notes-math-misc-secretaryProblem

If you are interviewing candidates for a job, and after you see each candidate, and you must decide on the spot whether to hire them or to forever lose them, and you want to maximize the expected rank of the chosen candidate, and after interviewing a candidate, you can tell what a rank is amongst all of the candidates you have already seen, and you know how many candidates there are, and will chose the last candidate unconditionally if you get to them, then:

You should interview the first sqrn(n) candidates, where n is the number of candidates, and then hire the next one better than all that you have seen so far. ( http://en.wikipedia.org/wiki/Secretary_problem#Cardinal_payoff_variant )

notes: if your goal is, not to maximize the expected rank of the hire, but rather to maximize the probability of getting the very best hire, then you should go through n/e candidates first, which for most numbers is bigger than sqrt(n) ( http://en.wikipedia.org/wiki/Secretary_problem#Deriving_the_optimal_policy ). If don't know how many candidates there are but you know you have T time to look before you have to stop and hire someone unconditionally, and you want to maximize the probability of getting the best one, you should wait for T/e time and then hire the next one better than all of those ( http://en.wikipedia.org/wiki/Secretary_problem#1.2Fe-law_of_best_choice ).

I wonder what you should do if you don't know how many there are, and you want to maximize rank rather than probability of getting the best?