Kelly Can't Fail: A Zero-Variance Gambling Game
This article presents a card game called "Next Card Bet," where the Kelly betting strategy, typically known for its high variance, surprisingly exhibits zero variance. By counting the remaining red and black cards, the player calculates the optimal bet fraction to maximize the expected logarithm of their stake. Python simulations demonstrate consistent returns of approximately 9.08 times the initial stake across 10,000 runs, showing no variance. The article explains this zero-variance property by showing the Kelly strategy's equivalence to a portfolio strategy that distributes bets across all possible card arrangements. This portfolio strategy guarantees a specific return, independent of the card order, thus explaining the zero variance of the Kelly strategy in this unique game.