=\mathbb{E}[\bar{x}|Y_1]-2\gamma_1\mathbb{C}ov(\bar{x},\bar{z}_1|Y_1)$
Similarly, by solving the maximization problem of agent-$2$ \begin{equation}y_2(y_1)=\frac{1}{1+\omega_2}P_2+\frac{\omega_1\omega_2}{(1-\omega_2)(1+\omega_2)}y_1+\frac{\omega_2\omega_3}{(1-\omega_2)(1+\omega_2)}y\end{equation} where $P_2=\mathbb{E}[\bar{x}|Y_2]-2\gamma_2\mathbb{C}ov(\bar{x},\bar{z}_2|Y_2)$
By solving the system of $(y_1,y_2)$ we obtain that $$y_1^*=\frac{(1-\omega_1)(1-\omega_2^2)}{1-\omega_1^2-\omega_2^2}P_1+\frac{\omega_1(1-\omega_1)\omega_2}{1-\omega_1^2-\omega_2^2}P_2+\frac{\omega_1\omega_3}{1-\omega_1^2-\omega_2^2}y$$
and
$$y_2^*=\frac{\omega_1(1-\omega_2)\omega_2}{1-\omega_1^2-\omega_2^2}P_1+\frac{(1-\omega_1^2)(1-\omega_2)}{1-\omega_1^2-\omega_2^2}P_2+\frac{\omega_2\omega_3}{1-\omega_1^2-\omega_2^2}y$$
$\textbf{My question}$
My question is, how can I build the Banach fixed point argument for the aforementioned problem where the set of strategic choices is in the intercept of the demand functions?
Well, jsut to help your post and may someone else come and make clear what you need to check. Every agent needs to solve the following problem $$y^*_i=\max_{y_i}g_i(y_i)$$ So you will obtain a two solutions for $y_1$ and $y_2$ and by solving the system of them, you take the pair of $(y_1^*,y_2^*)$. The question is why sould this be a Nash Equilibrium and is this the unique equilibrium pair of strategies? The answer is deinitely yes for the second part of my-your question, and it is implied by the lienarity of the derivative of $g_i(y_i)$ with respect to $y_i$. So we talk about a unique solution. Why this is a N.E. or how the Banach fixed point theorem is going to be applied here, it is something else that I can not tell, because I do not know. I hope someone else could have a full answer, but in first place, this is a part of your question to be answered.