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The Kelly Criterion for Polymarket Weather Trading: Position Sizing That Doesn't Blow Up Your Bankroll
Most weather traders spend enormous effort on the forecast — which model to use, how to calibrate probabilities. They spend almost no time on how much to bet once they’ve decided a trade is worth taking. In binary markets with high variance, position sizing is as important as the edge itself.
The Problem Kelly Solves
Imagine a trade where the 68–69°F bucket in Tokyo is priced at $0.35 and your calibrated model says the true probability is 0.50. You have a 15% edge. How much of your bankroll should you bet?
Some traders bet a fixed dollar amount. Some bet a fixed percentage. Some bet whatever “feels right.” None of these are optimal in any rigorous sense. The Kelly Criterion answers this question precisely: given your probability estimate and the current odds, what fraction of your bankroll maximizes the expected growth rate of your wealth over many repeated bets?
The Kelly Formula for Binary Markets
For a binary bet with p = your estimated win probability, q = 1 − p, and b = net odds received per dollar risked:
f* = (bp − q) / b
For Polymarket YES bets, where price is the current YES price:
f* = (p − price) / (1 − price)
Worked Example
- Market: Tokyo 68–69°F bucket
- Current YES price: $0.35
- Your model probability: 0.50
- Edge: 50% − 35% = 15%
Kelly YES fraction: f* = (0.50 − 0.35) / (1 − 0.35) = 0.15 / 0.65 ≈ 0.231
Full Kelly says: bet 23.1% of your bankroll. On a $5,000 bankroll, that’s $1,155. That feels large — and it should. We’ll get to why you shouldn’t actually bet this much.
What Kelly Is Actually Optimizing
Kelly maximizes expected log-wealth — equivalent to maximizing the geometric mean of outcomes over many repeated bets. This is the correct objective for compounding money over time. Maximizing expected dollar value leads to strategies that bet enormous fractions and risk frequent ruin.
Kelly is the betting fraction that, in the long run, produces more wealth than any other fraction — including betting more than Kelly. This is the criterion’s mathematical guarantee.
Why Full Kelly Is Dangerous in Practice
1. Your probability estimate is uncertain
Kelly assumes your probability estimate p is exactly correct. It isn’t. Your calibrated model might say 0.50, but the true probability could be anywhere from 0.40 to 0.60. If the true probability is 0.42, your apparent 15% edge is actually 7% — and full Kelly on a 7% edge is a much smaller bet. Betting 23% when the true edge is 7% is overbetting by more than 3×.
2. Correlations between simultaneous positions
A weather bot doesn’t place one trade at a time — it places 10–15 simultaneous trades across multiple cities. If several are correlated (a heat wave over East Asia simultaneously affects Tokyo, Shanghai, and Seoul), treating each bet independently understates total exposure. Per-bet Kelly applied to 10 correlated bets produces an aggregate exposure much larger than full Kelly on the composite position.
3. Variance and psychological comfort
Full Kelly produces violent bankroll swings. Even with genuine edge, a 20–30% drawdown before recovery is common. In practice, most people abandon a strategy during a painful drawdown before the edge has time to manifest. Fractional Kelly reduces drawdown and makes it psychologically sustainable to continue through losing streaks.
Fractional Kelly: The Practical Standard
Apply a multiplier to the Kelly fraction, typically 1/4 to 1/2 Kelly:
- 1/4 Kelly (0.25×): For new strategies or uncertain calibration. Very conservative, small drawdowns, slow but stable compounding.
- 1/3 Kelly (0.33×): For validated strategies with some track record.
- 1/2 Kelly (0.5×): For well-validated strategies with confirmed Brier score and 200+ trade history.
Applying 0.25× to the example above: f* = 23.1% × 0.25 = 5.8% of bankroll. On a $5,000 bankroll: $290 per trade. Still meaningful, but survivable through losing streaks.
Open-source weather bots typically use 0.15× Kelly — extremely conservative, appropriate for unvalidated strategies where actual edge is highly uncertain.
Hard Caps: Why Dollar Limits Matter
Even with fractional Kelly, apply an absolute dollar cap per trade. This protects against two failure modes:
Kelly blow-up on large bankrolls. As your bankroll grows, fractional Kelly positions grow proportionally. A 1/4 Kelly bet on a $100,000 bankroll might be $3,000+ on a single weather bucket — enough to move the market against yourself.
Overconfidence after winning streaks. Hard caps prevent the compounding-confidence problem. A practical cap structure:
- Per-bucket cap: $50–$150 (most open-source bots use $100)
- Per-market cap: $300–$500 total across all buckets in a single city/date market
- Per-day cap: $1,000–$2,000 maximum total exposure
Set these before you start trading and treat them as hard rules, not guidelines.
Kelly Across Different Edge Sizes
| Your Prob (p) | Market Price | Edge | Full Kelly (f*) | 0.25× Kelly |
|---|---|---|---|---|
| 0.45 | 0.35 | 10% | 15.4% | 3.8% |
| 0.50 | 0.35 | 15% | 23.1% | 5.8% |
| 0.55 | 0.35 | 20% | 30.8% | 7.7% |
| 0.60 | 0.35 | 25% | 38.5% | 9.6% |
| 0.45 | 0.40 | 5% | 8.3% | 2.1% |
| 0.55 | 0.40 | 15% | 25.0% | 6.3% |
| 0.70 | 0.55 | 15% | 33.3% | 8.3% |
| 0.80 | 0.70 | 10% | 33.3% | 8.3% |
Key observation: full Kelly gets very large very quickly. A 25% edge means betting 38.5% of bankroll — almost certainly overbetting. 0.25× Kelly produces conservative, bankroll-safe bet sizes for typical weather market edges.
Kelly for NO Positions
When your model probability is lower than the market price (the bucket is overpriced), the correct trade is buying NO shares. The Kelly formula for NO:
f* (NO) = (price − p) / price
Example: The 70–71°F bucket is priced at $0.55 but your model says 35% probability. f* (NO) = (0.55 − 0.35) / 0.55 ≈ 36.4%. Apply 0.25× Kelly: 9.1% of bankroll.
Kelly in a Multi-Bucket Laddering Strategy
When simultaneously buying YES on multiple adjacent buckets, use a portfolio-level Kelly rather than per-bucket Kelly — since exactly one bucket wins, the positions are not independent. In practice, sophisticated bots approximate this by:
- Sizing each bucket independently at a conservative fractional Kelly (e.g., 0.15×).
- Applying a total-market cap so the sum across all buckets in one event doesn’t exceed a set dollar limit.
- Adjusting for the correlation: if you buy YES on buckets B and C, you’re partially hedged — one of them will win, reducing net variance relative to independent binary bets.
The Non-Negotiable Rule: Validate Before You Size
Kelly is meaningless if you don’t have a real edge. Plugging a wrong probability estimate into the Kelly formula produces a guaranteed-losing bet sized with false precision.
Before applying Kelly with any real capital:
- Backtest your model against historical station data and historical Polymarket prices.
- Paper-trade for 30–60 days to confirm the backtest edge holds in live markets.
- Compute your paper-trade Brier score — if it’s worse than 0.25 on binary outcomes, your calibration isn’t ready for live money.
Only then should you apply Kelly sizing to live trades — starting at 0.15× or 0.25× and scaling up as your live Brier score confirms real calibration.
Kelly with a well-validated edge is a compounding machine. Kelly with a fake edge is just an efficient way to lose your bankroll faster than a fixed-bet system would.