LessWrong (30+ Karma) “What Is The Basin Of Convergence For Kelly Betting?” by johnswentworth
Nov 20, 2025
Join John S. Wentworth, a decision theory expert, as he dives into the intricacies of Kelly betting. He explains the mathematical underpinnings of independent bets and how log returns adhere to the Central Limit Theorem. Wentworth contrasts utility functions dominated by typical versus tail outcomes, showcasing how different strategies lead to varied risks. He also outlines conditions where Kelly betting fails and presents real-world examples of these failures, making complex theories accessible and engaging.
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Central Limit Intuition For Kelly
- Log returns add across independent multiplicative bets, so total log return approximates a normal distribution for large T.
- Utilities that care about typical/median outcomes converge to maximizing expected log return (Kelly) as T grows.
Which Utilities Favor Kelly
- Different utility functions are dominated by different parts of the return distribution: some by upper tail, some by lower tail, and a middle class by the typical outcome.
- Agents whose utilities lie in that middle 'basin' will asymptotically follow Kelly betting.
Contrasting Linear And Threshold Utilities
- Linear utility u(W)=W is dominated by the upper tail, causing agents to accept near-certain ruin for tiny chances of huge payoffs.
- A binary threshold utility prefers avoiding downside and will reject those huge-tail gambles.