# the secretary problem is a useful model for selling a house

Realtors sometimes think that the optimal solution is to convince their clients to accept the first offer made on their house. The marginal increase in the realtor’s fee is tiny if the sellers wait to get a small increase in the selling price (a 3% commission on the extra \$2000 that the sellers are holding out for is a measly \$60. The realtor may invest more than \$60 to better market the house while waiting for a slightly better offer to arrive. See the video from the Freakonomics documentary below for more on this subject.

We ended up using the optimal Secretary Problem policy to sell our house. It wasn’t our plan on the offset, but it’s what happened. An optimal policy to the Secretary Problem maximizes the probability of ending up with the best offer. The idea is to first estimate the number of offers you would expect to receive, at least in the timeframe that you have to sell a house; let’s call this n. Then you observe and reject the first n/e offers. After that, you accept the first offer that is the best you’ve seen so far. I thought n would be small (2-3), but there was a lot of traffic in our house and I had to increase my estimate of n to maybe 6.

The first person to look at our house made us an offer almost immediately. It was a good offer, but the buyer wanted to close a month earlier than we were ready, which would lead to some substantial costs on our end (not counting the stress of having to make immediate moving plans and find temporary housing). The net offer was good, but not good enough for us to sell our house. We let it go. At the time, this seemed like our n/e, which meant that we should accept the next offer that was better than our first one.

We had a second offer on our house that eventually worked its way up to match the net value of the first offer. It was too low, so we rejected it (and almost gave our realtor a heart attack in the process). We then had a third offer on the house that was not worth entertaining. Later that day, we received our fourth offer. After some negotiation, it became our best so far, and we accepted it.

It’s not often that I get to personally collect empirical evidence to validate an OR model. I’d like to say that it was fun, but it was mostly a stressful experience. I’m glad it worked out in the end. During the process, it was helpful to know that math backed us up on our offer rejections. We ended up with that extra \$2000 (less \$120 for both realtors).

Comment: When I say we “rejected” offers, I mean that we counter-offered with what we were willing to settle for and were turned down. Accepting/rejecting offers is a little more complicated when selling a house as compared to the secretary problem, where there is no negotiation.

Incidentally, the Wall Street Journal recommends using the Secretary Problem for finding a rental [Link].

#### 3 responses to “the secretary problem is a useful model for selling a house”

• Marc-André Carle

That is a great application of OR in one’s personal life. It must have took a lot of self-confidence to reject a reasonably good offer, but both you and the model were right in doing so. Glad this story ended up well for you and your family; buying/selling a house can be a thrilling but very stressful experience.

• Jeffrey W. Herrmann

Laura,

Thanks for this great example of a problem that I always cover in my engineering decision making course. n/e is the approximately optimal policy as n gets large. For small n, it is not hard to find the optimal number: for six alternatives, one should consider at least the first three.

For those who want to learn more, see, for example, Chun, Young H., “Sequential search and selection problem under uncertainty,” Decision Sciences, Volume 31, Number 3, pages 627-648, 2000, at http://onlinelibrary.wiley.com/doi/10.1111/j.1540-5915.2000.tb00937.x/abstract

• Alex Montana

Rejecting the first n/e is very bad advice, as you now have a high chance of rejecting the best offer. The Cardinal Payoff Variant is more appropriate to maximize expected value, where you’d reject sqrt(n) offers, https://en.wikipedia.org/wiki/Secretary_problem#Cardinal_payoff_variant

Although, offers aren’t a random distribution… a good agent should be able to help you understand the expected distribution, and you should stop once you get an offer with a greater than 50% probability of being greater than all the remaining offers.