Indexing the archive…
Your Universe of Digital Possibilities
Multiply your stake by a random factor, repeat — W ← r·W is the whole law. Ask the crowd how it went and the answer is the arithmetic mean, which compounds at ln⟨r⟩ and climbs. Ask any one member and the answer compounds at ⟨ln r⟩, which is always less — and for the coin on this stage, negative. The average of everyone is the biography of no one; the law never said who was averaging.
Multiply your stake by a random factor, repeat — the whole law. No strategy, no memory, no choice. The line never says who takes the average of the outcomes — over copies of you, or over your own tosses — and that silence is worth a sign flip.
The crowd’s average compounds the arithmetic mean of the factor; any one life compounds the geometric mean, which is always smaller. When ⟨r⟩ > 1 > e⟨ln r⟩ the same coin enriches the ensemble and ruins almost every member of it — both verdicts exact, both forever.
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 skew made precise: the mean rides above the median by eσ²t/2. A symmetric shock to returns is an asymmetric shock to price — upside unbounded, downside floored at zero.
This is the first half of a pair about what one line of law does not say — here, who is averaging. Over copies, the coin compounds ln⟨r⟩ and the verdict is growth; over one life’s tosses it compounds ⟨ln r⟩ and the verdict is ruin; the gap is the −σ²/2 volatility drag that The Oracle’s geometric Brownian motion carries in its exponent, and the reason The Wager’s Kelly rule maximises log-wealth rather than wealth. Bernoulli met the same wedge in 1738 — the St. Petersburg game’s infinite mean that no one will pay twenty coins for — and reached for utility; Peters’ 2019 rereading needs no psychology at all, only the honest average. The Ledger watches the same multiplicative arithmetic concentrate a whole economy; and the other half of this pair, The Ratchet, plays the wedge in reverse — two averages that lie downward, alternated into a win.