The Kelly Criterion, adapted for fixed-odds dice betting
The Kelly Criterion is the closest thing gambling math has to a celebrity formula — a single equation, built in 1956 by a Bell Labs physicist, that answers one specific question: given an edge, exactly how much of your bankroll should you actually bet? Here's what it says, and why almost nobody should use it at full strength.
What Kelly actually optimizes for
Kelly doesn't try to maximize your winnings on any single bet — it maximizes the long-run growth rate of your bankroll across many bets. That distinction matters: a bet size that maximizes one outcome can easily be a bet size that eventually ruins you over a long enough series of them.
The formula, in plain English
The classic form is f* = p − q/b, where p is your probability of winning, q is your probability of losing (1 − p), and b is the net odds you're being paid (a $2 payout on a $1 bet is b = 1). The result, f*, is the fraction of your bankroll Kelly recommends staking.
The intuition underneath the formula: your edge (how much better than fair your odds actually are) divided by the payout variance (how big the swings are) — bigger edge, bet more; bigger swings for the same edge, bet less.
Why full Kelly is too aggressive for dice
Kelly assumes you know your true edge with certainty. In practice you never do — your estimate of your edge is always somewhat off, and on a game engineered with a fixed house edge, that "edge" you're solving for is really just how effectively rewards and structure offset the built-in disadvantage, not a clean statistical edge on the roll itself.
Feed Kelly a slightly overconfident edge estimate and it recommends a bet size large enough to produce brutal drawdowns even when your long-run math is directionally correct. This is the single biggest reason full Kelly blows up real bankrolls in practice.
Fractional Kelly: the practical version
The standard fix is fractional Kelly — betting some fraction of what the formula recommends, commonly a quarter to a half of full Kelly. If the formula says stake 20% of your bankroll, fractional Kelly staking 25–50% of that recommendation puts you at 5–10% instead.
Fractional Kelly gives up some theoretical growth rate in exchange for dramatically smaller drawdowns — a trade that's almost always worth making when your edge estimate isn't perfectly precise, which it never is.
This is the same underlying logic behind DiceRoller's risk-management layer: it doesn't size bets off a single optimistic edge calculation, it sizes them conservatively against your actual risk profile and the previous day's result, covered in more depth in wager strategy vs profit strategy. The math is the same family of idea — it's just applied with the caution the formula itself demands.