Kelly Criterion Betting: The Mathematical Formula for Optimal Bet Sizing
April 13, 2026 · PolyMath Team · 10 min read
Most bettors think about two things: which side to bet, and how much to bet. The second question is just as important as the first — and most people get it badly wrong.
Bet too little and you leave compounding gains on the table. Bet too much and a run of variance wipes out your bankroll before your edge plays out. The Kelly Criterion is the mathematical answer to "how much should I bet?" — and it's the framework used by professional gamblers, quantitative traders, and serious Polymarket participants.
What Is the Kelly Criterion?
The Kelly Criterion (also called the Kelly Formula, Kelly Strategy, or Kelly Bet) is a mathematical formula that calculates the optimal percentage of your bankroll to risk on any bet with a known edge. Optimal here means maximizing long-run bankroll growth — not maximizing the expected return of a single bet.
It was developed by John L. Kelly Jr., a researcher at Bell Labs, in a 1956 paper titled "A New Interpretation of Information Rate." He was working on information theory and signal transmission — but gamblers and investors quickly recognized the formula's power.
The core insight: betting a fixed fraction of your bankroll, sized to your edge, produces faster bankroll growth than any other strategy over time.
The Kelly Formula
The basic Kelly Formula is:
Where:
- f* = the fraction of your bankroll to bet (Kelly fraction)
- b = the net odds received on the bet
- p = your estimated probability of winning
- q = your estimated probability of losing (= 1 - p)
Example: Polymarket Binary Bet
For binary prediction markets like Polymarket where shares are priced between $0 and $1, the adjusted Kelly formula is:
Where c = current market price (cost per share).
📊 Worked Example
Market priced at 50¢ (50% implied). Your estimated true probability: 60%.
p = 0.60, q = 0.40, c = 0.50
f* = 0.60 - 0.40 × (0.50/0.50)
f* = 0.60 - 0.40 = 0.20 (20% of bankroll)
Why Kelly Maximizes Long-Run Growth
Imagine a coin that lands heads 60% of the time. You start with $1,000. What fraction should you bet per flip?
| Strategy | Outcome |
|---|---|
| Bet 100% | First loss wipes to zero. Done. |
| Bet 10% | Safe, but slow — leaving edge on the table |
| Bet 20% (Kelly) | Optimal — maximum geometric mean growth |
| Bet 40% | Losing streaks cause catastrophic drawdowns |
The key insight: Kelly optimizes geometric growth, not arithmetic expected value. A bet with +EV can still destroy your bankroll if sized wrong. Betting more than Kelly is provably worse than Kelly over the long run.
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Half-Kelly and Fractional Kelly
Full Kelly betting is theoretically optimal but assumes your edge estimate is perfectly accurate. In practice, you're always working with uncertain probabilities. The solution: fractional Kelly.
Full Kelly (f*)
Theoretically optimal. Only use when you have high confidence in your probability estimate.
Half-Kelly (f*/2)
Professional standard. Loses ~25% of growth rate but cuts variance dramatically.
Quarter-Kelly (f*/4)
Ultra-conservative. Preserves most growth advantage with very manageable variance.
💡 Rule of Thumb for Polymarket
You're trading against informed counterparties. Your probability estimates have inherent uncertainty. Half-Kelly is usually the right starting point — the more uncertain your edge, the smaller fraction you should use.
Kelly in Prediction Markets: Key Differences
Price Is Your Odds
In Polymarket, shares are priced between $0 and $1. A YES share at $0.35 implies 2.86:1 odds on YES resolution. Use the current market price as your odds input in the Kelly formula.
Convert Fraction to Shares
Kelly gives you a fraction of bankroll to allocate — not a share count. Convert: (bankroll × Kelly fraction) ÷ share price = shares to buy. Example: $1,000 bankroll, 15% Kelly → $150 allocation → 428 shares at $0.35.
Account for Fees
Polymarket charges ~2% on winnings. This erodes your edge. Adjust your effective probability slightly downward (p_adj = p - 0.02×p) before running Kelly to get a more realistic fraction.
Portfolio of Positions
With multiple simultaneous positions, sum all Kelly fractions. If total exceeds 100%, scale all positions down proportionally. Correlated positions (e.g., multiple markets on the same election) should be sized more conservatively.
Common Kelly Criterion Mistakes
✗ Overestimating Your Edge
The most dangerous mistake. If you think you have 10% edge but actually have 2%, you'll bet 5x too much. Always sanity-check your estimates against the market price.
✗ Using Full Kelly with Uncertain Probabilities
Full Kelly on uncertain estimates means regularly over-betting positions where you've overestimated your edge. Half-Kelly is the professional standard.
✗ Applying Kelly to a Single Bet
Kelly optimizes a long-run series of bets. The advantage only materializes statistically across many bets with consistent edge. A single Kelly bet has high variance.
✗ Ignoring Bankroll Correlation
Multiple positions correlated to the same underlying event break Kelly's independence assumption. Size correlated positions more conservatively.
✗ Forgetting to Rebalance
Kelly produces a fraction of current bankroll — not a fixed dollar amount. If your bankroll grows from $1,000 to $1,500, your Kelly dollar amounts should grow proportionally.
Step-by-Step: A Real Polymarket Trade
Scenario:Market "Will the Fed cut rates in June?" — currently priced at 42¢ (42% implied). Your research suggests true probability is closer to 55%. Bankroll: $2,000.
Step 1
Identify inputs
Market price (c) = 0.42 · Your probability (p) = 0.55 · Bankroll = $2,000
Step 2
Calculate Kelly fraction
f* = 0.55 - 0.45 × [0.42/0.58] = 0.55 - 0.326 = 0.224 (22.4%)
Step 3
Apply Half-Kelly
f_half = 0.224 / 2 = 11.2%
Step 4
Calculate shares
Dollar allocation: $2,000 × 0.112 = $224 → $224 / $0.42 = 533 YES shares
Step 5
Verify EV is positive
EV/share = (0.55 × $0.58) - (0.45 × $0.42) = $0.319 - $0.189 = +$0.13 per share · Total expected profit: +$69
Strategy Comparison Over 200 Bets
Simulating 200 bets with 5% edge, $10,000 starting bankroll:
| Strategy | Median Final Bankroll | 10th Percentile |
|---|---|---|
| Full Kelly | $47,000 | $8,200 |
| Half-Kelly | $31,000 | $11,500 |
| 10% flat fraction | $18,000 | $9,800 |
| Flat $500/bet | $15,000 | $5,500 |
Half-Kelly produces 66% of Full Kelly's median growth — but its 10th percentile (bad luck scenario) is higherthan Full Kelly's. The variance reduction more than compensates for most real-world traders.
Conclusion: Kelly Is the Foundation of Serious Prediction Market Trading
The Kelly Criterion doesn't tell you which markets to trade. That requires research, probability estimation, and judgment. What Kelly gives you is the answer to the second half of every trade: how much to bet once you've found an edge.
Without Kelly, even traders with genuine edge will eventually over-bet into ruin or under-bet into irrelevance. The formula is mathematically provable — there's no strategy that does better over a long series of bets with accurate probability estimates.
For Polymarket specifically: use Half-Kelly, account for fees, and recalibrate as your bankroll changes. The compounding advantages are real — but only if you have the discipline to follow the math.
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