I have the following fixed point equation: for all $p\in[0,1]$
$$V(p) = \min_{\lambda_1,\lambda_2\in[0,1]}\max\{pr,(1-p)r,\beta\mathbb{E}_{a,y'}[V(f_{\lambda=(\lambda_1,\lambda_2)}(p,a,y'))]\}$$ where the expectation is given by \begin{align*} \mathbb{E}_{a,y'}[V(f_{\lambda}(p,a,y'))] &=(\gamma_1\lambda_1p + (1-\gamma_2)\lambda_2(1-p))V\left(\frac{\gamma_1\lambda_1p}{\gamma_1\lambda_1p + (1-\gamma_2)\lambda_2(1-p)}\right)\\ &+((1-\gamma_1)\lambda_1p + \gamma_2\lambda_2(1-p))V\left(\frac{(1-\gamma_1)\lambda_1p}{(1-\gamma_1)\lambda_1p + \gamma_2\lambda_2(1-p)}\right)\\ &+(\gamma_1(1-\lambda_2)(1-p) + (1-\gamma_2)(1-\lambda_1)p)V\left(\frac{\gamma_1(1-\lambda_2)(1-p)}{\gamma_1(1-\lambda_2)(1-p) + (1-\gamma_2)(1-\lambda_1)p)}\right)\\ &+((1-\gamma_1)(1-\lambda_2)(1-p) + \gamma_2(1-\lambda_1)p)V\left(\frac{(1-\gamma_1)(1-\lambda_2)(1-p)}{(1-\gamma_1)(1-\lambda_2)(1-p) + \gamma_2(1-\lambda_1)p}\right)\\ \end{align*}
where $r>0$, $\beta\in[0,1)$, and $\gamma_1,\gamma_2\in[0,1]$ are given.
I can use the fact that $V(\cdot)$ is convex in $p$. I can also prove that the expectation term, $\mathbb{E}_{a,y'}[V(f_{\lambda}(p,a,y'))]$, is convex in $p$ for any $\lambda=(\lambda_1,\lambda_2)$.
Question: Any ideas on how I would go about solving for $V(\cdot)$? It seems like it may be possible to get somewhere analytically due to the nice properties of $V$ but even a numerical procedure would suffice.
Attempt: Based on Paul's comment, the fixed point equation can be written as:
$$V(p) = \max\{pr,(1-p)r,\beta\min_{\lambda_1,\lambda_2\in[0,1]}\mathbb{E}_{a,y'}[V(f_{\lambda=(\lambda_1,\lambda_2)}(p,a,y'))]\}$$
but I'm not too sure how to represent the solution of the inner minimization problem, $\min_{\lambda_1,\lambda_2\in[0,1]}\mathbb{E}_{a,y'}[V(f_{\lambda=(\lambda_1,\lambda_2)}(p,a,y'))]$, in terms of $V(\cdot)$.
I tried to draw a picture of what $V(p)$ might look like. Let $g(p) = \beta\min_{\lambda_1,\lambda_2\in[0,1]}\mathbb{E}_{a,y'}[V(f_{\lambda=(\lambda_1,\lambda_2)}(p,a,y'))]$.
If $\beta$ is small enough, then it appears the solution of the fixed point is $V(p)=\max\{pr,(1-p)r\}$, but I am seeking a solution for all values of $\beta\in[0,1)$. Note that even though $g(p)$ is convex, the shape of $V(p)$ may be complex (it's possible to draw a convex function $g(p)$ that intersects the lines $pr$, $(1-p)r$ at most 4 times).
Update: I'm not actually sure if Paul's suggestion holds in general. Wouldn't the minimizer $(\lambda_1,\lambda_2)$ depend on $\pi$?