Week 11: Bankroll management and the Kelly criterion

Why the size of a bet matters even when its EV is fixed and positive, formalized as maximizing long-run exponential growth rather than per-bet expectation

3 May 2027

A question EV alone can’t answer#

Every week so far has asked “is this bet positive-EV?” This week asks a different question about a bet that’s already known to be positive-EV: how much of the bankroll should go on it? Betting more of a fixed edge doesn’t just scale the reward — it changes the shape of the risk, in a way per-bet EV doesn’t capture at all.

Why per-bet EV isn’t the right target for a repeated bet#

A bankroll compounds: a loss doesn’t just cost the amount staked, it costs the ability to stake that amount again. Week 4 already showed that a positive-EV bet can lose most individual attempts. Repeated many times, what matters isn’t the average of the outcomes — it’s the product, since each bet is sized off whatever the bankroll has become. That means the right long-run objective is the expected log of bankroll growth, not the expected bankroll itself.

The Kelly criterion#

For a bet that wins with probability p and pays b to 1 on a win, the fraction of bankroll that maximizes expected log-growth per bet is:

f* = (b·p − q) / b, where q = 1 − p

A concrete repeated bet#

A bet with a 60% win probability, paying even money (b = 1):

f* = (1×0.6 − 0.4) / 1 = 0.2 — bet 20% of the bankroll each time.

The per-bet growth rate at fraction f is g(f) = p·ln(1 + b·f) + q·ln(1 − f). At f* = 0.2:

g(0.2) = 0.6·ln(1.2) + 0.4·ln(0.8) ≈ +2.01% bankroll growth per bet, compounding.

Overbetting a real edge#

Doubling the stake to f = 0.4 (still a smaller-than-full bankroll bet, and every individual wager is still +EV in raw dollar terms):

g(0.4) = 0.6·ln(1.4) + 0.4·ln(0.6) ≈ −0.24% — the bankroll now shrinks on average, despite every single bet remaining favorable. Betting too large converts a real edge into long-run ruin; this is the actual content behind “size matters,” not just a caution.

Half-Kelly, and the trade this course keeps finding#

At f = 0.1 (half of f*): g(0.1) ≈ +1.50% per bet — lower growth than full Kelly, but meaningfully lower variance in the path bankroll actually takes to get there — not re-derived here in full, though it follows directly from the same growth-rate function. The same pattern week 8 found between GTO and exploitative play shows up again: the growth-maximizing size and the size a real bankroll can survive holding aren’t automatically the same choice.

What Kelly assumes, and doesn’t cover#

Kelly assumes p and b are known exactly and the bet is repeated independently. Real poker decisions rarely offer either — win probabilities are estimated (week 6’s whole subject), and results within a single session are correlated, not independent draws. Kelly is the right tool for sizing a known repeatable edge (a poker player’s overall session or bankroll allocation across games), not a substitute for estimating the edge itself.

Before next week#

The course now has, in order: how to price a single decision (weeks 1–3), how to reason about the size and reliability of that price (weeks 4–6), how two rational players settle on a stable joint strategy (weeks 7–8), how more players and a tournament structure change the calculation (weeks 9–10), and now how much of a bankroll a known edge is actually worth risking. Week 12 closes the course by putting all five pieces to work: it grades the claims week 1 opened with, then solves a real game outright.