# Tag Archives: secretary problem

## 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].

## robust Christmas shopping

I usually more or less finish my Christmas shopping before Black Friday, so I usually do not worry about optimal shopping strategies. I enjoyed reading about Aurelie Thiele’s latest paper on robust timing of markdowns.  Her blog post summarizes her new paper (she provides a link to the paper), where Aurelie and her collaborators propose a method for dynamically timing markdowns over a finite horizon using robust optimization models.

This got me thinking about how to reverse-engineer the process to get the best deals.  If you’re not willing to do Black Friday shopping (the getting up at 3am and waiting in long lines do not make up for the extra savings), then shopping for a few key items is like a stopping problem (e.g., the Secretary Problem), where a series of deals are offered (some at the same time, such as in the Sunday ads) and the consumer eventually decides to purchase exactly one item with a deadline of December 24.  When should I purchase the item?  I suppose it depends on how retailers adjust the prices from one time period to the next.

These are a few rules of thumb that I’ve learned (beware: no modeling or algorithms used here).  With a bad economy, I see no drawback to a wait and see approach, since sales continually improve and store coupons become available as shoppers stay home.  I have found that in the last three years or so, the pre-Christmas sales (not just Black Friday) are often better than the post-Christmas clearance sales.  So I stock up and do some personal shopping before the holidays.  However, the clearance sales become lucrative in February and insanely good in March (for clothes, at least).

How do you do your Christmas shopping?

Related posts: