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Like Yvain's parents, I am planning on moving house. Selling a house and buying a house involve making a lot of decisions based on limited information, which I thought would make a set of good exercises for the application of Bayesian reasoning. I need to decide what price to list my house for, determine how much time and money to put into fixing it up, choose a new home and then there's the two poker games of the final negotiations of the sale.

(I logged onto Less Wrong having just made the decision to consider posting this article, so I was kind of weirded out at first by the title of Yvain's post; but then I was relieved that the topic was somewhat different. I am used to coincidences but on the other hand they push me a little paranoid on my spectrum and I'll feel less stable for a few hours. I already know Google tracks me and who knows what algorithms could be running given a bunch of computer scientists...?)

House Story

tldr; We're listing at the appraised value +10%.

A few years ago, we purchased a beautiful house. 'We' is my husband and I and my parents. We purchased the house because it includes a guest house where my parents can retire. However, my mom continues to postpone retirement and in the meantime my husband and I decided we would a) like more light, b) like a shorter commute and c) could purchase two homes we prefer for the price of this one -- my parents would enjoy a house on the water. (Great post and spot on about the features that matter, Yvain!)

I would be happy to sell the house for +5%, covering real estate fees and new flooring we put in. However, three houses in the cul de sac have sold this year for +10% and so we listed it at that price too. Our house is bigger than theirs but not as nice (they have granite and impressive entrances and we don't). On the other hand, having the guest house makes us special.

Via agent and potential buyer feedback, we're coming to realize that we might be lucky to sell the house for +5%. At this price level, people prefer a house that is impressive and in perfect condition.

Primary Bayesian Question

My primary question is the following: how should we decide to modify our listing price as we get more information?

First, I've read that if a house is priced correctly you'll get an average of one offer every 10 showings. So far we've had 2 showings without an offer. After how many showings should we reduce the price?

Second, the other three houses sold in 6 or 7 months. After how many months should we reduce the price?

Keep in mind, we don't have to move and I estimate that I would be willing to stay in this house for about +3% per year. In other words, I would be willing to wait 2 years for a higher offer if I could sell it for +3% more by doing so.

I anticipate that after posting this I will be embarrassed that it is so pecuniary. On the other hand, this makes it concrete and the problem in general doesn't have too many emotional factors. Any money we make over the first +5% can be used as a down payment for our next house after we pay our parents back. (I did feel embarrassed, so I took out the dollar values and replaced with relative percents.)

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New Comment

You might want to consider a list price which is precise to \$100 or \$1000 rather than a round number, because the anchoring effect seems to be stronger when a price is more exact. The original paper is behind a paywall, but there's a summary in this article.

They looked at five years of real estate sales in Alachua County, Florida, comparing list prices and actual sales prices of homes. They found that sellers who listed their homes more precisely—say \$494,500 as opposed to \$500,000—consistently got closer to their asking price. Put another way, buyers were less likely to negotiate the price down as far when they encountered a precise asking price.

First, I've read that if a house is priced correctly you'll get an average of one offer every 10 showings. So far we've had 2 showings without an offer. After how many showings should we reduce the price?

This isn't enough information: you also need the number of showings necessary for a home mispriced by 5% upwards to get one offer.

Once you have that number, assign a prior. You think that the guest house is sufficient to justify the price, and have feedback from other people that it isn't. You like your opinion, but they have experience- let's put a prior of .5 that the house is correctly priced and .5 that the house is mispriced upwards. (You should pick numbers that reflect your beliefs before any showings.) Decide what threshold probability you need to lower the price of your house. (You can work this out from your time-price preference if you want to ensure consistency, but that doesn't seem very valuable. There's another way to figure this out below that's pretty cool, but is a short-term method.)

Every showing, you receive an offer or don't. If your house is mispriced, let's say the probability of an offer is 1 in 20- .05, whereas if your house is correctly priced, it's .1. The probability of not receiving an offer is thus .95 or .9. Let's call your current probability that the house is priced correctly PC, and the probability it's mispriced is 1-PC.

Each time you show without receiving an offer, PC is now (.9*PC)/(.9*PC+.95*(1-PC))=(.9*PC)/(.95-.05*PC). After two showings with that prior and those probabilities, you should believe there's a 47% chance the house is priced correctly, and a 53% chance it's mispriced. When you get an offer, PC is now (.1*PC)/(.1*PC+.05*(1-PC))=(2*PC)/(1+PC). This gives us another stopping condition we can use: "At what point will I believe it more likely the house is mispriced than correctly priced, even if I get an offer the next showing?" You can work this out (I did it in Excel, and can send you the file if you want to play around with other numbers) and figure out that after 13 showings without an offer, you'll believe PC is .33, and if the 14th showing results in an offer PC will be only .48.

(Edit: Note that this means that if your prior is only .33 that the house is correctly priced now, and you agree with the .1 vs. .05 numbers, then taking more data may not be necessary. More data will of course get you better knowledge- the stopping condition I put forward is arbitrary- but the whole decision problem of when to adjust your offer is a somewhat difficult game theoretic one.)

Thank you -- this is exactly the sort of Bayesian analysis I'm looking for.

The probability of having chosen a correct price is related, but most useful for my purposes is updating an estimate of how long it will take me to sell the house at the current listed price as information arrives in the form of showings-without-offers.

Correctly priced homes sell in 10 showings, on average. As the number of showings increase to 13, then 14 without a sale, I understand this lowers the probability that I've chosen a correct price. How should my estimate of the average number of showings needed be updated? I agree that an important piece of information I'm missing is how a price that is inflated by a certain amount increases the needed number of showings. Can I also estimate this as I go?

I agree that an important piece of information I'm missing is how a price that is inflated by a certain amount increases the needed number of showings. Can I also estimate this as I go?

Sort of. The problem here is how you define your prior matters a lot. The following math will be a little sloppy, but should work well enough. A somewhat reasonable way is to assume that the probability of an offer is somewhere between 0 and 0.1- and let's just assume it's equally likely for all of those probabilities, so you start off with a uniform prior. (Here is where I wish I had a whiteboard, so I could start drawing stuff). You essentially have a probability density over probabilities- you think the density at .05 is 10, same at .1 and 0. The chance you assign to the rate being between .04 and .05 is the integral of 10 from .04 to .05- which is .1. (Knowing calculus is necessary to use this method, but I can give you some results from it with no calculus necessary).

Then you show the house, and don't get an offer. Now, this is possible if the probability of an offer is 0.1- but it's even more likely if the probability you get an offer is 0. You can see that the densities are going to get multiplied by (1-x), as that's the probability you don't get an offer for each probability. You need to renormalize it (since the integral of probability densities should be 1), but what we're really interested is in the center of mass of this probability distribution.* It starts out at .05, and drops down as you show more without getting an offer. The formula is a little ugly to stick into a comment, but I've added it to the same excel sheet. If you currently think the probability density of offer chances has a weighted mean at 0.048 (what it would be after 2 showings with the prior mentioned above), then 1/.048=20.73; you expect it'll take 21 more showings, on average, to receive an offer. (Notice that, if you knew the chance was definitely .1 of receiving an offer, you would always expect about 10 more showings on average.) After 20 showings without an offer, you think the weighted mean is .034, and it'll take 30 more showings.

You can use this to figure out when it's worthwhile to reduce the expected number of showings left down to 10. If you do a showing a week and believe the 3% / 52 weeks number, that means you would be willing to wait 83 weeks in order to get a price that's 5% higher.** We want to find out when your expected number of showings left is 93, which will happen after 91 showings (i.e. 89 more than you've done now). The math underlying 91 uses a bit of sloppiness, so don't put too much weight by that particular number. The method I used should escape most of the contamination possible by including 0, but that's sort of what you're worried about (it could be the house will never sell at a price too high, rather than selling with very low probability).

*Really, we're interested in the center of mass of the inverse of this probability distribution, but because of the prior we chose that's a worthless number. (If there's a .1 chance centered at 0, .01, ..., .09, the average time until you get an offer is infinity, because there's a 10% chance it'll never happen. That's not particularly useful, though, and so instead we're just calculating the mean offer rate, and figuring out how long it would take at the mean offer rate (22.2 showings with those clusters).

**I'm using 1.1/1.05=1.047 minus 1 = .047/.03*52=82.5. You could also do 5/3*52=87, or there's probably something else that's more rigorous than this. Doesn't make too much difference.

I'm really liking this bayesian case study. Could you put the excel spreadsheet you made for it on Google Docs (or something) and post the link here?

Sorry this took so long- it was about 5 minutes of formatting that I kept putting off. You can find the spreadsheet here. The bolded numbers are numbers that you can change to play around with. The first page has two areas- the left one is "what are my probabilities on hypotheses A and B given that the house has been shown X times and has received 0 offers", and the right one is "what would be probabilities on A and B be if I show the house X times, receive 0 offers, and then receive one offer?"

The second sheet is "given that I started with a uniform prior over success rates between 0 and 0.1, how many more times do I expect I need to show the house to receive an offer?" while updating on the information that you haven't received an offer yet.

Excellent. This is an example of the usefulness of Bayesian reasoning, and it can be generalized to any situation where you are trying to use an observation of the form 'it-hasn't-happened-yet' to update your estimate of it's rate of occurring.

So, paraphrasing what you said, I first choose a reasonable range of probabilities for something-happening and a very rough estimate of my probabilities for those probabilities over that range. (For example, I think my house would sell in between 1 out of 50 and 1 out of 10 showings, and the probability should increase linearly over that range with some slope.) Second, each observation that something-hasn't-happened should update my probabilities as you described.

This is very interesting to me, that I can do something with the 'information' I get after each showing without an offer, and these calculations give me something to do while I'm waiting. (Besides continuing to stage my house, which I continue to work on as well even though I suspect I am in the region of diminishing marginal returns for that.)

I finally formatted my spreadsheet as a google doc; you can find it here.

Note! See represenatativeness. Don't think anything is wrong if it doesn't sell after 50 showings.

List it at the low end of what you're willing to accept, so that you receive multiple offers. Their attempts to out-bid each other should drive the price higher than any of them would have been willing to accept as an initial offer. Human psychology strongly favors things that are desired by other people over objectively similar things that have been sitting on the table for multiple years. If you go too low, though, you'll be fighting against the anchoring effect.

(Take this with a grain of salt; I don't have direct experience with selling houses. I do have a bit of experience with the effects of social proof, and it's shockingly powerful.)

Edit: A bit of googling seems to indicate that this is common advice.

This is interesting. How do I make sure I get multiple offers? There's a limited number of people that want a house in this region for this price range with a guest house -- I would estimate 1 person per month. When someone makes an offer, you usually need to decide within a couple days.

Have you considered trying to do some home improvement projects to improve the house? If "nice" entrances make houses sell for disproportionally more than the cost of those entrances, it would seem as though it makes a lot of sense to invest the time and money (or just money if you want to contract it out) to improve the entrance to your house.

Yes, it's amazing what a few inexpensive plantings and restaining the door has done. On the other hand, I can't change the architecture of the house and that is what is much more modest than that of the neighbors. (Think cottage verses pillars.)

We're currently considering updating the kitchen, but since the kitchen is actually terrifically functional (in my opinion) it might be better to sell the house for less to someone looking for a bargain than to put the money in that.

... but yes, I should research this idea more. I can continue by looking at houses that are similar to mine in higher price ranges and see what the differences are, in order to see which fancy improvements, if any, are consistent with my house's architecture. Thanks!

Have you watched one of those "reality TV" shows where they bring in a "consultant" to help people sell their home?

Trite as they may be as entertainment, they usually cover the basic items of procedural knowledge that apply to every house sale. Maybe it says a lot about TV audiences that there seems to be a need to go on for hours and hours about detail consequences of these things that anyone smart could work out for themselves, but if you watch just one or two I'm pretty sure you'll get useful advice.

For instance, one thing that stuck in my mind (and resonated with my experience on the other side, as a buyer) was that the more personalized, the more "you" your house reflects, the less a buyer will be tempted to make an offer. Sellers should remove items like family photos and well-loved but weird gadgets.

the two poker games of the final negotiations of the sale.

That's not the only way to think about negotiation and it's quite often not the best one.

I would be happy to sell the house for +5%, covering real estate fees and new flooring we put in.

I don't think you can say that for sure until you know about a place to move into.

I don't think you can say that for sure until you know about a place to move into.

Why do you say that? The extra information would be the price of my next house, or whether I'm able to find it, etc?

Yes, you want to trade the present house for a new house and money. Saying you'll be happy with a certain sale price that's pegged to the purchase price and unrelated to the cost of a new home sounds like loss aversion and the sunk costs fallacy.

Keep in mind, we don't have to move and I estimate that I would be willing to stay in this house for about +3% per year. In other words, I would be willing to wait 2 years for a higher offer if I could sell it for +3% more by doing so.

At +3% per year wouldn't you wan't +6% after waiting 2 years?

Yes, I meant +6% after waiting 2 years.