Resolution of the St. Petersburg Paradox via Asymptotic Capital-Growth Optimization

The St. Petersburg lottery has an infinite expected payoff, yet no one will pay much to play it. For three centuries the gap has been closed by a concave utility—ad hoc, and, as Menger showed, defeated by inflating payoffs. We give an objective resolution with no psychological primitive, treating the lottery as a repeated multiplicative process whose time-average growth rate, not its ensemble average, governs wealth. At price C the discrete Kelly criterion gives an exact characterization of the growth-optimal fraction, Pi≥1[C(1 − f) + f2i]−1 = 1, with the solvency bound f < C/(C − 2). One midpoint (Euler–Maclaurin) expansion delivers the rest. Its first moment is the growth profile G(f) ∼ (f/C ln 2)[1 − ln(f/ ¯ f)] about the reference scale ¯ f = (C/√2)2−C, from which the optimal fraction f∗ ∼ ¯ f, the maximal growth rate G(f∗) ∼ 2−C/(√2 ln 2) and the ruin threshold f0 with f0/f∗ → e follow as special values: overbetting by the price-independent factor e means almost-sure ruin. Its second moment gives the universal ratio V (f∗)/G(f∗) = π2/3, a growth-to-volatility ratio peaking at ¯ f/e—one of three critical stakes a factor e apart—and a signal-to-noise horizon near 18 doubling times. All are confirmed numerically. The resolution has two parts: since the expectation diverges the game is favorable at every finite price, so a growth-optimal strategy always exists; but the growth rate collapses geometrically, the doubling time grows like 2C, and the schedule a compounding target would demand—one round every Δt ∝ 2−C—ceases to be executable at moderate prices. Divergent expectation and finite growth rate are different functionals of one lottery; the paradox was only ever their conflation.