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Why positive expectations, hard work, and clever optimisation aren’t enough.
Imagine I offer you a game. Every round, an independent toss of a fair coin determines what happens to your money. Heads: you gain 60%. Tails: you lose 50%.
A quick calculation gives an expected return of 5% per round. A positive expected return! Some people would call that a good investment.
Let’s play twice. Start with €100. Win the first round and you’re at €160. Lose the second and you’re at €80. Lose first, then win, and you’re still at €80.
The order doesn’t rescue you. A win and a loss leave you 20% poorer.
Over repeated independent rounds, the expected wealth across many possible trajectories grows by a factor of 1.05 per round. Yet the typical compounded trajectory heads in the other direction. Its long-run growth factor per round is the geometric mean:
√(1.60 × 0.50) ≈ 0.894
The expected log-growth rate is 0.5 ln(1.60) + 0.5 ln(0.50) ≈ −0.1116 per round, equivalent to a long-run compounded loss of about 10.6% per round. This is why a strategy can look attractive on an expectation spreadsheet while behaving terribly for someone who has to keep playing it.
“Avoid ruin above all. The average doesn’t happen to you.”
It’s a useful aphorism, not a literal statement that averages never describe individual experience. The underlying distinction is between averages over an ensemble of possible outcomes and growth along a sequence of multiplicative outcomes. Those need not agree. The mathematical issues are developed in Peters’ work on ergodicity economics.
The game, as described, never takes your balance to exactly zero in a finite number of rounds. So saying it guarantees bankruptcy would be wrong. But repeated full-stake play drives typical wealth toward zero, and real people have bills, collateral requirements, and thresholds below which a theoretical recovery is no longer useful.
That changes how I think about risk.
A profitable opportunity can still be a terrible position
The first question isn’t simply whether the payoff has a positive expectation. It’s how much to expose to that payoff, for how long, and what happens if the bad outcomes arrive early.
John Kelly approached a version of this problem in his 1956 paper on information and capital growth. Instead of maximising expected wealth after one round, his criterion maximises expected logarithmic growth for repeated bets under specified conditions.
For our 50/50 game, if I can bet any fraction f of my wealth, the two multipliers become 1 + 0.60f and 1 − 0.50f. Maximising their expected logarithm gives f = 1/6, or about 16.7% of wealth per round—not 100%.
That’s a rather different answer from the average payoff is positive, so take the bet.
Kelly sizing relies on knowing the outcome probabilities and payoffs, and on valuing long-run log growth. Those assumptions can fail spectacularly in real markets. But the underlying insight travels well: a good opportunity and a good position size are not the same thing.
When the consequences are irreversible, preserving the ability to try again may matter more than maximizing what looks best on average.
Yesterday’s bill is not tomorrow’s reason
Suppose you’ve spent six months building a product. You’re tired, the market has moved, and the evidence suggests a different direction would be better.
You can hear the internal argument already: But I’ve put six months into this.
Unfortunately, the calendar isn’t impressed.
“Sunk costs aren’t costs. They’re history.” As a slogan, that overstates it: sunk costs were real costs when you paid them. The precise decision-theoretic claim is narrower. If a past expense cannot be recovered, and is identical across the options available today, it should not by itself influence which option you choose now.
The time spent may have taught you something about the market, your competence, or the product’s prospects. That information can legitimately affect your forecast of future costs and benefits. What cannot justify continuing is the bare fact that you already paid.
Try a thought experiment. A stranger offers you the exact same project, in its present state, with no history attached. Would you commit your next month to it?
If the answer is no, your previous effort is not an argument for saying yes.
This is a normative principle of rational choice under a specified objective, not a psychological theorem proving what people actually do. The interesting psychology is why we’re so tempted to throw more time after time we’ve already spent.
Doing more at once is often doing things more slowly
Founders and engineers have another favourite trick: start more work when work isn’t moving quickly enough.
Five features are in progress, so we launch another three. Three clients are waiting, so we open ten more conversations. Everyone is busy; surely progress will follow.
A queueing result called Little’s law offers a useful reality check:
Average work in progress = Throughput × Average time in system
Or, in the usual notation, L = λW.
Imagine a team that completes five tasks per week and has twenty tasks in its system on average. Under stable operating conditions, a task spends an average of four weeks in that system.
Now suppose the team starts enough additional work to sustain forty tasks in progress, but throughput stays at five per week. Average time in the system rises to eight weeks.
That’s not a motivational speech against multitasking. It’s a relationship among averages in a stable system, proved by John Little in 1961. Adding work might improve throughput in an underutilised system, so the law doesn’t prove that starting more tasks always makes everything slower. It does prove that, at unchanged throughput, higher average work in progress means higher average time in the system.
For a team drowning in half-finished projects, this is a helpful distinction. The goal isn’t to look maximally occupied. It’s to improve the flow of completed work.
And once you start asking what really limits a system, a more philosophical question appears: why should the optimisation method you happen to prefer be the right one for this problem?
There is no magic algorithm that wins everywhere
Machine learning makes it extraordinarily easy to become attached to a method. You see a successful model, copy the architecture, tune the parameters, and hope the success transfers.
But optimisation has an unfriendly result: “There’s no best method for every problem.”
The No Free Lunch theorems of Wolpert and Macready show that, for their black-box optimisation setting and the relevant uniform averaging over possible objective functions, improvements on some problem classes are offset by poorer performance on others. Without restricting what the world might look like, no search algorithm enjoys a universal advantage.
The assumptions are crucial. We do not encounter objective functions sampled uniformly from every mathematically possible function. Real-world problems have exploitable regularities, and algorithms can absolutely be better or worse for the distributions we care about.
In fact, that’s the point. Useful performance comes from assumptions, inductive biases, and domain knowledge that fit the problem.
The same is true of human decision strategies. A checklist that is excellent for a routine operational decision may be hopeless for navigating a new market. A strategy designed for repeated independent bets may be inappropriate for an irreversible personal choice.
There is no shame in having a favourite tool. The mistake is forgetting what makes it work.
Optimise for the life you’re actually living
These four ideas share a theme that took me longer than I’d like to admit to appreciate.
Expected value describes a distribution, not necessarily the life of someone moving through it. Sunk costs describe what has happened, not what should happen next. A large backlog describes activity, not necessarily progress. And an excellent algorithm is excellent only relative to a class of problems.
Each asks us to identify the actual objective, the actual constraints, and the consequences of getting the assumptions wrong.
An impressive average is not enough if your path to it is one you cannot survive. Neither is a busy calendar, a past investment, or a fashionable method.
Practical wisdom is often less about finding a number to maximise than about noticing what must remain possible after you’ve made your choice.