The Edge Was Never the Hard Part: Bet Sizing, the Kelly
Criterion, and Survival on a Favorable Coin

In one line: 61 finance-trained people got a coin rigged in their favor. 28 percent still went broke by betting too much.

Abstract. I revisit a controlled experiment in which sixty-one finance-trained subjects received a coin biased to land heads 60% of the time, thirty minutes, and a $25 stake, with winnings capped at $250. Despite a disclosed and guaranteed edge, 28% went bankrupt, about a third finished below their starting stake, and only one in five reached the cap; the average payout was $91. The failures trace to bet sizing rather than to the edge. For a 60/40 coin the growth-optimal Kelly fraction is 20%; long-run growth is single-peaked in bet size, asymmetric about the peak, and crosses zero near 39%, so a constant 40% stake on a favorable coin compounds negatively. Three sizing rules follow: size from the stressed edge estimate, hold the gates fixed, and treat survival as the objective.

1. Introduction

In 2016, sixty-one finance-trained subjects were each given a coin that lands heads 60% of the time, thirty minutes, and twenty-five dollars, under a single instruction: bet on flips and keep the proceeds, capped at $250. The game is constructed to favor the player.

Twenty-eight percent went completely broke. About a third finished below their starting stake. Only one in five reached the cap. The average payout was $91. The subjects were handed the edge explicitly, with the win probability disclosed; the one variable left in their control was how much to bet, and it was bet size that ruined them (Haghani and Dewey, 2016).

2. The growth-optimal bet fraction

2.1 The Kelly fraction

The relevant result dates to 1956, when Kelly framed a gambler with an edge as a communication channel carrying signal. For an even-money bet, the growth-optimal fraction of the bankroll equals twice the win probability minus one. For a 60/40 coin that fraction is 20%. Long-run growth as a function of bet size traces a single-peaked curve: zero at a zero stake, a maximum near 20% (about 2% compounded per flip), and a second zero near 39%. A constant 40% stake on the same guaranteed-winning coin produces negative growth of roughly a quarter percent per flip. The edge is unchanged; only the size is wrong.

2.2 Asymmetry and volatility drag

The curve is asymmetric. Betting half-Kelly, 10% rather than 20%, retains roughly three-quarters of the maximum growth rate for about half the swings. Underbetting is inexpensive; overbetting is ruinous. The distinction between mean and median outcome is large. Over 300 flips at full Kelly on a $25 start, the average outcome exceeds $3 million while the median is about $10,000. Only one path is realized, and it is the logarithm of wealth that compounds along it. The gap between mean and median is volatility drag, the same mechanism by which a series with a positive average return converts $100k into $93k.

3. Sizing doctrine

Three house rules descend from the shape of the curve.

1. Kelly is highly sensitive to the edge estimate. Overstating the edge moves the chosen stake past the peak onto the downslope. The sizing doctrine therefore begins one step earlier: the edge entering any sizing calculation is the stressed estimate, gross return halved and costs doubled. The raw backtest never enters. A margin of safety on the input is worth more than precision in the formula.
2. The asymmetry is why the gates never loosen. A false survivor promoted by a loose gate does more than waste capital; it invites Kelly-scaled bets on an edge of zero, which the curve identifies as negative growth.
3. Survival is the strategy. Returns compound multiplicatively, and a single zero anywhere in the chain collapses the product. The drawdown rules and pre-committed ladders exist because there is no recovery from zero, whatever edge remains.

4. Provenance

One of the experiment’s authors was a founding partner of Long-Term Capital Management, the canonical sizing catastrophe. The cleanest demonstration of the sizing lesson was designed by someone who had lived its most expensive version.