Here are two integral equations that correlated with each other
For $w\in[0,w_0]$: \begin{equation} x^*_w\triangleq\begin{cases} &-\infty,~\mbox{if }\nexists~x\geq 0~\mbox{such that }y^*_v+p-v\leq x\leq y^*_v+p-h-w,~\mbox{for some }v\in[0,v_0]\\ &\\ &\displaystyle\underset{x\geq 0}{\arg\max}~G^s(x;w)=\frac{1}{v_0}\int_0^{v_0}\mathbf{1}_{\{v+x-p-y^*_v\geq 0\}}[y^*_v+p-h-x-w]^+dv,~\mbox{otherwise} \end{cases}. \end{equation} and for $v\in[0,v_0]$: \begin{equation} y^*_v\triangleq\begin{cases} &-\infty,~\mbox{if }\nexists~y\geq 0~\mbox{such that }x^*_w-p+h+w\leq y\leq x^*_w-p+v,~\mbox{for some }w\in[0,w_0]\\ &\\ &\displaystyle\underset{y\geq 0}{\arg\max}~G^c(y;v)=\frac{1}{w_0}\int_0^{w_0}\mathbf{1}_{\{y+p-h-x^*_w-w\geq 0\}}[x^*_w+v-p-y]^+dw,~\mbox{otherwise} \end{cases}. \end{equation} where $v_0>w_0>0$ are constants and $p>h>0$ are constants. $\mathbf{1}_{\{E\}}=1$ if $E$ occurs, is an indicator function. $[x]^+=\max\{x,0\}$. In the function $G^s(x;w)$, it contains another function $y^*_v$, which is the optimal solution to $\underset{y\geq 0}{\max}~G^c(y;v)$ and in the function $G^c(y;v)$, it contains another function $x^*_w$, which is the optimal solution to $\underset{x\geq 0}{\max}~G^s(x;w)$.
I think there is no standard way of solving this type of problem (a potential way of solving it could be to directly take derivative with respect to $x$ for the first expression, but that involves the derivative of indicator function and $[...]^+$ function). Usually, we conjecture a solution $x^*_w$ and $y^*_v$, then we verify that indeed the conjecture is correct. I am now try to prove some properties of $x^*_w$ and $y^*_v$ by just looking at the definition of $x^*_w$ and $y^*_v$. For example, it can be shown that $x^*_w$ is weakly decreasing in $w$ and $y^*_v$ is weakly increasing in $v$.
The property I want to show (hope it is correct) is that, I believe $x^*_w$ is linearly decreasing in $w$ with a possible truncation at $0$; $y^*_v$ is linearly increasing in $v$ starting from zero. Specifically, my hunch is that For those $x^*_w$ and $y^*_w$ that are non-negative \begin{align*} x^*_w=\begin{cases} c-dw,~&\mbox{if }w\leq c/d\\ 0,~&\mbox{if }w>c/d \end{cases},~y^*_v=\begin{cases} 0,~&\mbox{if }v\leq -a/b\\ a+bv,~&\mbox{if }v> -a/b \end{cases}, \end{align*} where $a,b,c,d$ are non-negative function of $p,h,v_0,w_0$. And depends on the value of $p,h,v_0,w_0$, $c/d$ could be less than $0$, between $0$ and $w_0$, or greater than $w_0$. If $c/d<0$, then this means $x^*_w\equiv 0$; if $c/d>w_0$, then this means $x^*_w=c-dw$. Similar situation for $y^*_v$ and $-a/b$.
It is also very important to identify the condition on $w$ and $v$ such that $x^*_w$ and $y^*_v$ are non-negative.