I'm interested in learning how to optimally solve a multiple stopping problem with a known payoff distribution, like the following:
You are observing a sequence of forty $(40)$ opportunities, each with a payoff independently drawn from the following distribution:
- $50\%$ of the opportunities have payoffs between the values $1$, which is the minimum value, and $50$ (inclusive);
- $45\%$ of the opportunities have payoffs between the values $51$ and $75$ (inclusive); and
- $5\%$ of the opportunities have payoffs between the values $76$ and $100$, the maximum value (inclusive).
(You may assume that probability within each range of values is uniform.)
You are equipped to take advantage of only four of these opportunities. That is, as you observe each opportunity in the sequence, you may choose to "stop" and receive the current opportunity's payoff. Specifically, you may choose to do so at maximum four times. If you choose not to "stop" at any given opportunity, then you permanently miss that opportunity and do not receive its payoff; so, once passed, an opportunity may not be recovered.
You must decide the optimal strategy for using your four available "stops," to maximize your expected payoff from this sequence.
Additionally, how would your strategy change if three of your available stops each had only a $60\%$ chance of successfully taking advantage of a given opportunity and its payoff?
Many thanks in advance for the help!
What you have here is a variant of a variant of the Cardinal Payoff Variant of the Secretary Problem.