Optimal stopping
for sexist scenarios –
an odd solution.
The Secretary Problem is the ‘fear of missing out’ in mathematical form. It encompasses the idea of deciding when to stop looking through a list of options when you can’t go back to a previously dismissed option and don’t know what future options might hold. It presents the scenario that you’re hiring a secretary and want to hire the very best. You consider each candidate in turn and must decide to hire or reject them immediately after their interview. You know how good they are and how good all those applicants before them were but you’ve no way of knowing how good the remaining applicants are – the best may be yet to come. When do you hire someone, when do you decide which is best?
It’s a problem involving optimal stopping theory – choosing when to take an action to maximise the expected reward and/or minimise the expected cost. When do you stop rejecting applicants to get the best secretary?
The shortest proof to the problem is the odds algorithm devised by F. Thomas Bruss in 2000. The proof equates to the idea that when hiring a secretary under the strict conditions of the problem, you should view a certain number of applicants and then hire the next applicant that scores higher than any of those. The number of applicants to reject first is defined by ~ n/e where n is the total number of applicants and e is the base of the natural logarithm. This proof of the problem is likely to select the single best applicant from the total pool of applicants around 37% of the time, regardless of how many applicants there are.
Curiously The Secretary Problem has been known by a number of different names, most of which involve men picking between women in some way: the marriage problem, the sultan’s dowry problem, the fussy suitor problem, the googol game, and the best choice problem.
Further reading:
Bruss, 2000, Sum the odds to one and stop – https://doi.org/10.1214%2Faop%2F1019160340
The Secretary Problem – https://en.wikipedia.org/wiki/Secretary_problem