Indexing the archive…
Your Universe of Digital Possibilities
You have a real edge on a repeated bet — a probability and a payoff that favour you on average. The only question left is how much of the bankroll to stake each round. Too little wastes the edge; too much lets an ordinary run of losses compound you toward zero even though the game is in your favour. The Kelly fraction is the exact answer: the one stake that grows wealth fastest, no more and no less.
Bet the fraction f* of your bankroll set by your edge over the odds — win probability p, loss q = 1 − p, payoff b to 1. No edge, no bet; a big edge at short odds, bet big. This one fraction maximises long-run growth.
The exponential rate your wealth compounds at when you stake fraction f. It is a hill: zero at f = 0, peaking at f*, and back to zero at the no-growth point f = 2f* — past which a favourable game loses money to volatility drag.
Bernoulli’s idea, made a law by Breiman: value log-wealth, not wealth. Maximising expected log-wealth is exactly maximising the geometric growth rate — and it almost surely beats every other strategy over the long run, while a bigger expected dollar return can march the median to zero.
The same criterion for a lognormal asset (The Oracle’s GBM): stake the excess return μ − r over the variance σ². Thorp carried this from blackjack to the market — the growth-optimal portfolio is the Sharpe ratio, squared and scaled.
Bernoulli’s 1738 resolution of the St. Petersburg paradox already knew wealth should be judged by its logarithm, not its face value — utility diminishes as you have more. Kelly’s 1956 paper, written inside Bell Labs, reframed that log-wealth growth rate as a channel-capacity problem: bet size is a code, and the fraction that maximises long-run growth is exactly the fraction Shannon’s capacity picks out, which is why The Code(INST·20) and this instrument share a formula. Breiman (1961) proved Kelly betting is asymptotically optimal — almost surely beats any other strategy given enough rounds — and Thorp (1962) took it out of the journals and onto the blackjack table. The continuous-time limit is The Oracle’s (INST·24) geometric Brownian motion, where the optimal fraction collapses to the clean f* = μ/σ². The Ledger(INST·72) is this instrument’s population-scale mirror: wealth there, as here, is a conserved multiplicative quantity that a single bad rule can concentrate or destroy. And every growth-hill in this chart is the discrete cousin of the additive-vs-multiplicative contrast The Walk(INST·19) draws first.