You're looking for a flat. Offers come one after another and you have to decide on each straight away, or someone else will take it. You can't go back. When do you stop looking and say “I'll take it”?
A gameThe flats and the points are made up. The game shows how a decision rule works, not how lucky you'll be. What you enter isn't stored or sent anywhere unless you save your result yourself.
You'll see 20 flats. Can you pick the best one?
Each flat has a score from 1 to 100. You don't know the scores of the ones to come. You can't go back.
Mathematicians call this the secretary problem. It first appeared in print in February 1960 in Martin Gardner's column in Scientific American, and the statistician Dennis Lindley published the first solution in a scientific journal a year later. When offers arrive in random order, you can't go back and you want the very best, your best move is to just look at roughly the first 37% of offers and then take the first one that beats everything so far. With 20 flats, that means looking at seven without taking any. You'll then find the very best flat in 38% of cases. Without more information, you can't do better.
The rule usually “fails”. In 62% of cases you won't pick the best flat. In 35% of cases the best one was among the first seven, and the rule forces you to take the very last offer, whatever it's like.
The 37% rule is built for “the best or nothing”.
Is a good flat enough? Stop looking sooner. If what matters is that your flat ranks as high as possible on average, then among rules of this kind it's better to look at just three flats. The average rank of the flat you pick is then 4.2 instead of 5.0, and you only have to take the last offer in 15% of cases.
People stop sooner than the maths says. In lab experiments, people stopped searching earlier than was optimal (Seale and Rapoport, 1997; Bearden, Rapoport and Murphy, 2006).
In real life, though, that need not be a mistake.
The puzzle doesn't count the time and nerves that searching costs, and if you're not only after “the very best”, the previous paragraph says you should stop sooner even by the maths.
What to take from it. Decide in advance whether you're looking for the best or for good enough. The psychologists Sheena Iyengar, Rachael Wells and Barry Schwartz (2006) followed final-year university students looking for a job after graduation. Those who, by questionnaire, tend to always want the best ended up choosing jobs with starting salaries about 20% higher. But they were less satisfied with their jobs, and the search cost them more bad feelings.
How it works
The problem: the number of offers is known in advance (20), offers come in random order, you decide on each one straight away and you can't go back. The 1–100 scores in the app are only there to make it clearer. The rule doesn't use them; it only checks whether a flat beats all the previous ones. You can see them, though, and if you use them (say, take the first flat that is the best so far and has at least 90 points) you find the best in about half of the cases; with full information the theoretical limit is roughly 58% (Gilbert and Mosteller, 1966).
The rule “just look at the first r, then take the first one that's better”: it finds the very best with probability (r / n) · Σ 1 / (i − 1) for i from r + 1 to n (for r = 0 it's 1 / n). It peaks at r = 7: 38.4%. You have to take the last offer with probability r / n, because the best was among the first r. The app computes the average rank exactly: the rule stops at the i-th flat with probability r / ((i − 1) · i), and the flat it picks is then the best of i random ones, with an average rank of (n + 1) / (i + 1). The best average rank (4.2) comes from looking at three or at four flats. The app shows three, because then you end up with the last offer less often. Only rules of this type are compared. A rule that towards the end also accepts the second- or third-best so far reaches an average rank of about 3.0 for 20 flats (Chow, Moriguti, Robbins and Samuels, 1964). We also checked the numbers by simulation.
The experiments: Seale and Rapoport had people play the classic problem, where you only see the ranking. Bearden, Rapoport and Murphy used a version that pays not just for the best but according to the rank of the chosen applicant, so closer to “good enough”. In both, people stopped earlier than was optimal.
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Sources
- Ferguson, T. S. (1989). Who Solved the Secretary Problem? Statistical Science, 4(3), 282–289. History of the problem: Gardner, Scientific American, February 1960; Lindley, D. V. (1961). Dynamic Programming and Decision Theory. Applied Statistics, 10(1), 39–51.
- Seale, D. A., Rapoport, A. (1997). Sequential Decision Making with Relative Ranks: An Experimental Investigation of the “Secretary Problem”. Organizational Behavior and Human Decision Processes, 69(3), 221–236.
- Bearden, J. N., Rapoport, A., Murphy, R. O. (2006). Sequential Observation and Selection with Rank-Dependent Payoffs: An Experimental Study. Management Science, 52(9), 1437–1449.
- Iyengar, S. S., Wells, R. E., Schwartz, B. (2006). Doing Better but Feeling Worse: Looking for the “Best” Job Undermines Satisfaction. Psychological Science, 17(2), 143–150.
