Combinations beat single skills

Suppose ten thousand people work in a field. Being the best of them is a lifetime's project with poor odds, and most of the advice anyone receives is some version of attempting it anyway.

Here is the arithmetic of a different route, and then the honest account of when it does not work.

The arithmetic, spelled out

Ten thousand people. Take the top quarter at the first skill and two thousand five hundred remain. Of those, take the top quarter at the second and six hundred and twenty-five remain. Take the top quarter at the third and you are down to roughly one hundred and fifty-six.

Now the specialist. One in a hundred at a single skill leaves one hundred people out of the ten thousand.

One hundred is a smaller number than one hundred and fifty-six, so on this arithmetic the specialist is still rarer. Push the stack to four skills and it drops to thirty-nine, comfortably rarer than the specialist. Push the specialist to one in a thousand and they are rarer again.

The point is not that one always beats the other. It is that rarity is multiplicative across independent dimensions and only additive within one, and multiplication gets there faster from a much lower starting effort.

Predict

The multiplication above assumes the three skills are independent. In practice they usually are not. What does that do to the estimate?

Choose one answer

Nothing much — correlation is a second-order effectIt makes the combination less rare than one in sixty-fourIt makes the combination rarer, because the skills compete for practice timeIt depends on the field and cannot be signed

Lock in answer

The condition everyone skips

Rarity is necessary and it is nowhere near sufficient.

There are combinations of three skills that nobody on earth possesses and that nobody would pay a currency unit for. Rarity only becomes a wage when somebody needs the combination and cannot easily get it, which is the demand factor and the scarcity factor from two lessons ago, and both of them have to hold at once.

The test is unpleasantly concrete: is there a role, a client or a problem where having all three at once solves something that having any two does not? If the answer is that the third skill is merely nice to have, the stack is a hobby with a business justification attached.

What the payoff looks like over time

A stack is not a raise. It is a change in the rate at which what you can charge moves, and the difference between those two things is the whole reason this is in a track about arithmetic.

The calculator below is a general compounding engine. Read the starting balance as what a year of your work is priced at today, the annual rate as the rate at which that price rises, and the final balance as what a year of your work is priced at fifteen years later. Nothing is being saved here — the amount added each year is set to zero, because this is a level compounding, not a pot filling.

Compound growth

≈ Hypothetical model, not a forecast

Grows a starting balance plus a fixed yearly contribution at a constant rate.

It assumes

• The rate at which a price rises is treated as constant, which no real career has ever been.
• Nothing is added or saved here — this compounds a level rather than accumulating a pot.
• Every figure is nominal, so rising prices generally have not been taken off.
• It assumes the work continues to be wanted for the whole fifteen years, which is the subject of the next lesson.

It ignores

• Tax on growth, income or withdrawal
• Platform fees, spreads and trading costs
• Inflation — every figure shown here is nominal
• Any year in which the return happens to be negative

Find the annual rate at which the price of a year of your work doubles inside fifteen years. Note how small that rate is, and then note how large the gap is against a rate two points lower.

Annual return300%

Final balance

62,319

Total paid in

40,000

Growth

22,319

Growth as a share of the final balance

0.358

• Price of a year of work

◷ Not there yet — keep moving the controls.

The rate that doubles a level in fifteen years is under five percent a year, which is a small enough number that no single year would feel like anything was happening. That is the property worth taking away: the payoff from a stack is invisible annually and large cumulatively, which makes it structurally difficult to notice while it is working and impossible to attribute afterwards.

Check

Two people are both in the top quarter at three things. One is paid far more. Which explanation is most consistent with everything in this lesson?

Choose one answer

Their combination removes a handoff that somebody is currently paying forTheir three skills are less correlated, so the combination is rarerThey are actually in the top few percent at one of the threeThey are better at negotiating

Check answer

Where the specialist wins

Nothing above argues against depth, and there are two situations where the specialist route is clearly stronger.

The first is any field with extreme scale at the top. Where the best performer's output is distributed to everyone — a musician, a researcher whose result is copied everywhere, an athlete — the reward for being first is not slightly better than being fiftieth, it is many multiples better. Rarity of the ordinary kind does not compete with that.

The second is any field where competence has a hard floor set by consequence. Nobody wants a surgeon who is in the top quarter at surgery and also good at two other things. The impact factor dominates, and the market for adequate is very small.

The stack argument applies most strongly in the wide middle, where being extremely good is not required, several capabilities have to meet, and the meeting is currently done by a handoff between people.

›Why the third skill is the expensive oneOptional

Getting into the top quarter at anything takes a while. Getting into the top quarter at a second thing takes about as long, and by then several years have passed and the first skill has been earning nothing extra the whole time.

That is a real cost and it is why most stacks stop at two. The third skill is paid for out of a period when the second one was already producing an income, which makes the opportunity cost of learning it much higher than the first two.

It is also why the stacks that appear in practice are often assembled by accident — a career change, a redundancy, an unusual first job — rather than by design. Almost nobody spends four deliberate years on a third capability whose payoff is a growth rate they cannot observe.

Worth remembering

• Rarity multiplies across independent skills and only adds within one, so combinations get rare faster than specialisms deepen.
• Assuming three skills are independent overstates the rarity, because related skills correlate positively.
• A rare combination pays only where someone needs all of it at once; otherwise it is a collection rather than a stack.
• Stacks typically pay by removing a handoff, and the value captured is roughly the size of the loss at that boundary.
• The payoff arrives as a change in a growth rate, which is invisible in any single year and large over fifteen.