Indexing the archive…
Your Universe of Digital Possibilities
Vega’s booth sells one ticket: a coin flips until tails, and the pot doubles on every head — 2, 4, 8, 16… Its expected value is infinite. You name the price you would pay, and whatever you name, you are wrong twice: too high for your ledger, too low for the mathematics.
A ticket that pays 2k with probability 2−k adds one coin to its expectation for every depth, forever. No single price is fair; Feller showed that n tickets together are fair at about n·log₂ n, so the running mean climbs like log₂ n with no ceiling — the more you buy, the more each is worth.
Value the payout by the logarithm of the wealth it leaves, and the row of ones becomes a row that shrinks geometrically and sums: at wealth 100 the ticket is worth 7.89 coins — The Wager’s growth-optimal idea, applied to one ticket.
Here the band is the exact distribution of 100 tickets’ total payout, convolved and truncated with its overflow carried, shifted by the prices you fixed before each flip.
Expected value is not a price. The ticket is worth infinity on average and single digits to any ledger that has to survive it, and both numbers are exact. Bernoulli answered with utility — the same 1738 idea The Wager turns into Kelly’s rule and The Ensemble into time against the crowd; Feller answered with a price that depends on how many you buy. The Martingale next door changed the shape and kept the mean; this game keeps the shape and loses the mean. And The Envelope’s vault, uncapped, is this booth.