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.
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.)
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?
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.