Indexing the archive…
Your Universe of Digital Possibilities
Two games of chance, each rigged a hair against you. Played alone, either one bleeds your capital at a rate a three-state Markov chain predicts to four decimals. Played in alternation — by coin toss, or in the rhythm A·B·B·A·B — the same two games climb. Nothing is hidden: the win is a flashing Brownian ratchet in accounting clothes, and the averages that promised you would lose were both telling the truth — about the games the alternation never plays.
Game A is a plain biased coin; game B picks its coin by whether your capital is divisible by 3 — a vicious coin on multiples of three, a generous one elsewhere. Played alone, each drifts your capital down. The law never mentions alternation.
The drift per round, exactly: π₀ is how often the capital sits on a multiple of 3 — the stationary state of a three-state Markov chain. Game B digs its own trap (π₀ high, the bad coin often); mixing in A shuffles π₀ down, and the same formula changes sign.
Overdamped particles in a sawtooth potential that switches on and off, s(t) ∈ {0, 1}. On: they slide into the teeth. Off: they spread as pure diffusion. On again: the asymmetry harvests the spread — net transport uphill against the load F, paid for entirely by the switching.
This is the second half of a pair about what one line of law does not say — here, which average you consulted. Each game’s losing rate is a true time average — of that game played forever. Alternation plays neither game forever: game A re-randomises the capital that game B keeps parking on multiples of 3, so B’s vicious coin comes up less often than its own stationary maths assumed, and π₀ drops far enough to flip the drift’s sign. Parrondo built the games in 1996 as a discrete flashing ratchet — Ajdari & Prost’s 1992 machine, where switching a sawtooth on and off pumps The Walk’s diffusion uphill. Smoluchowski (1912) and Feynman (1963) had already ruled the unswitched version out: in one heat bath a passive pawl rectifies nothing — the win is paid for by the switching, exactly as The Demonpays Landauer’s bill for its sorting. And the twin of this instrument, The Ensemble, runs the same silence the other way — averages that lie upward instead.