I am an engineering student who has just started researching. Because of lack of mathematical knowledge, some evidence is difficult to understand. If you can handle this, give me some hints.
First, the formulas used in the proof process.
$v_k(\theta)$ = \begin{cases} \lambda p - c , & \mbox{if }\theta \mbox{ $\geqq h_k$} \\ 0, & \mbox{if } \theta \mbox{ $< h_k$} \end{cases}
$p_k(\theta' \mid \theta)$ = \begin{cases} 1 - \alpha , & \theta \geqq h_k \text{ and } \theta' = \min\{K_k, \theta + 1\} \\ \alpha , & \theta \geqq h_k \text{ and } \theta' = \theta - 1 \\ \alpha, & \theta = h_k \text{ and } \theta' = 0 \\ 1, & \theta < h_k \text{ and } \theta' = \theta + 1 \\ 0, & \text{ otherwise} \end{cases}
$r(\theta, a)$ = \begin{cases} \min\{K_k, \theta +1\} , & \mbox{if } a = H \text{ and } \theta \geqq h_k \\ \theta - 1, & \mbox{if } a = L \text{ and } \theta \geqq h_k + 1 \\ 0, & \text{ otherwise} \end{cases}
$v^{\infty}_k(\theta^{(t_0)}) = \mathbb{E}( \sum_{t=1}^\infty \delta^t v_k(\theta^{(t)})) = v_k(\theta^{(t)}) + \delta \sum_\theta' p_k(\theta' | \theta^{(t_0)})v^{\infty}_k(\theta') $
Based on the above formulas, the contents of the proof are made, but this part is not understood.
At, $\theta \geqq h_k +1 $ and $a = H $ so consumes a cost c. So, Workers r increase $min\{K_k, \theta +1\}$ with the probability $ 1 - \alpha $ and decreases to $\theta - 1$ the probability $\alpha$. Hence workers function $V_\theta (H) = \lambda p - c + \delta[(1-\alpha)v^{\infty}_k(\min\{K, \theta_+1\}) + \alpha v^{\infty}_k(\theta-1)]$
I understand above function. But, I can't understand this part. $ a = L $ , worker saves cost c. And, workers r increase and decrease becom $\alpha \ and \ 1 - \alpha $.
$V_\theta (L) = \lambda p + \delta[(1-\alpha)v^{\infty}_k(\theta-1) + \alpha v^{\infty}_k(\min\{K, \theta_+1\})]$
I think $V_\theta (L) = \lambda p + \delta[(\alpha)v^{\infty}_k(\theta-1) + (1 - \alpha) v^{\infty}_k(min\{K, \theta_+1\})]$ is right function.
plz give me any advice for me.