The section on experimental studies touches on it briefly, but it's important to note that costs involved in the selection process are not considered in the standard construction of the problem.
From wikipedia:
> In large part, this work has shown that people tend to stop searching too soon. This may be explained, at least in part, by the cost of evaluating candidates.
and then
> For example, when trying to decide at which gas station to stop for gas, people might not search enough before stopping. If true, then they would tend to pay more for gas than they might had they searched longer.
So you might pay more for gas if you don't use the optimal strategy, but waste more money in fuel costs searching for the cheaper price than you save buy purchasing at that price.
Perhaps most interestingly, there is a formulation of the problem that requires a decision to be made within a certain time period, from an unknown number of candidates who arrive over that time period. If you know (or can estimate) the arrival times of the candidates, you can use a very similar strategy to achieve optimal results.
Essentially, wait until you have seen 1/e of expected candidates in the time frame you have allowed for (based on the arrival density function you know or estimated) then pick the next best option.
This puts a limit on the amount of searching you do, and so provides an optimal strategy with a bounded limit on how long you search for; it provides a bound on the cost involved in the search assuming the cost is related to time taken.
In the searching for gas example, you could use this strategy if you knew roughly how often you pass a gas station, and how long you are willing to search for.
There's a TV show in the UK that uses a spin on this - it's called 4 rooms.
The premise is that people come on the show with, what they consider to be, a valuable artifact. They then have the chance to take it to 4 collectors who will offer them a sum of money for said artifact.
The aim is to come away with the best offer you can get - but you only get one shot with each collector, you can't go back to a previous one because all other offers have been lower.
Are the collectors aware of the order in which they talk to the seller? That is, is the 4th collector aware that he's the "last chance" for the seller?
According to the article, with such a small n, the contestant should visit ~1.47 of collectors before picking the best one. That's hard to do in practice.
More precisely; reject the first offer but note the price, accept any offer that beats the first one, or accept the final offer if none beat the first.
This gets posted on HN occasionally, and when it does I post this fun blog post by Michael Trick, a CMU Operations Research professor, who used the Secretary problem to pick his wife:
When you live in a major city and the dating options are boundless, there is always someone around the corner that is "perfect" or more perfect than the current person that you are dating. Next thing you know, you're in your late 30s and still single.
This only works if you stick with every partner long enough to get a perfectly accurate assessment of them as a long term mate, that your assements are perfectly objective, independent, and stable, and that you know n.
The amount of ifs, ands, and buts in this caveat means it's time to defer to one of my favorite xkcd's of all time. [0]
I always feel bad when I see these posts. I'm 26 and in a major city, but finding dates is a serious problem for me. My looks are only slightly below average (4/10) but my personality is probably the deal breaker.
Having been where you've been, may I suggest that the dealbreaker is that you're treating dating like a World of Warcraft inventory checklist? I wish someone had told me that when I was saying the same things.
FWIW, I do think there are similarities between dating and RPGs that I wish I had understood before
* you can improve your character: confidence is important, if you feel ugly sometimes you can do stuff about it (i.e. go to gym). Ditto if you are a boring talker, or if you are a poor listener.
* grind. If you are, like I was, a guy that has a couple hours on a weekend to meet a new possible date it's hard to find the "right" one. Try to increase the chances (go out more, frequent new places) and do try more.
This is fair enough and true. Back when I was in my early twenties and late teens I used to grind all the time (going out to bars and talking to women). And although I experienced more than my fair share of rejection/disinterest, I achieved plenty of success and abundance.
These days I am just lazily sending out messages on OKCupid. And when I do go out to bars, I almost never approach.
If you want excitement, you'll get excitement.
If you want stability, you'll get stability.
If you want to confirm an opinion of yourself as undateable - guess what, that's what you'll get.
Most people say they want one thing but actually don't work towards that thing. They just pay it lip service.
Hmm, I wonder if this could be used to find a decent pub whilst in a strange city?
It's always a problem when you're on holiday and you know there are n pubs around, but you don't want to spend all your time going around and checking each and every pub 'cos that gets tedious. The question is - how many pubs should I visit before I give up?
As a general rule is this saying you should visit n/e pubs and then just pick the next best one?
It would. But you'd need to know, or at least guess, how many pubs there were in the city (to know when you'd reached n/e), and it would also depend on whether the pubs were randomly distributed or not. If there's nice end and a trashy end of town, you could easily have exhausted all of the good pubs before you hit the n/e.
Yes, I guess you'd have to use real world factors to try and trim down your n before you start with the n/e thing. Although if you're in a strange place you probably won't know that much about which areas are good or not.
Quite interesting that discarding the first n/e candidates produces a 1/e probability of choosing the optimal one...
As for real-world applications, the only one that comes readily to mind is rolling up a D&D character, supposing one has only the patience for some predetermined n rolls. Somewhat ironically it is not a realistic fit for hiring, as there is seldom a need to accept or eliminate candidates immediately, interviewing more than 5-6 people for a role is torturous, and an interviewer uses his experience interviewing and interacting with people generally to have no mathematical prejudice against hiring the very first pretty-good option.
> Quite interesting that discarding the first n/e candidates produces a 1/e probability of choosing the optimal one...
Freeform thoughts from here:
1/e is the probability of failing (every time) if you try a 1/n chance n times, where n is large.
To get the optimal choice, there are two (generously defined) requirements:
- #1 does not occur in the discarded group
- #1 occurs before everyone else who was not discarded, but exceeds the maximum of the discarded group
The second one seems hairy to me, so I'll wrap things up here. ;p
> As for real-world applications, the only one that comes readily to mind is rolling up a D&D character
This was introduced to me as a marriage problem, and wikipedia also notes it by that name. The analogy never seemed inaccurate to me; it's frowned upon to evaluate candidates simultaneously and rare to accept someone other than the latest candidate.
The function that approximates the best ideal solution can be approximated with the log function. It's kind of like why the taylor polynomials means that e^pi*i = -1
The | notation means "given". So x|b means "x, given b".
For these abstract formulations, they are reduced to already-existing calculations..
Finally, you get the big sigma symbol. This is called a Summation. Summations tend to be able to be approximated using integrals, since an integral is the area under the curve, which is also a summation of sorts.
Many integrals can be defined using the "log" function, which has base "e". In this case, when you add in "infinity" for the summation, it converges towards 1/e.
Hope this helps some. If it doesn't, well, I was fired from my job TA'ing calculus =B so..
Very interesting! I assume this is posted because it parallels how YC does their interview admissions. I think the YC method has some differences to this problem. YC has already seen the applications for groups, and probably already developed some kind of best to worse ranking of the applicants. The interview most likely functions as a confirmation that the teams live up to their great application. Additionally, YC expands its class to fit the number of qualified applicants. In this problem there is single spot to fill.
Edit: It is also totally possible many people upvoted this article only because it is interesting. If that is true ignore my post.
Interesting. It describes roughly the strategy that I've settled on when looking for accommodation to rent. I think it also has the added advantage of making you feel you've made a good decision.
I, like many others, used to have the attitude that "puzzles are for interviews". But recently, I have started seeing that there can be practical applications for seemingly academic ideas.
For example, I know a proprietary trader who uses a concept similar to this from the field of Optimal Stopping for his trading system exclusively, i.e., his entire trading strategy, managing tens of millions of dollars for a big bank, is basically an application of Optimal Stopping to the markets!
There's a large body of work on variations of the secretary problem. The one I know of which is most relevant to your question is section 4 of the following:
http://www.cs.cornell.edu/~rdk/papers/secArt4.pdf
Section 6 is particularly interesting, where you're further restricted - you want to choose multiple secretaries, but there's certain constraints on the sets you can choose. For example, the secretaries might be edges in a graph, and you can't pick a subset of edges which would result in a cycle (this is a "graphic matroid," which is an example of a mathematical object known as a matroid). The reason why this formulation of the problem is interesting is because the best known algorithm is O(sqrt(log k))-competitive (where k is the rank of the matroid), whereas it is conjectured that an O(1)-competitive algorithm exists.
I worked on a version on this problem where the goal is to maximize the probability of hiring all top M candidates while using a similar single threshold strategy.
And what about preferring the second best candidate to the average candidate... say if you interviewed 95% of applicants after rejecting the first 37%, and then you find a candidate who is almost, but not quite, as good as the very best one?
One solution might be to try and find the the best candidate after (N/M)/e candidates, rather than N/e candidates. Then find the best candidate after (N_remaining/(M-1))/e candidates. That is recurse, by finding one candidate at a time, and discarding less then N/e each time to ensure you get enough candidates.
I worked on several iterations in the dating space for about a year, and this type of stuff was pretty interesting to me. Wrote a bit more about it here: https://medium.com/unfinished-thoughts/2cac1e6cc7b4
I'm going to be the dumb guy ranting here and say that I dislike this word problem since external knowledge of the world can change your strategy. I might be stopping too soon because the time cost of evaluating candidates is far too high compared to the work that needs to be done immediately. The sooner I get someone in, the sooner that work gets done, the less behind we all get, the less workload for the new secretary, the better he/she will perform, and the better the first impression.
You might also be able to recognize a rock star secretary immediately or recognize obviously incompetent people. There's a competence threshold that's present here and has to be addressed if you dissect the analogy. Going for 'best' here has diminishing returns beyond a certain level of competence. The immediate need to decline/accept also really doesn't make sense if we're trying to explain this with a hiring analogy.
However, the Game of Googol nails it and I really like this approach much better when explaining this problem. It's a game with arbitrary rules so I can't easily use my worldly experience biases to solve the problem.
I think the difference is that you're reading it as practical word problem, when the intent is to present a math problem.
I first heard about this problem in a politically incorrect variation: a tribe chief is given the opportunity to choose a wife from a pool of n women, presented in a random order, one at a time. He can choose any woman, but can't choose a woman who he has already passed up. What's the strategy for the chief to choose the most beautiful wife?
Do you really need to ask? Are you seriously incapable of seeing that some people might be offended (stupidly, if you ask me, but that doesn't make them any less so) by a story where a woman is apparently picked like cattle by a feudal lord?
I suspect he meant to say that the analogy accurately depicts something that really happens. An assessment that, incidentally, is pretty politically incorrect in itself.
Ah, yes, what an accurate depiction of the "tribe."
Seriously, if you want to go the accurate route, probably best to pick a specific people where that's what actually happens. Not having done that, the problem as stated advances stereotypes of peoples that are referred to as "tribes" as being primitive and misogynistic.
Perhaps use "nation" instead, or better yet, formulate the problem differently.
I think that when presenting a problem like this, it's not unreasonable to assume that the audience is capable of the abstract thinking that is needed to distill the underlying problem from the application in which it is presented.
Come to think of it, maybe I'd argue even stronger: when I was younger, I had a hard time understanding programming problems that were explained in terms of 'foo / bar' classes or function names. 'Applied' examples with cars and wheels made more sense, even if I could easily have come up with 'not every car has 4 wheels' counterarguments. Nowadays, I like foo / bar better, because they expose the problem directly, without an additional layer of real-world application, but only because I've gotten better at abstract reasoning.
As I became more experienced, I started understanding abstract concepts better because I have already seen more real world things. I can now quicky think of my own examples in my head.
(Aside: This is why I think you should work first, study later. University deals with abstract concepts, but without experience these abstract concepts often mean nothing to you)
If you play that game with any kind of prior belief about the values that you're about to see, then the theoretical solution is not longer valid.
What you're describing with the rock star vs incompetent people scenario is about bringing prior knowledge in there. You see a rock star as first candidate. You figure that, in your lineup of 10 people, odds are that no one will be better than that rock star. So you pick him/her.
This makes the theoretical solution at best a kind of intuitive justification in practical settings, but you shouldn't apply it mechanically without using your judgment.
From wikipedia:
> In large part, this work has shown that people tend to stop searching too soon. This may be explained, at least in part, by the cost of evaluating candidates.
and then
> For example, when trying to decide at which gas station to stop for gas, people might not search enough before stopping. If true, then they would tend to pay more for gas than they might had they searched longer.
So you might pay more for gas if you don't use the optimal strategy, but waste more money in fuel costs searching for the cheaper price than you save buy purchasing at that price.
Perhaps most interestingly, there is a formulation of the problem that requires a decision to be made within a certain time period, from an unknown number of candidates who arrive over that time period. If you know (or can estimate) the arrival times of the candidates, you can use a very similar strategy to achieve optimal results.
Essentially, wait until you have seen 1/e of expected candidates in the time frame you have allowed for (based on the arrival density function you know or estimated) then pick the next best option.
This puts a limit on the amount of searching you do, and so provides an optimal strategy with a bounded limit on how long you search for; it provides a bound on the cost involved in the search assuming the cost is related to time taken.
In the searching for gas example, you could use this strategy if you knew roughly how often you pass a gas station, and how long you are willing to search for.