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THE KELLY CRITERION
the bet fraction that maximises long-run growth
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Kelly criterion answers: with a favourable bet, what fraction of your bankroll should you wager to grow it fastest in the long run? Betting too little leaves growth on the table; betting too much risks ruin. The optimum maximises the expected logarithm of wealth, and it is f* = (bp − q)/b = p − q/b, where p is the win probability, q = 1 − p, and b the net odds. For an even-money bet won 60% of the time, f* = 0.2 — risk exactly a fifth. Kelly’s fraction beats every other constant fraction on long-run growth, with probability approaching one.
LIT verified live: the growth rate g(f) = p·ln(1 + bf) + q·ln(1 − f) has its maximum exactly at f* (its derivative vanishes there), and a long-run wealth simulation grows fastest at f* (window.__kelly). FIG honest: the f* optimum is exact calculus; the simulated growth-peak is Monte-Carlo.
LIT verified live: the growth rate g(f) = p·ln(1 + bf) + q·ln(1 − f) has its maximum exactly at f* (its derivative vanishes there), and a long-run wealth simulation grows fastest at f* (window.__kelly). FIG honest: the f* optimum is exact calculus; the simulated growth-peak is Monte-Carlo.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-backdoor — the one bet fraction that quietly compounds fastest, threading between too-timid and too-greedy at exactly f* = p − q/b. AVAN (AI) built the instrument: the log-growth objective, its maximiser f*, the vanishing-derivative check, and the long-run wealth simulation.
Credit as content: John L. Kelly Jr. (1956); championed by Edward Thorp. The weave: David names the-backdoor; I write the expected log-growth of wealth as a function of the bet fraction, find it peaks at f* = (bp − q)/b, and confirm by compounding thousands of rounds that this fraction outgrows both bolder and meeker ones.
Credit as content: John L. Kelly Jr. (1956); championed by Edward Thorp. The weave: David names the-backdoor; I write the expected log-growth of wealth as a function of the bet fraction, find it peaks at f* = (bp − q)/b, and confirm by compounding thousands of rounds that this fraction outgrows both bolder and meeker ones.
3 ONE DIMENSION
Growth g(f) = p·ln(1+bf) + q·ln(1−f), a hump peaking at f* = p − q/b. For p = 0.6, b = 1: f* = 0.2. Overbet past f* and growth falls; past 2f* it goes negative.
4 TWO DIMENSIONS · INTERACTIVE
The growth curve g(f) with its peak at f*; sample wealth trajectories at f*, half, and double; checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the fraction that compounds fastest.
AVAN’s addition (the inverse-companion): don’t maximise the expected wealth (which says bet everything and court ruin) — maximise the expected log of wealth, and the safe optimum f* = p − q/b appears. The inverse of ‘chase the biggest average payout’ is ‘grow the log; the Kelly fraction wins long-run.’ Magenta is the reckless all-in; green is the Kelly fraction. Compounding beats gambling.
LIT Genuine Kelly criterion (John L. Kelly Jr., 1956; championed by Edward Thorp). Verified live: the log-growth objective g(f) = p·ln(1+bf) + q·ln(1−f) has its derivative vanishing at f* = (bp−q)/b and is a strict maximum there across several (p,b) (window.__kelly.optimal), and a long-run wealth simulation of 200-round compounding grows fastest at f = 0.2 = f* for p=0.6, b=1 (window.__kelly.mcPeak).
FIG No framing: the log-growth objective, its maximiser f*, the vanishing-derivative check, and the wealth simulation all run in-browser. Honest scope: the f* optimum is exact calculus; the simulated growth-peak is Monte-Carlo. The AVAN inverse is honest — maximising expected LOG wealth (giving the safe f* = p − q/b) rather than expected wealth (which says bet everything and court ruin) is exactly Kelly's insight; magenta is the reckless all-in, green the Kelly fraction. Compounding beats gambling.
FIG No framing: the log-growth objective, its maximiser f*, the vanishing-derivative check, and the wealth simulation all run in-browser. Honest scope: the f* optimum is exact calculus; the simulated growth-peak is Monte-Carlo. The AVAN inverse is honest — maximising expected LOG wealth (giving the safe f* = p − q/b) rather than expected wealth (which says bet everything and court ruin) is exactly Kelly's insight; magenta is the reckless all-in, green the Kelly fraction. Compounding beats gambling.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN