The Kelly criterion1 provides a mathematically proven way to determine the optimal fraction of your bankroll to wager each time, so you can maximize your overall winnings. It does this by focusing on long-term growth rather than a single big win, balancing your gains against the risk of ruin. Put simply, if you know your chance of winning and the payout odds, the Kelly formula tells you exactly what percentage of your bankroll you should bet to see the fastest compound growth over many plays. For example, if a coin has a 60% chance to pay even money, the Kelly bet would be 20% of your bankroll every time. Over many bets, using Kelly leads to significantly higher final outcomes than betting arbitrary fractions or going all-in.
Kelly Bet Calculator
Using the Kelly criterion to size bets optimally
To understand the utility of the Kelly criterion, give this scenario a try without using the Kelly calculator above.
Imagine being handed $25 and told you can place even-money bets on a coin that lands heads 60% of the time. You get about 300 flips in 30 minutes, with a chance to walk away with as much as $250. Sounds like a promising opportunity—yet in an actual study with these exact rules, 28% of participants lost everything, and on average people walked away with just $91. Only one in five players hit the $250 cap, while others made puzzling choices like betting it all at once or even backing tails. 2 With these surprising outcomes in mind, how would you have played it differently? The $250 cap has been removed so you can feel free to maximize your winnings.
Did you win? Did you go bust? Do you think you chose a good strategy? Why not take a look at how you would do if you played through this scenario 100 times? We’ve added the $250 cap back in.
Multi-Run Betting Simulator (Capped at $250)
When the Kelly criterion is not optimal
If you played around much with the above scenario (winnings capped at $250), you might have noticed that the number given by the Kelly criterion isn’t actually the optimal bet size.
Several Kelly criterion assumptions that are violated by the experimental setup. In particular, Kelly assumes (i) an effectively infinite sequence of repeated bets, (ii) no upper bound on winnings, so that additional gains are always valuable, and (iii) an objective of maximizing expected logarithmic wealth.
In the experiment, the game ends after a short, finite time and (most importantly) imposes a hard $250 payout cap, meaning gains beyond that point have no value while losses below the cap remain costly. In this scenario an optimal strategy should depend on the player’s current bankroll and remaining time, reducing risk near the cap rather than maintaining a fixed Kelly fraction throughout.
An improved strategy would use elastic bet sizing that accounts for the diminished proportion of bet Gained on win near the cap. Put simply, in the above scenario, if you have $240, it doesn’t make sense to bet more than $10.
While the Kelly criterion is not the true optimum in scenarios with a hard cap on winnings or a finite number of bets, it remains a remarkably good approximation. Even when the Kelly assumptions are violated, the resulting optimal strategy often lies close to the Kelly criterion. In capped or finite horizon settings, the exact optimum becomes state dependent and time dependent and is highly sensitive to the specific payoff structure, horizon, and objective. As a result, there is no simple closed form replacement for Kelly, and it instead serves as a practical and intuitive benchmark that is near optimal across a wide range of realistic scenarios.
- https://en.wikipedia.org/wiki/Kelly_criterion ↩︎
- Haghani V, Dewey R. Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin. arXiv [q-finGN]. Published online 2017. http://arxiv.org/abs/1701.01427 ↩︎
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