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At the beginning of last year, I was faced with having to make a decision that needed to be both immediate and final: picking our next place to rent in Melbourne. When faced with choices like this, the most difficult thing to handle is the potential regret of making the wrong decision. I was worried that the place I just inspected was worse than the place that I was planning to inspect next.

In life, we are frequently faced with situations like this. For example, you might be thinking whether or not to immediately accept the job offer you receive, or go to the next interview and compare it with the offer on the table. In dating apps, you are faced with the choice of whether to swipe left or swipe right.

Sequential searches like these are a classic explore-versus-exploit dilemma: the longer you search, the more options you see, but the risk grows that the best one will slip away.

So is there a way to figure out the best option to pick from? According to math, the answer is yes.

The problem

In mathematics, this problem is usually framed as the secretary problem. Imagine you are an employer interviewing N job applicants one at a time. You rank applicants from best to worst, but you must either hire or reject an applicant immediately. In 1966, Gilbert and Mosteller pioneered a strategy that maximises the chance of selecting the absolute best candidate.

First, decide how many candidates you are willing to interview. Let this be N number of candidates. Then, reject the first X number of candidates outright, while keeping track of the best of these. And then at a certain point S, you need to stop rejecting them and start evaluating whether a candidate is the best you’ve seen so far. During this evaluation phase, you accept the first subsequent candidate who is better than everyone you’ve seen so far. But where is point S? At what sample point should you start evaluating?

The authors found that the optimal sampling fraction lies at a value of 1/e or 37%. So explore and reject 37% of options you have just to get a sense of what’s (or in this case, who’s) out there, and choose the first candidate that is better than all the 37% of candidates you have interviewed. And if you follow this rule, mathematically, you’ll pick the very best candidate about 37% of the time.

Mathematics meets humans

Mathematical optimality is neat, but how do real people behave? In laboratory tasks, participants seem to adopt threshold-based strategies: they watch the early part of the sequence to learn what “good” looks like and then commit once the current option crosses some internal threshold. However, their thresholds are rarely at 37%.

In 2020, a paper published in Proceedings of the National Academy of Sciences tested how closely human behaviour matches the mathematical optimum. In the classic rank-order secretary problem, we know that the optimal strategy is to examine the first 37% of options and then choose the next one exceeding the maximum seen so far. Yet when psychologists looked at empirical data, people tended to set their cutoff earlier and would start accepting candidates sooner than 37%. Similarly, Maime Guan and colleagues showed that people use threshold rules but often set them sub-optimally. Some participants in their research were more cautious, waiting for very high values, while others were quick to settle. The model suggests that factors like risk aversion or personality could influence your personal threshold and deviate it from the optimum point. Baumann et al. (2020) also argued that in environments where good options are plentiful, people tend to stop earlier and make a decision faster than the optimum point; while in scarce environments they search longer. Such behaviour makes sense - although not mathematically the most ideal - because you are less bound to keep looking when bargains are plentiful, but hold out longer when they are rare.

Another equally interesting factor that pushes people to undersample is their expectations. A 2024 study in Communications Psychology asked participants to search through smartphone contracts and choose one of the top three deals. The task provided full information of each contract and allowed participants to sample up to twelve options. The authors found that participants often undersampled and stopped too early even when the optimal strategy required gathering more information. Computational modelling indicated that this bias was not due to laziness but to pessimistic expectations about future options. When the model recommended increasing the sample rate, people were reluctant to do so and maintained low sampling rates because they assumed - again incorrectly - that later options would be worse.

In other words, some of us give up evaluating future options because we expect the future to disappoint.

Implications

As tempting as it is to treat the 37% rule as a magic number, we should know that that number is valid when options are presented randomly and their total number is known. Also, humans are not computers. We use heuristics and carry expectations about how good future options will be. Understanding these tendencies can help us make better decisions. Consider these three tips when you need to choose one of something out of a pool of many:

1. Define your N. The solutions presented by Gilbert and Mosteller require knowing or estimating the number of options. Decide in advance how many you’re willing to consider and commit to it.

2. Set a deadline. Spend roughly a third of your allotted search to learn what’s out there. During this phase, don’t commit but take notes on what qualifies as “good.” Once you’ve reached your exploration limit, pick the next option that exceeds the best you’ve seen.

3. Recognise your biases. Are you someone who hates missing out? Remember research suggests that risk tolerance influences how high you set your threshold. You might wait too long and miss good opportunities, or you might settle too early if you’re pessimistic about the future.

That’s all for this week. See you next Saturday 😄 

With love,

Krish

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