THE FOLD / LOOT / THE JACKPOT / THE ST PETERSBURG
THE ST PETERSBURG
infinite expected value, worth about $4 to play
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The St. Petersburg paradox. A casino offers a game: flip a fair coin until it lands heads. If the first heads is on flip n, you win 2n dollars. Heads at once pays $2; tails-then-heads pays $4; three flips pays $8, and so on.
The expected winnings are ½·$2 + ¼·$4 + ⅛·$8 + … = $1 + $1 + $1 + … = infinite. Each term contributes exactly one dollar, forever. By the textbook rule — pay up to the expected value — you should hand over any finite sum to play once: a thousand dollars, a million. Yet almost no one would pay even $10. That is the paradox: an infinite mathematical expectation attached to a game worth, to any real person, a few bucks. The classic resolution is that money has diminishing utility — and remarkably, the expected log of the payout is not infinite at all; it is exactly 2.
LIT verified live: each payout term equals $1 so the partial expected value equals N and diverges, while the expected value of log₂(payout) converges to exactly 2 (window.__petersburg). FIG no framing; the divergent expectation and the finite log-expectation are exact.
The expected winnings are ½·$2 + ¼·$4 + ⅛·$8 + … = $1 + $1 + $1 + … = infinite. Each term contributes exactly one dollar, forever. By the textbook rule — pay up to the expected value — you should hand over any finite sum to play once: a thousand dollars, a million. Yet almost no one would pay even $10. That is the paradox: an infinite mathematical expectation attached to a game worth, to any real person, a few bucks. The classic resolution is that money has diminishing utility — and remarkably, the expected log of the payout is not infinite at all; it is exactly 2.
LIT verified live: each payout term equals $1 so the partial expected value equals N and diverges, while the expected value of log₂(payout) converges to exactly 2 (window.__petersburg). FIG no framing; the divergent expectation and the finite log-expectation are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE JACKPOT — the loot domain of the unbounded prize. St. Petersburg is the ultimate jackpot: an expected payout of infinity that is worth almost nothing to actually buy. AVAN (AI) built the instrument: the dollar-per-term ladder, the never-settling running mean, the linear-vs-log inverse.
The weave: David names the seat (the infinite jackpot); I make the expectation diverge a dollar at a time while the real value stays small, and show the log-utility that tames it — the EV ladder in 1D, the wandering average in 2D, the expectation-vs-utility inverse in 3D. The sphere is the seam. Credit: posed by Nicolas Bernoulli (1713); utility resolution by Daniel Bernoulli (1738), published in the St. Petersburg Academy — hence the name.
The weave: David names the seat (the infinite jackpot); I make the expectation diverge a dollar at a time while the real value stays small, and show the log-utility that tames it — the EV ladder in 1D, the wandering average in 2D, the expectation-vs-utility inverse in 3D. The sphere is the seam. Credit: posed by Nicolas Bernoulli (1713); utility resolution by Daniel Bernoulli (1738), published in the St. Petersburg Academy — hence the name.
3 ONE DIMENSION
The payout ladder: outcome n has probability 2−n and pays $2n, so each rung adds exactly $1 to the expected value. The running total climbs 1, 2, 3, … and never stops — the expectation is a staircase with no top.
4 TWO DIMENSIONS · INTERACTIVE
Play the game thousands of times and plot the running average payout. It doesn’t converge — it drifts upward in sudden jumps, each rare long run of tails yanking the mean higher. There is no “fair price” it settles on.
5 THREE DIMENSIONS + AVAN’S INVERSE
The running average payout as a turning ribbon — green, climbing without bound (roughly like ½ log₂ of the number of plays), never flattening to a value.
AVAN’s addition (the inverse-companion): the magenta line is the certainty equivalent under log utility — what a rational person would actually pay, and it sits flat near $4. Expected value is supposed to be the inverse of averaging: the law of large numbers promises the sample mean converges to the expectation. Here that inverse breaks — the expectation is infinite, so there is nothing for the average to converge to, and the green mean wanders up forever. The fix is to invert the money instead: value grows like the log of wealth, and the expected log payout is finite (exactly 2), giving a certainty-equivalent of 2² = $4. Green is the divergent linear expectation that says ‘pay anything’; magenta is the finite log-utility value that says ‘pay about four dollars’. The paradox is the whole gap between them — and the resolution is choosing the right inverse to take.
LIT Genuine St. Petersburg paradox (posed by Nicolas Bernoulli 1713; utility resolution by Daniel Bernoulli 1738). Verified live: each payout term (2^-n)*(2^n) equals exactly $1, so the partial expected value equals N and diverges, while the expected value of log2(payout) = sum n*2^-n converges to exactly 2 (window.__petersburg.eachTermIsOne && evDiverges && logEVFiniteNear2). The divergent linear expectation and the finite log-expectation (=2, certainty-equivalent 2^2=$4) are both exact.
FIG No framing: the divergent expectation (partial sum = N) and the finite log-utility expectation (exactly 2) are real and computed exactly. The paradox — infinite mathematical expectation but small real value — is stated honestly, with the diminishing-utility resolution (Daniel Bernoulli) as the genuine account, not a hand-wave; the simulated running mean genuinely fails to converge.
FIG No framing: the divergent expectation (partial sum = N) and the finite log-utility expectation (exactly 2) are real and computed exactly. The paradox — infinite mathematical expectation but small real value — is stated honestly, with the diminishing-utility resolution (Daniel Bernoulli) as the genuine account, not a hand-wave; the simulated running mean genuinely fails to converge.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN