The Math of making the most optimal decision

You have one spear and a single stone tip lashed to the end of it. You're in the hills of what people will much later call Cantabria, in the green north of Spain, near a cave one day called Altamira. Below you, down a slope of holm oak and grey rock, a red deer is grazing. It hasn't seen you.

That stone tip is the work of a patient hand and a piece of good stone, which is not lying around everywhere. You worked it down flake by flake and bound it to the shaft with sinew and pine pitch. Lethal but fragile at the same time. Throw, miss and hit limestone instead of the deer and it will shatter into pieces. You feel its weight neatly balanced, but also that it is not just a spear. It's every calorie you and the people waiting near the cave will eat or won't.

The deer is side-on, but too far. You could throw now. At this range you'd almost certainly miss, the point would find rock and that would be the end of it. So you don't. You press flat against the warm stone, wait and it steps closer. Better. Closer again. Now it's in range. A real cast, not a sure one though. Throw?

It might even come closer. A few more steps and you almost cannot miss. Why spend your one tip on a maybe when a near-certainty might be a breath away? So you wait. You wait for the perfect moment.

Then the deer lifts its head. The ears turn. It has caught something off the wind, probably your scent and is gone. You understand exactly what you've done. The cast you had a moment ago was the best one you were ever going to get and you spent it waiting for a better one.

There are two ways to lose here and you have just felt both. Throw too soon, miss and the point bursts on the rock for nothing. Wait too long and it sits whole and useless in your hand while the deer is already gone. Either way the cave goes hungry. ​The argument can be made that when you let the deer go, you keep your spear intact and can throw it into the deer that passes tomorrow. So you keep optionality. But that has to be balanced against how long you and your tribe have gone without eating. Because the spear is only one part of the equation. The arm that has to be strong enough to throw it is the other.

When does "I could do better" have to surrender to "This is as good as it gets"?

Optimal Stopping Theory

Seventeen thousand years on, we traded spears for munitions or markets, but the question stays the same, deciding when to stop considering our options and committing to one.

All these problems share the same underlying structure. A decision-maker observes a process that unfolds over time and carries some degree of randomness. Acting only on what is known at each moment, the decision-maker must choose how to maximise reward or minimise cost while encountering the available options one after another, with no opportunity to return to an option once it has passed (Hill, 2009).

Let's say you have to pick the highest possible number from a set of 25 cards, where you don't know the range of possible ones. How would you approach the task?

As you might have felt intuitively, it makes sense to get a feel for the playing field before committing to any one option. But there is a precise mathematics behind this, one that can tell you exactly which strategy gives you the best outcome. It rests on a long history in the field and goes by several names, the marriage problem, the secretary problem or simply the best-choice problem (Ferguson, 1989).

The 37% rule

Optimal Stopping Theory states that you should reject the first 37% of all candidates by default and after that window, go with the one that is better than the best one you experienced so far. This is mathematically proven to be the best strategy for finding the optimal candidate.

You only look at candidates and reject them by default, but let the tallest set the bar. After the first 37%, the rule takes the first one that is taller than the set bar.

The rule turns up across very different domains. Even some fish use a version of it to find a mate. You have probably done much the same without ever naming it. When a behaviour shows up in a creature whose line split from ours hundreds of millions of years ago, the wiring is likely old.

Care to play the game of picking the highest cards again? Now knowing the most successful strategy.

Flaws of the 37% rule

Although mathematically proven to be the best strategy, it comes with some flaws as the following circumstances can also happen.

This time the tallest slip of all sits inside the first 37%, so you can't take it. As nothing after the window ever reaches it, you have to take the last slip you are handed.

Or the possibility that in the first 37% you only saw rather mediocre candidates and then have to go with the marginally better one, missing out on the best candidate by a wide margin.

Here the first 37% is full of weak slips, so the bar sits low. The first slip that beats it is barely better than what you already saw and the rule stops there. The real best arrives a few slips later, far taller but you have already committed to one candidate.

Moving the gate to a different axis

The 37% rule only operates on ranks, technically called ordinal values, so whether the thing in front of you beats what came before. When picking the highest card, you saw the exact number but not the possible range from which they were drawn. ​Draw a -20, 30, 0.5, ... then 90, you still don't know if 90 is big, as the next card you draw could be 9,000,000

If you knew the set you draw from is 0 to 100, drawing a 90 would be a cardinal value, because you would know not only that it beats the cards before its ordinal value (rank) but where it sits on the whole scale cardinal value (magnitude). A 90 out of 0 to 100 lands in the top tenth of everything that could possibly be drawn, so any future card has only a one in ten chance of beating it.

Circling back to the two circumstances when the 37% rule failed. The second one was, if you just experienced losers at the beginning and then with a skewed understanding of the market took the only marginally less bad option. The first one was, when encountering the best option very early, you could not take it as gated behind the look only. A slip that clears the bar by a hair and a slip that towers over everything before it look the same to the 37% rule, but not if you can leverage the magnitude of it.

Gilbert and Mosteller worked this out in 1966. With the cardinal value you can set the bar simply at the level you want. Something clears it, great, you lock it in, even if it is the very first slip you see. The gate is no longer a moment you have to reach, just a level you have to beat. With full information the best-choice win probability rises to about 0.58 (Gilbert & Mosteller, 1966), against 1/e ≈ 0.37 in the rank-only problem.

Let's say you are out for a new job. Before the first interview you decide you will say yes to anything above 120k with the right team. The first offer that beats it gets taken, whether it lands on the fifth interview or the very first. You are not interviewing the early ones just to learn the market.

On the left the gate sits on the time axis, the 37% rule. On the right the gate sits on the value axis, a level you set yourself.

Maybe you already caught it. This presupposes that you know that cardinal value and by that the shape of the distribution you are drawing from. The mean and the spread. With the job offer example, things you can approximate well enough. Salary bands for a role are public, so you walk in already knowing roughly the range. But with something like finding a life partner you cannot. There is no published scale for how good a person could be for you, no mean and no spread to set a bar against, so you are back to ranking each one only against the ones who came before. Gilbert and Mosteller's work, therefore, also comes with limitations. Furthermore, there are structural problems that also need to be addressed.

Structural Problems

As elegant as it looks on paper, it starts to break when you put it into practice.

First: For the 37% rule to work you need to know how many candidates you will encounter. Without that number, while you are still searching, there is no way to tell when you crossed the 37% mark. Bruss (1984) got around this with the "1/e-law of best choice". The idea is to count time instead of heads. Skip the first 37% of your search window, then take the next record. But the uncertainty does not vanish, as it only moves. Now you have to know when your window closes and trust that candidates arrive evenly across it. If they bunch up early or late, "37% of the time" stops matching "37% of the candidates" and the rule misfires.

Second: It treats the candidates as draws from a fixed distribution. In real life they shift. They get scarcer when competitors fish the same pond. They get better when you become able to attract higher quality. Sometimes both at once. Your first instinct is probably to move the bar. If options improve, you raise your standard, if they dry up, you lower it. But a run of three bad candidates might not mean the pool is drying up. It might just be noise. Mid-search you cannot tell the two apart.

Third: You're judging the candidate over time on different metrics, because searching teaches you what you actually want. This one is big enough that it needs its own essay.

Sit with that third problem for a moment. If every candidate teaches you something about what you are looking for, then no candidate is only a candidate. Leading to a question that might help us even when the theory broke after impact with reality.

Every candidate has two values

Is the candidate itself or the data point the candidate provides more valuable?

Early candidates are, on average, worth more as data than as catches. Late candidates are the reverse, with little left to learn and almost no runway to apply it. The deer on the slope is meat. It is also a fact about how close the herd will let you get this year. Which of the two matters more depends on so much more than just the deer itself.

Two guiding thoughts pull against each other.

The first pulls you toward acting. A lesson learned early has a long run of later decisions to pay off in, again and again, so the cheap time to get things wrong is at the start. A wrong turn at twenty-two is tuition. The same wrong turn at fifty-five is just a wrong turn, because there is less life left to spend what it taught you. The actress Tallulah Bankhead put it with no arithmetic at all. "If I had my life to live over again, I'd make the same mistakes, only sooner." (Bankhead, 1952)

The second pulls you toward waiting. Looking is cheap. Committing is expensive, because committing blind throws away every later choice the lesson would have sharpened. Take the first job you are offered or marry the first person you date and you lock in a verdict before you have any idea what good looks like.

SHORTLONG

HORIZON

Early bars are mostly data, late ones mostly catch. Set the horizon, to see them change as rough approximations.

With those two pulling against each other, the answer was never going to be a number. It depends on how much game is left and what time horizonTime HorizonsWhy dying is often hard, but sometimes easyRead the essay↗ you expect to experience.

The question that remains

The hunter on the slope was never solving the famous problem. He had one tip, a moving target, a distribution he could only guess at and a clock set by how empty the cave already was.

If there is a heuristic to carry off the slope, it might be the idea: Spend the early ones to learn. Spend the late ones to win. But for each individual situation the question remains: Is one more look worth more than what it costs?