St. Petersburg Paradox (2D Cascade) — Why Expected Value Can Diverge

The Game

Flip a fair coin until it lands heads. If heads first appears on flip n, the payout is 2ⁿ coins. Watch each game cascade down the probability ladder and play it thousands of times per second.

Parameters

Games per tick 40

Flip cap (truncation) 16

Entry fee you'd pay 8 coins

Animate falling tokens on the ladder

Live Readouts

Games played

Running average

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Max payout seen

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Theoretical E[X] (cap)

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Average − entry fee (your net)

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How it works

Each game: flip a fair coin repeatedly; P(first heads on flip n) = (1/2)ⁿ. Payout for that game is 2ⁿ coins, so the expected contribution of every single flip-count is exactly:

P(n) · payout(n) = (1/2)ⁿ · 2ⁿ = 1   for every n

E[X] = Σ 1 + 1 + 1 + … = ∞

That is the paradox: the game's expected value is genuinely infinite, yet no rational player would pay more than a modest entry fee — because in practice a single trial almost always pays very little (50% of games pay just 2 coins), and only astronomically rare runs pay astronomically much. The ladder view above makes this visible directly: a token exits at rung n with probability (1/2)ⁿ, and its exit dot is sized by log(payout) — most tokens peel off in the first few rungs, but the rare deep ones are worth exponentially more.

The flip cap slider truncates the game after a maximum number of flips (a coin that never lands heads still gets paid out at the cap). With a hard cap C, the divergent sum becomes finite:

E[X]_capped = Σ(n=1..C) 1  +  1   =   C + 1

Lower the cap and the running average visibly converges toward C+1, per the Law of Large Numbers. Raise the cap toward infinity and convergence gets slower and slower — the bottom rungs of the ladder get rarer to reach but each is worth exactly as much in expectation as every rung above it.

Probability ladder — each token is one live game

low n (cyan) high n (amber)

Payout frequency (log scale)

Running average, over time