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 replacing money with 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. Applying the discrete Kelly criterion at price C, we prove an exact characterization of the growthoptimal 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∗)], from which the optimal fraction f∗ ∼ (C/√2)2−C, the maximal growth rate G(f∗) ∼ 2−C/(√2 ln 2) and the ruin threshold f0 = e f∗ follow as special values: overbetting by the price-independent factor e means almost-sure ruin. Its second moment is V (f) ∼ (π2/3)f/(C ln 2), giving the universal ratio V (f∗)/G(f∗) = π2/3, a growth-to-volatility ratio peaking at f∗/e—so that f∗/e, f∗, ef∗ are three critical stakes a factor e apart—and a reliability horizon near 18 doubling times. All are confirmed numerically to four figures. 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, the horizon on which favorability is observable is longer still, and a finite-horizon agent rationally refuses costly tickets. Divergent expectation and finite growth rate are different functionals of one lottery; the paradox was only ever their conflation.