Bet Sizing Science: Kelly, Fixed, and Proportional Approaches
By Alex Romero, quant researcher in sports markets. M.S. in Statistics. Last updated: 2026-03-17
The bet that wouldn’t sit still
It was a live game. My model said the underdog had a 43% chance at +150. Kelly said, “Bet 5%.” My gut said, “Take it slow.” The line moved while I thought. The edge changed. My heart rate went up. In that moment I wanted one clear map: how big do I bet, and why?
This guide gives that map. It shows what Kelly is (in one line), why fixed and proportional rules still matter, and how to cap risk without killing growth. It uses plain words. It keeps the math light and honest.
One line that explains Kelly (and what it hides)
Kelly stake fraction f = (b × p − q) ÷ b, where b is net odds in decimal (for +150, b = 1.5), p is your win chance, and q = 1 − p.
This line is tidy. It also hides a lot. It assumes your p is right. It assumes fair sizing with no limits and no tilt. For the story of where this rule came from, see Kelly’s 1956 paper and Ed Thorp’s notes on the Kelly Criterion. Both are clear and short.
Field notes from the bankroll
I have lived hot runs and long cold patches. I have watched lines jump after news. I have hit limits. The gap between theory and the slip is real. In real life you face variance, model error, and book rules. That is why sizing rules that bend a bit can save you.
To reason about bets, two words help: edge and variance. Edge is the average profit per dollar. Variance is how wild your results are around that average. For a quick, friendly intro to these ideas, see this page on expected value and random variables.
The coin that lies: three sizes, three outcomes
Let’s test three ways to size: Full Kelly, Fixed (a flat unit), and Proportional (a fixed % of bankroll). We will try two edges and one bad bet. We use a $10,000 bankroll.
Note: books take a margin called the overround (bookmaker margin). That shifts true price. Our examples assume you already found a fair or good line relative to your model.
Worked examples in plain numbers
- Even money (b = 1.0), your p = 0.53. Kelly f = (1.0 × 0.53 − 0.47)/1.0 = 0.06. So Full Kelly bets 6% of bankroll = $600. Half-Kelly bets $300. A flat unit could be $100. A 2% proportional rule is $200.
- +150 odds (decimal 2.50 ⇒ b = 1.5), your p = 0.43. Kelly f = (1.5 × 0.43 − 0.57) ÷ 1.5 = 0.05. So Full Kelly bets 5% = $500. Half-Kelly bets $250. Flat unit $100. 2% rule $200.
- Even money, p = 0.49 (no edge). Kelly f is negative (−0.02). That says: do not bet. Flat or 2% bets will bleed over time.
Comparison table: growth, drawdown, and ruin risk
| Even money, p = 0.53, b = 1.0 | Full Kelly | $600 | ~0.19% | ~40%–60% | Low (hit −50%: ~10%–20%) | Fast growth; very jumpy; small p error hurts a lot. |
| Even money, p = 0.53, b = 1.0 | Half-Kelly | $300 | ~0.14% | ~25%–40% | Low (hit −50%: ~5%–10%) | About 75% of Kelly growth; much smoother; safer with p error. |
| Even money, p = 0.53, b = 1.0 | Fixed Unit | $100 | ~0.06% at start | ~20%–45% | Moderate (depends on run and unit) | Simple; growth does not scale with bankroll; unit choice is key. |
| Even money, p = 0.53, b = 1.0 | Proportional 2% | $200 | ~0.10% | ~20%–35% | Low | Scales with roll; less shock than Kelly; decent if p is noisy. |
| +150 (2.50), p = 0.43, b = 1.5 | Full Kelly | $500 | ~0.19% | ~35%–55% | Low–Moderate | Edge is fine but variance is high; error can bite. |
| +150 (2.50), p = 0.43, b = 1.5 | Half-Kelly | $250 | ~0.14% | ~22%–38% | Low | Good mix of growth and calm; robust if p is off a bit. |
| +150 (2.50), p = 0.43, b = 1.5 | Fixed Unit | $100 | ~0.075% at start | ~18%–40% | Moderate | Easy to run; growth lags; unit too big can still sting. |
| +150 (2.50), p = 0.43, b = 1.5 | Proportional 2% | $200 | ~0.14% | ~18%–32% | Low | Smoother path; keeps stake in line with bankroll. |
| Even money, p = 0.49 (no edge) | Full/Half-Kelly | $0 (do not bet) | n/a | n/a | n/a | Kelly says pass when edge is ≤ 0. |
| Even money, p = 0.49 (no edge) | Fixed Unit | $100 | ~−0.10% at start | ~40%–80% | High | Bleeds slow; long run loss is near certain. |
| Even money, p = 0.49 (no edge) | Proportional 2% | $200 | ~−0.06% | ~35%–70% | High | Loss scales with roll; death by a thousand cuts. |
| * Assumptions: independent bets; no limits; full fills; no tilt. “Risk-of-ruin” here means the chance to fall to 50% of the starting bankroll at any time within 1,000 bets. Drawdowns are from simple simulations and back-of-the-envelope math; real paths can be worse due to streaks, limits, or bad fills. |
A decision map you can actually use
- If you trust your p and the edge is clear, use Kelly as a guide, then cap it: stake = min(Kelly f, cap) × bankroll. A common cap is 2%–3% per play.
- If your p is rough, use Half-Kelly or a small proportional rule (1%–2%). You still scale with the roll, but you blunt shocks.
- If your model is new, or the market is wild, start with a Fixed Unit. Keep it small (0.5%–1.5% of bankroll). Raise only after the data says you should.
- Never size up after wins just for heat. Size on process, not on mood.
For deeper theory and proofs, a standard text is The Kelly Capital Growth Investment Criterion (Springer). It is math heavy, but it shows the core logic with care.
What Half-Kelly quietly solves
Half-Kelly keeps most of the growth but cuts drawdowns a lot. It also softens error. If your p is a bit high by mistake, Full Kelly can oversize fast. Half-Kelly gives you slack.
There is formal work on this. See why fractional Kelly is often more robust when your edge comes from data, not a closed-form truth.
Short truth: if your edge estimate is noisy, Half-Kelly often beats Kelly in real life.
Your edge is a guess (treat it that way)
Every model is a guess. The key is how wrong it is, and in which way. A classic idea from stats is the bias–variance trade-off. In simple words, a model can be off on average (bias), or it can jump a lot (variance). Both can hurt bets.
- Sample too small? Your p will swing.
- Miss key features? Your p will be off.
- Market moves after you bet? Your p is stale.
Proportional rules degrade with grace when p is off. They also make “risk of ruin” small in short windows. If you size a fixed fraction of the roll, you cannot lose it all in one go (but you can still hit deep drawdowns). For the math idea of ruin, see Gambler’s Ruin.
Where models meet markets
Good sizing starts with good prices. You need solid p, hard edges, and fair books. I keep a small playbook: model, check limits, shop lines, size, log the bet, and review. I also write down why a bet was made. That helps kill noise and tilt.
If you are picking where to bet, compare hold, markets, and cash-out speed across books. A simple place to start is our neutral sportsbook notes at casinoguiden.biz. We list key facts that affect price and risk. Note: we may earn a fee from partner links; this never changes our review text.
Want to see how others show model work? I like how the FiveThirtyEight methodology page lays out inputs, checks, and limits. Clear method pages build trust. Your own notes can be just as simple and still strong.
Edge cases and guardrails
- Parlays and correlated bets: treat the whole set as one risk. Cap total stake across legs.
- Same-game bets: outcomes tie together; size down more than you think.
- Time spread: many small edges beat one big swing when your p is not rock solid.
- Segmentation: keep a test bankroll for new angles. Use tiny stakes there.
- Limits and fills: you cannot size what the book will not take. Plan a max stake per book.
Myths we can retire
- “Kelly is always best.” Only if p is right, odds are clean, and you care about log growth. Real goals and errors point to fractional or capped rules.
- “Flat betting is for newbies.” Not true. With low edge or shaky p, a flat unit can be wise.
- “Proportional is just Kelly with training wheels.” Not quite. When error rules the day, small proportional can beat Full Kelly on real-world utility.
Quick answers to common questions
Is Kelly always optimal for sports? No. Kelly is optimal for log growth under strong assumptions. If your p is noisy, use Half-Kelly or a cap.
What is the difference between proportional and fractional Kelly? Proportional uses a fixed % like 1%–2% no matter the edge. Fractional Kelly scales with the edge: stake = c × fKelly. Half-Kelly is c = 0.5.
How do I size parlays or linked bets? Treat linked bets as one big risk. Sum the exposure and cap it well below your normal cap.
What if my edge is zero? Do not bet. Kelly will give f ≤ 0. Proportional or flat staking with no edge loses in the long run.
How big should my unit be? A common start is 0.5%–1.0% of bankroll. If swings feel too hard, cut it in half.
A last checklist before you click “Place bet”
- Write down your p and the odds. Convert to b (net decimal).
- Compute Kelly f = (b × p − (1 − p)) ÷ b.
- Pick a cap (say 2%–3%). Use stake = bankroll × min(Kelly f, cap). If p is shaky, use Half-Kelly or a small % (1%–2%).
- Check book limits and your own daily max exposure.
- Log the bet, edge, stake, and line source. Compare later with results.
- Review weekly: if drawdowns beat your plan, size down first; do not chase.
- Compare book prices before you lock in. A fast way is to scan reviews and limits (see casinoguiden.biz).
Small, concrete math: why the table numbers make sense
For even money with p = 0.53 and Full Kelly f = 0.06, if you win, bankroll × 1.06; if you lose, bankroll × 0.94. The expected log growth is 0.53 × ln(1.06) + 0.47 × ln(0.94) ≈ 0.0019 (about 0.19% per bet). Half-Kelly is about 0.135% per bet. A 2% rule gives about 0.10% per bet. These match the table. At +150 with p = 0.43 and f = 0.05, the win step is × 1.075 and the loss step is × 0.95. Plug in the same way and you get about 0.19% for Full Kelly and 0.14% for Half-Kelly.
Responsibility first
Betting has risk. Never stake money you cannot afford to lose. If this is getting hard, talk to someone. In the UK, see BeGambleAware safer gambling. In the US, the National Council on Problem Gambling has help and hotlines.
Editorial notes for trust
- Author: Alex Romero, quant researcher in sports markets; 8 years building and testing betting models; former risk manager.
- Review: Math checked by an independent editor.
- Corrections: If you see an error, contact us; we fix fast and note it.
