This question concerns the 1979 paper "Optimal search for the best alternative" by Martin Weitzman.
The setup is as follows:
There are $m$ boxes containing finite-expectation rewards $x_i \sim F_i$. Opening box $i$ costs $c_i \in \mathbb R$; if box $i$ is opened, its reward is observed after time $t_i \ge 0$.
The decision maker starts with a reward of zero. At each point in time he can decide to stop search and consume the highest reward observed so far or continue searching by opening a box.
Time is continuous and discounted at rate $r \ge 0$; define $\beta_i = e^{-r t_i}$
The goal is to to decide which boxes to open in what order to maximize discounted expected net reward.
The reservation price $z_i$ of box $i$ is defined by
$$ z_i = -c_i + \beta_i E\max(z_i, x_i). $$
The reservation price exists and is unique whenever $c_i > 0$ or $\beta_i < 1$; assume in the following that this condition holds for all boxes.
The paper's main result is that an optimal policy is:
Pandora's rule: Continue search while the highest reserve price $z$ among the remaining closed boxes exceeds the highest reward $y$ observed so far, in that case open a box $j$ with $z_j = z$. Terminate otherwise.
Note that this result implies that if there are several boxes which have a highest reservation price, it does not matter which box is chosen. This leads me to my question:
Question: Suppose that at some step there are several boxes that are tied in the sense that they all have a maximal reservation price. Why does the same discounted expected net reward result, no matter in what order these boxes are chosen?
The proof in the paper does not appear to address this question because in my reading of it, only deviations to boxes with strictly smaller reservation prices are considered.
Maybe the answer to my question is obvious but I would be very grateful for an explanation. Thank you!