Kelly Can't Fail

Optimality and Expected Return

• Multiple commenters ask whether the Kelly strategy is truly optimal in this specific fixed-deck game.
• Consensus in the thread: under the specified rules and “sensible” strategies (notably, always betting everything once only one color remains), all such strategies share the same expected return (~9.08× the initial stake).
• Kelly is highlighted as special because it achieves this expected return with zero variance in the continuous-bet model; alternatives can have the same EV but much higher variance.
• Some argue that accepting higher variance without higher expected return is just “gambling.”

Alternative Strategies and Intuition

• Several alternative strategies are discussed, such as:
  • Doing nothing until only one color remains, then going all-in each time.
  • Toy versions with small decks (e.g., 2 red/2 black) to build intuition and inductively generalize.
• These strategies are shown or simulated to have the same EV as Kelly in this game but different variance profiles.
• A concise inductive proof is sketched to show the Kelly payoff formula holds regardless of card order.

Dependence, Information, and Assumptions

• Some question whether card dependence (non-iid draws) breaks Kelly assumptions or allows a better strategy.
• Others clarify: information gained from flips is independent of the amount bet, and the described Kelly-style rule already updates bets based on the changing composition of the deck.
• There is debate around independence in real-world analogies (e.g., coin tosses, trading), with some insisting real processes are not iid.

Discrete Stakes, Rounding, and Practical Limits

• A major subthread notes that the theoretical result assumes infinitely divisible stakes.
• With cent-level rounding, very long streaks of one color can drive the stake to zero unless the initial bankroll is very large; simulations and APL code explore concrete thresholds.
• Simple rounding of Kelly bets performs poorly; a dynamic-programming strategy can guarantee about 8.08× with discrete units, less than the continuous 9.08×.
• Commenters connect this to Martingale-like issues and real-world constraints: finite bet granularity, counterparty solvency, and transaction costs.

Real-World Use and Caveats

• Kelly’s use in gambling and investing is discussed, with emphasis that:
  • Real bankroll definition and risk tolerance matter.
  • Probabilities are often estimated and non-stationary, so full Kelly can be too aggressive; practitioners tend toward fractional Kelly.
  • Known paradoxes and changing odds (e.g., Proebsting’s paradox) highlight that “Kelly can’t fail” only holds under strict, often unrealistic assumptions.