Thursday, October 15, 2009

The secretary problem applied to selling your house

This problem goes under various names: the secretary problem, the sultan's dowry problem, the 'choose a fiancée' problem, the beauty contest problem.

In the beauty contest variant, you are the judge. You are presented with the contestants one-by-one. After evaluating each contestant, you must either declare that contestant to be the winner, at which point the contest ends; or disqualify the current contestant from any further consideration and move on to the next girl.

The question is, which choosing strategy makes it most likely you will choose the best, most beautiful candidate?

Clearly, if you choose the first girl you see, you are rejecting all the rest without seeing them: hardly smart.

On the other hand, if you just keep on rejecting until you come to the last one, you're stuck with her. How likely is it that she's going to be the very best of the lot?

Clearly a better plan is to keep looking (and rejecting) for a while to get a sense of the general level of beauty, and then choose the next contestant you see who is more beautiful than all the ones you've previously rejected - hopefully before you run out of candidates!

If there are n girls in the competition, it turns out that the best strategy is to consider and reject the first 37% of n (exact figure is n/e) and then choose the next candidate who is the best so far.

This leads you to choose the girl who is actually the most beautiful 37% of the time (exact figure 1/e). Not stupendous odds, but reasonable under the constraints.

The reason for mentioning this is that selling a house can look like a version of this problem too. The viewers come one by one and make an offer (a dowry) to you.

If they're not interested, their value to you is zero, they're not beautiful at all.

Or they make make an offer which doesn't impress, they're not all that beautiful.

Each of these viewers is telling you something about the overall market - the space of possible offers for your house. So when should you accept an offer, realising that it's the best you're likely to get?

One of the issues with the secretary problem, in all its variants, is that you need to know the total number of applicants in advance. This translates to the total number of viewers you are prepared to tolerate.

Noting we are currently seeing an average of 2 viewers per week, for the sake of argument I'm going to take three possible viewing totals: 16, 24 and 32, corresponding to the house being on the market for two months, three months and four months.

If n = 16, you should see and reject bids from 37% of 16 = 6 viewers, and then accept a higher bid once you get it from a subsequent viewer.

If n = 24, you should reject bids from the first 9.

If n = 32 you should reject bids from the first 12.

The model therefore advises that we should see and 'reject' a few more - we have so far seen and 'rejected' seven viewers (six of whom did not make formal offers).

Of course, this also applies to buying a house where the message is: don't make an offer too early.