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

Abstract

The St. Petersburg paradox is usually read as evidence about preference: since expectation-maximization licenses any finite entry price, something must be wrong with valuing money linearly. We argue the inference is misplaced. Expected terminal wealth is the right functional for a payoff received once and consumed, not for a stake re-committed round after round, where what accumulates is a product rather than a sum. For a divisible, indefinitely repeated exposure — a protocol we defend rather than assume — the governing quantity is the time-average growth rate, and we solve that optimization exactly as a function of the entry price C. The solution carries no adjustable constant. An exact identity fixes the growth-optimal stake, which a solvency ceiling caps well below anything expectation-maximization registers; the optimal stake and the growth it attains are both of order 2^(−C); the stake above which wealth is almost surely destroyed exceeds the optimum by the factor e, independently of price; and variance over growth at the optimum equals π²/3, placing the horizon at which drift clears noise near eighteen doubling times. Two consequences follow. The game is favorable at every finite price, so nothing here produces refusal. But growth collapses geometrically, and a compounding target would demand one round every Δt ∝ 2^(−C), a schedule that ceases to be executable at moderate prices. No utility function is required, and none would serve: Menger's inflation defeats any unbounded utility, while bounded utility and cumulative prospect theory each meet the paradox by restricting a free function after the fact.