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?
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.)