Artificial Wasteland · Pattern

The Average That Never Arrives

A coin game with an infinite expected value — that no one in three centuries has been willing to pay much to play. The paradox isn't in the game. It's in the word average: this one game has three of them, and they disagree by everything between $2 and infinity.

Here is the whole game. A fair coin is tossed until it first comes up tails. The pot starts at $2. Every heads doubles it. The first tails ends the game and you take whatever is in the pot. So a tails on the first toss pays $2; heads-then-tails pays $4; two heads then tails, $8; and a first tails on toss k pays exactly 2^k — with probability 2^−k.

Now the question Nicolas Bernoulli asked in a letter in 1713, and his cousin Daniel answered in print from the St. Petersburg academy in 1738:¹ what is a fair price to play once? Multiply each prize by its chance and add:

01 — PLAY ITPlay the game, and watch the average refuse to settle

Flip by hand, or let it run thousands of times. Keep your eye on the average won per game — the one number that should, by every ordinary law of averages, settle down. It doesn't.

Play a few by hand and it looks tame: most games pay $2 or $4, you feel like you're losing. Then run it a hundred thousand times. The average per game climbs — past 10, past 15 — and it climbs alongside the pale-blue curve log₂(games), always a couple of dollars above it, never levelling off. Reset and run it again and the same thing happens at a different height, because somewhere in there a single game paid a colossal pot and dragged the whole average up with it. There is no number the average is heading toward. That is not a bug in the simulation. It is a theorem — and we'll name it below.

02 — THREE AVERAGESOne game, three answers, all exact

"The average" is not one thing. Here are the three that matter, drawn straight from the exact distribution — no sampling, no noise.

The median — $2. Half of all games end on the very first toss, so half of all games pay exactly $2 and no more. The typical game — the one in the middle — is the cheapest possible outcome. (87.5% of games pay $8 or less; 96.9% pay $32 or less.) By the standard of the median player, this game is nearly worthless.

The geometric mean — $4. Instead of averaging the prizes, average their logarithms — the right thing to do when quantities multiply, or when what you feel is proportional change, not absolute dollars. Then E[log₂ payout] = Σ k·2^−k = 2 exactly, so the geometric mean is 2² = $4. This is precisely Daniel Bernoulli's 1738 escape: he proposed that we value not money but its utility, which grows like a logarithm, and a logarithmic valuer prices this game at a small, finite figure.¹ Gabriel Cramér had reached almost the same place in 1728 with a square-root valuation (≈$2.9).²

The arithmetic mean — ∞. The one every textbook computes, and the one no one obeys. It is infinite not because any prize is infinite — every prize is a finite power of two — but because the tail of rare, enormous jackpots is just heavy enough that its contribution never converges. Which average is "right"? The paradox's lasting lesson is that the question has no answer in the abstract: it depends entirely on whether you play once, play with borrowed money, or play a great many times — and on who is holding the bank.

03 — THE FINITE HOUSEWho can actually pay infinity?

No real casino has infinite reserves. If the house can pay at most W dollars, the pot is capped there — and the fair price collapses from infinity to something you could carry in a wallet. Drag the bankroll and watch.

With the payout capped at 2^L — where L = ⌊log₂ W⌋ is how many doublings the bank can survive — each affordable round still contributes exactly $1, and the truncated tail adds one more, so the fair price is exactly L + 1. The consequence is the quiet dissolver of the paradox: the value grows like the logarithm of the bankroll. To make the game worth one more dollar, the house must double its entire fortune. A banker as rich as the whole world economy makes this "infinitely valuable" game worth about the price of a paperback.³

04 — THE AVERAGE THAT NEVER ARRIVESWhy the running total refused to settle

Back to instrument 01 — the average that kept climbing. Ordinary randomness obeys the law of large numbers: average enough coin flips, dice, or heights and the running average homes in on a fixed value. The St. Petersburg game breaks that law, because its true mean is infinite — there is no fixed value to home in on.

In 1945 William Feller found what happens instead.⁴ Add up the winnings S_n from n games. Then

So a "fair fee" exists only for a fixed session length: to play n games, pay about log₂ n per game. Play a thousand and roughly $10 each is fair; play a million and it's about $20; and as n grows the fair fee grows with it, without bound. The single-play fair price the Bernoullis hunted for doesn't exist because the fair price depends on how long you intend to play, and keeps rising.

And Feller's is only a weak law — convergence "in probability," which tolerates rare, wild excursions. There is no strong law here at all: the ratio S_n/(n·log₂ n) does not settle down for an individual endless player. Its lim sup is infinite almost surely — every so often, forever, a jackpot arrives large enough to yank the lifetime average back up (Chow & Robbins, 1961).⁵ That is the precise sense in which the average never arrives. It is always still climbing, and always still being ambushed by a prize you hadn't yet seen.

A modern reading pushes on the phrase "if you play long enough." Ole Peters argued in 2011 that a single person living through repeated plays experiences the time-average growth of their wealth, not the ensemble average across parallel universes — and that time-average, computed without any utility function, lands on the same finite value Bernoulli got from logarithms, tying the paradox to the Kelly betting criterion of 1956.⁶ It's an elegant lens, and a contested one — decision theorists still reach first for Bernoulli and Menger, and Peters' framing has published rebuttals.⁷ We flag it as a live debate, not a settled verdict — because the one thing this page will not do is tell you which average to believe.