Losing Most Bets, Winning the War

Last lesson gave you the two shapes: the convex smile (capped loss, open-ended win) and the concave frown (capped win, open-ended loss). The idea was elegant. This lesson makes it arithmetic. We’re going to take one convex bet and one concave bet, run each a hundred times, and add up every dollar on the way. No hand-waving, no “trust me, it works out.” A ledger.

And the punchline will feel wrong the first time you see it: the convex bet is going to lose four times out of five and finish the year up $120**, while the concave bet **wins nine times out of ten** and finishes down **$110. The winner of most battles loses the war. By the time you’ve counted the last dollar, “how often did I win?” will feel like the amateur’s question it is.

The convex bet, in full arithmetic

Here’s the analogy to hold onto: a convex bet is a venture portfolio in your pocket. Most of the little companies you back go to zero and cost you your small stake; one of them is a rocket. You don’t need to be right often. You need to survive being wrong until the rocket shows up.

Let’s make it concrete and countable. The bet:

• Lose $1 on 80% of tries.
• Win $10 on the other 20%.

Run it 100 times and lay out the ledger. Nothing hidden — every trial is in one of two rows:

Read that bottom-right cell and let it sting a little: +$120, earned by someone who was wrong 80 times out of 100. The 20 wins didn’t just cover the 80 losses; they buried them. The rare event was ten times the size of the common one, and that size did all the work.

We can compress the whole ledger into a single number — the expected value (EV), the average dollars per trial. Weight each outcome by how often it happens:

$\text{EV} = (0.20 \times \$10) + (0.80 \times -\$1) = \$2.00 - \$0.80 = +\$1.20$

That **+$1.20 per trial** is the engine. Multiply by 100 trials and you get the +$120 we counted by hand — the same number two ways, which is how you know the arithmetic is honest. The per-trial EV is positive, so the more you play, the further ahead you drift, even though most individual plays lose.

How many wins can you afford to miss?

Here’s the part that shows how much cushion a convex bet really has. We assumed a 20% win-rate — but what if you’re worse than that? What if the rocket is rarer than you hoped? Let w be your true win-rate. Each win pays $10 and each of the `1 − w` losses costs$1, so per-trial EV is:

$\text{EV} = (w \times \$10) + \big((1 - w) \times -\$1\big) = 11w - 1$

Now walk the win-rate down and watch the bet stay alive:

The bet is still profitable at a 15% win-rate. Still profitable at 12%. Still just barely ahead at 10%. It only turns negative when your win-rate falls below the break-even point, which we can solve for exactly by setting EV to zero:

$11w - 1 = 0 \quad\Rightarrow\quad w = \tfrac{1}{11} \approx 0.091 = 9.1\%$

So this bet keeps making money as long as you win more than about 1 time in 11. You could be wrong ninety percent of the time and still not lose. That enormous margin between “how often you actually win” (20%) and “how often you need to win” (9.1%) is the convex bet’s buffer — and it exists entirely because the win is 10× the loss.

When to use it

Reach for this arithmetic the moment someone dismisses an idea because it “usually fails.” Cheap experiments, early-stage bets, cold outreach, planting many seeds, submitting to long-shot opportunities — anything where the cost of a miss is small and fixed but the payoff of a hit is large and open-ended. Before you judge it by its hit-rate, compute the break-even win-rate. If your real odds clear that bar even with room to spare, the losing streak is a feature, not a bug.

Why the losing streak is brutal — and why to ignore it

There’s a catch the ledger hides, and it’s psychological, not mathematical. The +$120 is guaranteed in the total — but the total arrives in an ugly order. A 20% win-rate means that, on average, four out of every five tries hurt. You will sit through long strings of pure losses waiting for a win that pays for all of them.

Run the numbers on the streaks. At a 20% win-rate, the chance of losing several in a row is not small:

Ten straight losses with no win in sight will happen to you roughly one stretch in nine, purely by chance, on a bet that is mathematically excellent. This is where the money is actually lost — not in the arithmetic, but in the human sitting on top of it.

Two forces conspire to make you quit at exactly the wrong moment. The first is loss aversion — the well-documented fact that a loss stings roughly twice as hard as an equal gain feels good. Ten small $1 losses in a row don't feel like −$10; they feel like a verdict: this doesn’t work, I’m an idiot, stop the bleeding. The second is plain impatience: the win is rare by design, so the reward for your discipline is always somewhere off in the future, never in the row you’re staring at right now.

The learner who understands the arithmetic but not the psychology abandons the convex bet three losses before the win that would have paid for everything — and then, salt in the wound, watches someone with more patience collect it.

So what’s the discipline? Size each loss so small that the streak can’t hurt you. This is where the model you already learned — margin of safety — walks straight into asymmetry. The entire reason we set the loss at **$1** and not$50 is so that a brutal run of ten losses costs a survivable **$10**, not a ruinous$500. Cap the per-trial downside low enough and the losing streak becomes an inconvenience you can outlast rather than a knockout blow. The margin of safety is what keeps you at the table long enough for the convexity to pay off.

The concave mirror: pennies in front of a steamroller

Now flip every sign and watch the horror show. If the convex bet was a venture portfolio, the concave bet is writing insurance against a disaster that hasn’t happened yet — you collect a comfortable premium almost every day, and one day you pay a claim that eats a decade of premiums in an afternoon.

The mirror-image bet:

• Win $1 on 90% of tries.
• Lose $20 on the other 10%.

Look at that win-rate first, because it’s the whole trap: 90%. Nine times out of ten you win. It feels like a machine that prints money. Now run the ledger over 100 trials and watch the machine reveal itself:

There it is: a **90% win-rate that loses you $110**. The ten rare losses, at$20 apiece, wiped out the ninety comfortable wins and then took another $110 on top. As a single per-trial number:

$\text{EV} = (0.90 \times \$1) + (0.10 \times -\$20) = \$0.90 - \$2.00 = -\$1.10$

A per-trial EV of **−$1.10**. Every single play, on average, costs you$1.10 — and yet nine plays out of ten hand you a dollar and a smile. This is Nassim Taleb’s image made literal: picking up pennies in front of a steamroller. The pennies are real and frequent (the +$1 wins). The steamroller is real and rare (the −$20 losses). You bend down, collect a penny, and it feels like free money — right up until it isn’t.

Set the two bets side by side and the symmetry is almost cruel:

The bet that felt awful was the one making money. The bet that felt incredible was the one bleeding you dry. Comfort and correctness pointed in opposite directions — which is exactly why win-rate is such a treacherous scoreboard.

Sort the shapes: convex or concave?

Every real decision hides one of these two shapes. Here are six from the wild — drop each into the bin that matches its payoff. Ask the diagnostic question each time: when I’m wrong, is the loss capped and small, or open-ended and huge?

Why fat tails make the steamroller inevitable

You might think the concave bet is fine as long as you’re careful — just avoid the big loss, right? Wrong, and this is where the fat-tails course you already took comes home to collect. Two separate forces, both from that course, make the steamroller not just possible but inevitable over enough trials.

Force one: the rare loss is bigger and more common than a bell curve says. In a thin-tailed, bell-curve world, extreme events are astronomically rare — a “20-sigma” loss essentially never happens. But almost nothing that matters lives in that world. Markets, wars, pandemics, viral hits, cascading failures — all fat-tailed, where the extreme is far more frequent and far larger than the bell curve predicts. So the concave bettor who modeled their worst case as “−$20, once in a while" is using the wrong ruler. In a fat-tailed world the true worst case might be −$200, or −$2,000, and it arrives sooner than the tidy 10% suggested. The downside of a concave bet is systematically underestimated precisely because our instincts are calibrated to a bell curve the world doesn’t obey.

Force two: over enough trials, the rare event becomes near-certain. Suppose a ruinous loss has just a p chance on any single trial. The chance you avoid it for n straight trials is (1 − p)ⁿ, so the chance it eventually hits is:

$P(\text{ruin by trial } n) = 1 - (1 - p)^n$

Feed this a small p and watch the ruin creep up as the trials pile on. Take a 2% chance of a ruinous loss per trial:

A 2% risk sounds ignorable. Play the bet 100 times and there’s an 87% chance the ruinous event has already landed. Keep playing and it becomes a near-certainty. This is the cruel arithmetic under every “it’s never happened before” strategy: rare events aren’t safe, they’re just early. Given enough trials, the tail always comes.

Now put the two courses together. The concave bettor’s long green winning streak isn’t evidence of safety — it’s the tail loading. Each comfortable +$1 win is one more trial on the clock, one more roll of the dice against that p, quietly walking them toward the loss that erases everything. The turkey fed every morning for a thousand days grows more confident in the farmer right up to the afternoon before Thanksgiving. A smooth track record on a concave bet is not the absence of the steamroller. It’s the sound of it approaching.

And here is the whole course in one contrast. The convex bettor also faces fat tails — but for them the fat tail is on the upside, and their downside is capped by design so no single trial can end the game. Fat tails are a gift to the smile and a death sentence to the frown. That is why you engineer convexity: cap your downside so tightly that the worst trial is survivable, and leave your upside open so the rare giant win can find you.

Recap

Next up

You’ve now watched, in full arithmetic, the smile win the war while losing most of its battles, and the frown lose everything while winning nearly all of its. The takeaway is a habit: cap the downside, keep the upside open, and count in expected value, not win-rate. But so far we’ve only found convex bets in the wild. The real power move is to build them on purpose. Next up: Optionality: The Right Without the Obligation — how to pay a small, known cost for the right but not the obligation to a big upside, manufacturing the smile shape instead of waiting to stumble on it.