A guaranteed edge, and a third lost money
Haghani and Dewey gave subjects trained in finance a coin they were told was biased in their favour, a fixed stake, a clock and a cap. The win probability was disclosed. The one variable left in the players' hands was how much to stake per flip, and that variable ruined them.
| Measure | Value |
|---|---|
| Subjects, trained in finance | 61 |
| Coin, probability of heads | 60% |
| Time allowed | Thirty minutes |
| Starting stake | $25 |
| Cap on winnings | $250 |
| Went bankrupt | 28% |
| Finished below the starting stake | About a third |
| Reached the cap | One in five |
| Average payout | $91 |
The growth curve
Kelly (1956) treated a gambler with an edge as a channel carrying signal, and derived the stake that maximises the growth rate of a bankroll. At even money it is twice the win probability minus one. Growth against bet size is a curve with a single peak, zero at a zero stake and zero again further out, past which a coin still guaranteed to win compounds negatively. The edge is identical along that whole curve. Only the size changes.
| Bet fraction | Growth per flip in the long run |
|---|---|
| 0% | 0 |
| 10% (half-Kelly) | ≈ ¾ of maximum |
| 20% (Kelly) | ≈ +2% (maximum) |
| ≈ 39% | 0 |
| 40% | ≈ −0.25% |
The curve is not symmetric about its peak. Halving the stake gives up little growth and buys roughly half the swings, so underbetting is cheap and overbetting is ruinous. The average is also the wrong thing to plan around. The mean sits far above the median, only one path is realised, and along it the logarithm of wealth compounds. That gap is volatility drag.
| Measure | Value |
|---|---|
| Run at full Kelly from a $25 start | 300 flips |
| Average outcome | Exceeds $3 million |
| Median outcome | About $10,000 |
| Drag, worked illustration | $100k becomes $93k |
Sizing doctrine
Three rules for the pipeline behind this site follow from that shape.
- Size off the stressed edge. Kelly is acutely sensitive to the edge estimate, and overstating it moves the stake past the peak onto the downslope. Sizing here takes the stressed estimate, gross return halved and costs doubled. The raw backtest, the strategy's simulated result on historical data, never enters.
- The hurdles never loosen. A strategy with no real edge that a loosened hurdle promotes invites a stake sized by Kelly on an edge of zero, which the curve prices as negative growth.
- Survival is the strategy. Returns compound multiplicatively and one zero collapses the product. Drawdown limits and pre-committed ladders, size cuts fixed in advance, exist because nothing recovers from zero.
One of the experiment's authors was a founding partner of Long-Term Capital Management, the canonical sizing catastrophe. The cleanest demonstration of the lesson was designed by a man who had lived its most expensive version.
References
- Haghani, V., and Dewey, R. (2016). Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin. SSRN Working Paper.
- Kelly, J. L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal, 35(4), 917–926.