Threshold Rule

The Threshold Rule is simply accepting a result immediately if and only if the result is above a certain criterion.

Considering the #Secretary Problem, in a #complete information market, by using the Threshold Rule, you immediately accept an applicant if he/she is above a certain percentile.¹

Another great example is when rolling a die as explained in detail in this post Threshold Rule: Rolling a Die#.

Algorithms to Live By by Brian Christian and Tom Griffiths - Optimal Stopping
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2021-12-29

#Threshold Rule: Rolling a Die
- A rudimentary application of the #Threshold Rule is when rolling a die. A general rule of thumb for solving optimal stopping problems is thinking backwards or using backward induction.¹ A standard die with a maximum roll of 6 assuming there are equal chances of rolling any face of the die at any given roll, let \(k\) equals to the remaining retry. At zero remaining reroll, we set up the base case which has an average value of a die roll of 3.5 \(((1+2+3+4+5+6)/6)\), an average of all value of the die since any value would be better than nothing since its the last roll.
  1. Algorithms to Live By by Brian Christian and Tom Griffiths - Optimal Stopping
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Secretary Problem#
- Obviously, the secretary problem is impractical and does not apply to real life situations as there are too many variants. For one, applicant recall is always an option regardless if the interviewer further the search, provided the applicant is still available. Also, there is about 50/50 chance of being rejected by the applicants themselves which further complicates the optimal stopping. Furthermore, the interviewer knows nothing about the applicants other than the relative comparison to each other, but by how much? There is also a risk of passing up an exceptional applicants while in the calibration phase or before the optimal cutoff, and not to mention, a better one after hiring. Finally, a question of preparation over capabilities, as one could be so prepared than the ones who are more capable. There are a lot more variants the secretary problem oversimplified, but the fact we quantify this much is a good basis for other variations to come as did some brilliant mathematicians have, such as the odds algorithm¹ or the #Threshold Rule, after the seminal 37% rule.
  1. https://en.wikipedia.org/wiki/Odds_algorithm “Odds Algorithm”
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