Kelly Criterion for Chicken Crash Games
What the formula needs, and what it returns when the edge is not the player's
Kelly calculates stake size from probability and payout — but only produces a positive stake when the player has a positive edge.
Key Takeaways
Kelly needs two inputs: the payout at the chosen exit step, and the probability of reaching it. The multiplier on its own is not enough. If the expected value of the bet is negative, the formula returns a negative fraction — in practical terms, there is no positive Kelly stake. For the examples on this page we use an illustrative RTP range of 95%–98%, corresponding to a 2%–5% theoretical house edge.
What the Kelly Criterion Is
Kelly calculates the fraction of a bankroll to stake when there is a measurable mathematical edge. It is bet sizing derived from the edge, the probability of winning and the payout, and it is the standard answer wherever expected value is positive — which is why card counters and portfolio managers use it. Kelly answers a specific question: how much of the bankroll should be staked when the probability of winning and the payout are known. In step-based games, that means knowing both the multiplier at the selected exit step and the probability of reaching that step.
The Formula
f* = (bp − q) / b, where f* is the fraction of the bankroll to stake, b is the net payout odds (multiplier − 1), p is the probability of a successful exit and q is 1 − p. When bp exceeds q the result is positive and gives the share of the bankroll to stake; when it does not, the result is negative. For a step-based game, p is the probability of successfully reaching the selected exit step. The multiplier alone cannot be used to calculate Kelly.
Applying Kelly to Step-Based Games
| Theoretical Return | Player Edge | Kelly Implication |
|---|---|---|
| 95% | -5% | No positive Kelly stake |
| 96% | -4% | No positive Kelly stake |
| 97% | -3% | No positive Kelly stake |
| 98% | -2% | No positive Kelly stake |
| 100% | 0% | No growth advantage |
| Above 100% | Positive | Kelly can produce a positive stake |
For the examples in this guide, we use an illustrative RTP range of 95%–98%. Exact RTP and probability models can vary between games, versions and providers. A worked example, as arithmetic rather than as data from a particular game. Take a target multiplier of 2.00x and a probability of reaching it of 48.5%: b = 1, p = 0.485, q = 0.515, so f* = (1 × 0.485 − 0.515) / 1 = -0.03. Kelly returns -3%, which is to say there is no positive stake to place. Those two numbers are a theoretical return of 2.00 × 48.5% = 97% — the 97% row above.
Fractional Kelly as Damage Control
Fractional Kelly means staking part of a positive Kelly result: full Kelly is f*, half Kelly is f* / 2, quarter Kelly is f* / 4. It trades growth for a smoother bankroll curve when the edge is real but the estimate of it is uncertain. When f* is zero or negative there is nothing to take a fraction of, so a 1% stake cannot correctly be called half Kelly, nor a 0.5% stake quarter Kelly. When Kelly is zero or negative, using a small fixed percentage of bankroll is percentage-based bankroll management, not true Fractional Kelly — the bankroll management guide below covers it on its own terms.
Kelly vs Other Bet Sizing Methods
| Method | How It Works | Main Use |
|---|---|---|
| Kelly | Sizes stake from a measurable edge and probability | Positive-edge situations |
| Flat Betting | Same stake each round | Simple fixed exposure |
| Percentage | Fixed percentage of bankroll | Bankroll-based stake sizing |
| Martingale | Increases stake after losses | Loss progression |
| Anti-Martingale | Increases stake after wins | Win progression |
Frequently Asked Questions
Can the Kelly criterion be used in Chicken-style games?
Yes, but only if the probability of reaching the selected step and its payout are known. The multiplier alone is not enough.
Why can Kelly be negative?
Because the expected value of the bet is negative. A negative Kelly result means there is no positive bankroll fraction recommended by the formula.
What is fractional Kelly?
It means staking a fraction of a positive Kelly result. If Kelly is zero or negative, there is no positive Kelly stake to divide.
Is Kelly better than the 1% rule?
They answer different questions. Kelly calculates stake size from a measurable edge; the 1% rule is a practical bankroll-sizing guideline.
Can changing the exit step make Kelly positive?
Not simply because the multiplier is higher. Both the payout and the probability of reaching that step have to be considered.