Failure Regimes of the Kelly Ruin Boundary Approximation: A Three-Parameter Framework for Sequential Decision Problems under Compound Poisson Arrivals

This preprint examines the widely cited heuristic f₀ ≈ 2f* relating the Kelly optimal fraction to the ruin boundary. We show that this approximation arises from a second-order Taylor expansion of the log-growth function and is valid only in the small-bet regime. For large payoff asymmetries, the approximation may become infeasible and yield values exceeding the admissible interval.

The analysis is placed within a three-parameter framework (λ, p, κ) describing sequential decision problems under compound Poisson arrivals. Using Wald’s identity, we highlight the multiplicative role of arrival rate, classifier accuracy, and payoff asymmetry in determining long-run expected reward. Monte Carlo simulations illustrate the theoretical predictions.