The Crooked Coin

Here is the best offer you will ever get. I have a coin that lands heads 75% of the time, and I will let you flip it a hundred times. Before each flip you bet any amount you like. Heads, I match your bet; tails, I keep it. You start with $100,000.

How much do you bet?

Try it before reading on.

The expected-value trap

Every dollar you bet returns $1.50 in expectation, so if expected value is your only guide, you bet everything on every flip. Do that and you almost surely go broke. To survive a hundred all-in flips you have to win all hundred, and the odds of that are 0.75¹⁰⁰. Expected value still says bet everything, because the one universe where you win all hundred flips is rich enough to pull the average up. But you do not live in the average. You live on a single path, and the first tails on it wipes you out.

What keeps you alive is rational.

From amounts to fractions

A fixed bet has two problems. Bet a small amount and you are safe but slow: $10,000 a flip earns about $5,000 a flip on average, and since your winnings pile up on the side instead of being bet, the money grows in a straight line rather than compounding. Bet a large amount and a run of early tails can end the game before your edge has time to pay off. There is a deeper issue underneath both. A fixed dollar bet does not keep pace with your bankroll: $10,000 is a tenth of your money at the start, but only a hundredth once you reach a million. What you want to hold steady is not the number of dollars but the share of your bankroll at risk. Bet a fixed share and your wins compound on their own. So what fraction of your bankroll should you bet?

Kelly’s answer

Suppose you bet a fixed fraction f of your bankroll on every flip. After n flips with W wins and n − W losses, your bankroll is

$$B_{n} = 100{,}000 \times (1+f)^{W} \times (1-f)^{n-W}$$

Take the log of both sides and divide by n to get the growth rate per flip:

$$\frac{1}{n} \ln \frac{B_{n}}{100{,}000} = \frac{W}{n} \ln(1+f) + \left(1 - \frac{W}{n}\right) \ln(1-f)$$

Over many flips the fraction of wins W/n converges to the win probability p, and the losses to q = 1 − p. So the long-run growth rate is

$$g(f) = p \ln(1+f) + q \ln(1-f)$$

Maximize it by setting the derivative to zero:

$$g'(f) = \frac{p}{1+f} - \frac{q}{1-f} = 0$$

which rearranges to p(1 − f) = q(1 + f), and solves to

$$f^* = p - q$$

For this coin, f* = 0.75 − 0.25 = 0.5: bet half your bankroll, every flip. This is the Kelly criterion, from John Kelly at Bell Labs in 1956.

Underbet, overbet

Look again at the growth curve. It rises to a peak at the Kelly fraction, then falls, and past about 85% it drops below zero. The two sides of the peak are not the same, and the difference is survival.

You might think a bigger bet grows faster. It does on the luckiest paths, but not on the one you are likely to live: past Kelly the swings take back more than the bigger bet adds, so the typical result grows slower, not faster. It is the all-in trap in slow motion. Set the simulator to 90% and watch most runs spike, then collapse toward the floor. Underbetting is the gentler mistake, a little less growth for much smaller swings, which is why careful players sit at half of Kelly. So Kelly is the ceiling: bet less and you trade growth for safety, bet more and you get less growth and a real risk of ruin, the one loss you cannot undo. When unsure, bet less.

The humility discount

In this game you know the coin is 75%. In every real decision you estimate that number, and you estimate it with error. The shape of the curve tells you which way to lean. If your edge is smaller than you believe and you bet full Kelly on the imagined edge, you land on the steep side of the peak, where the damage is. If your edge is larger than you believe and you bet under, you give up a little growth. The mistakes are not symmetric, so the bet should not be either. This is why the people who live on this math, the card counters and the quants, mostly bet half-Kelly: a standing discount for the chance that their estimate of the edge is wrong.

Sizing is the other half

Expected value answers one question: is this a good bet? It says nothing about how much to put in, and the crooked coin shows the second question can matter more than the first. Every strategy in the simulator plays the same coin, and the outcomes run from billions to bankruptcy on sizing alone.

The same structure shows up wherever bets repeat and losses compound: sizing a position in a portfolio, setting a startup’s burn rate, staking a career on one company’s stock. Having an edge is only half of the decision. The other half is how much you put behind it, and whether you survive if it goes against you. An edge only turns into money if you are still in the game to collect it. Find the edge, then bet it in a size you can survive being wrong about.