Imagine making a series of irreversible decisions, and at each step, you have to decide whether to continue or stop. What is the optimal stopping rule to maximize your expected gain? This relates to the famous "Secretary Problem."
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The optimal stopping rule to maximize expected gain in the scenario you described, also known as the Secretary Problem or Marriage Problem, involves stopping at a specific point to select the best option. Here's the general strategy:
1. Estimate the total number of options (let's say N) that you will encounter in the series of irreversible decisions.
2. Calculate or estimate the value of m, where m = N/e (where e is Euler's number, approximately 2.71828).
3. Initially, observe and reject the first m-1 options without making a choice.
4. Starting from the mth option, continue evaluating the options and select the option that surpasses the value of the best option observed in the initial m-1 options.
5. If no option surpasses the best option from the initial m-1 options, select the last available option.
By following this strategy, you maximize your chances of selecting the best option while minimizing the risk of making a suboptimal choice too early. The expected gain using this stopping rule is approximately 1/e or approximately 0.368.
It's important to note that this strategy assumes random order and that you have no information about the quality or rank of the options beforehand.
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