Should I buy this house or wait for a better one — solved with a risk-preference model
…By the way, works for dating and ad bidding too!
TLDR: Here is an Excel based calculator, which will help you make a this important decision. Download it here, enable the “solver” plug-in and enjoy!
Anyone who’s been shopping for a home has experienced that time when a decision is imminent: do I take this one or wait for my dream option? This situation is not too unlike dating, when one needs to decide whether to stay with the partner, based on the experience so far, or move on. It turns out, there is a formula for what information-so-far you should have before making that decision, and it’s owed to a handful of mathematicians, according to a helpful discussion of the problem in Washington Post.
The Classic Solution
It turns out, the problem is a classic one — sometimes called “the secretary problem,” sometimes “the sultan’s dowry problem,” — presents the decision maker with a certain pool of options (houses or mates), that arrive in a random order and are up for grabs to the decision maker. According to the classic solution, one should review the first 37 percent (~1/e) of all options and then make a choice for the first option that beats the best one has seen so far. But is this really the best algorithm?
For starters, it requires a person to assess how many options they’d have in total. Although somewhat troubling, this could work in theory if you, say, give yourself 2 years to see options in total and then plan on seeing X number of them every week. Then the number of options would equal to the length of search times the option frequency. In this sense, the 37 percent would correspond to either the number of options or amount of time lapsed before the deadline.
However, the 37-percent law can lead to some perverse behaviors in edge cases. For instance, when all the options in the “test set” (the first 37 percent) were really bad, the person will end up accepting a really mediocre next option. By contrast, if all the options in the “test set” were amazing, the person would end up passing on the rest and end up with nothing in the end. This is because the classic solution assumes a uniform distribution of the arrival probability of candidates with different qualities. When this is violated, as life does not always guarantee uniformity, different levels of people’s risk aversion should factor into their decision to avoid these pitfalls. For instance, a risk-averse person may take an option as long as it’s good enough. On the contrary, a risk-taking person would wait longer, even if the search is in a very late stage.
Lastly, when does it happen that one is in full control of a transaction, either commercial or interpersonal? Deals fall through last minute because a better buyer (or suitor) comes along. Because of this, an additional risk layer must be added to reflect the possibility of just getting outbid (or dumped).
The new model, let’s call it the “Risk-Preference Based Decision Making Under Time Constraint Model,” works like this:
The Algorithm
1. Input the user’s timeline for decision making (how long will they be looking)
2. Input the present situation parameters:
a. How long has the person been in the market?
b. About the choice at hand (call it choice B): how good is it on the scale of 1–100? (100 being the best. Think of it as the percentile of B among all existing choices)
c. How many similar choices has the decision maker witnessed so far?
d. How many bids have been placed on such choices? (For dating, could be how many competitors there have been.)
3. Based on step 2, estimate the distribution of choices using the frequency of B (2.c/2.a) as the estimator of the probability of B; as well as the decision maker’s rating of B (2.b), which is an estimator of the cumulative probability of B. (see the diagram below)
— Note: assume a Normal distribution for choice values. The mean of that distribution (i.e. the values themselves) can be arbitrary since the score are based on the cumulative distribution of choices, reflecting their relative goodness rather than an absolute number.
4. Ask the decision maker about their best attainable option (call it choice A), which will also be a percentile, like B. (Could be phrased as: How would you rate the best option you can afford?)
5. Compute the associated value and probability of A, using the distribution of choices from step 3
6. Compute the probability of seeing A in the time left (result in step 5 times # of periods left)
7. Factor in the risk of losing due to competition (another buyer or suitor outbidding the decision maker for choice A) based on the estimated probability from the answer in 2.d
— Note: assume each contender has equal chances at your option; also the competition is uniform across all options because people only consider options that are “attainable” to them.
8. Input the person’s risk preference on the scale of 0–1 (with 0 being most risk averse and 1 being most risk tolerant)
9. Compute the expected utility of the gamble for choice A: Px^c. Use the Power Utility Function (x^c), with x being the “top attainable score” from step 4; the exponent (c) from step 8; and the probability (P) from step 7
10. Compute the expected utility of B: P1x1^c + (1-P1)Px^c. Use x1 from 2.b; c from step 8; and P1 from step 2.d
— Note: The alternative of losing B is going back to the search for that elusive aspirational choice (or the expected value of waiting for option A: Px^c)
11. Compare 9 and 10 to see which option wins
The Calculator
Here is the Excel-based calculator that does all these computations via a user-friendly UI. Open it up in Excel, give it a shot and let me know what you think! (Note: this calculator does require the “solver” plugin, which I use to estimate the distribution.) Tell me, is this a better way of solving the classic problem? Does the problem formulation make sense to you? Do the scores surprise you, or are within your expectations? I’d love to see your comments!
An Interesting Extension for Business
Just as people bid on houses, ad networks may bid on users who may arrive in a stochastic order. The same statistical model can be used to evaluate whether to show an ad to a certain user or wait for a better matching one as to not waste ad impressions.
Further research
Although I believe this model is a more realistic cut at an average person’s decision-making environment — compared to the classic formulation — and also considers risk-aversion (an important factor in all decision-making), certain deficiencies still remain. A discontinuous timeline function, which has a user-defined cutoff point, is unrealistic for real-world scenarios. Most people do not set stringent deadlines and certainly don’t meet the ones they do set. As such, to say that one must buy a house or marry by a certain time frame seems unreasonable, just as knowing exactly how many suitors you will have (per the classic formulation). An asymptotic penalty function in time would be a more reasonable model for this reason. For instance, the more time goes by, the less time you will have to enjoy what you get.
Another important improvement that would make the model more consistent with other statistical models would be to address the information confidence. Having seen 1 choice over 1 month vs. 100 over 100 months would produce the same estimated probability in this model, whereas clearly the second scenario should be treated differently. In a better version, the decision maker will get an associated confidence interval for every choice, and then factor in their lower-bound tolerance level in deciding whether to go with the present choice or wait for more information.
These improvements will be the subject of the next paper 😊
