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For a Polymarket share bought at price p, where your own estimated probability that the outcome occurs is q, full Kelly sizing commits (q − p) / (1 − p) of your bankroll to that position. That is the stake that maximizes expected logarithmic wealth growth over repeated bets, provided q and p are exactly right. Neither input is exact. The probability is an estimate, and the price is a quote you may not fill at, so the stake is only as sound as the probability model, the payoff assumptions and the execution price behind it.
Kelly is a sizing objective. It does not guarantee a profit on any market, and it cannot tell a bot whether its estimate is good. This guide covers the payoff model Polymarket’s contracts imply, the YES and NO formulas, a worked example with illustrative numbers, and the risk controls a bot needs beyond the formula.
What Kelly sizing optimizes
Kelly chooses the fraction of capital to stake that maximizes the expected logarithm of wealth after repeated bets, assuming the bettor knows the probability and payoff of each one. Optimizing the logarithm rather than expected profit rewards long-run compounding and penalizes ruinous losses, so full Kelly does not recommend betting everything whenever the expected value is positive.
The central practical limitation is estimation. Kelly assumes the probability is known, while a bot estimates it from finite data. The academic treatment of Kelly sizing and finite-data risk (a Humboldt University repository copy, not specific to Polymarket) studies the risk that arises when parameters are estimated this way, both with and without known process parameters. The formula itself shows why this matters: the stake grows with the gap between your estimate and the price, so any overstatement of that gap passes straight into position size.
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Neither that academic work nor Polymarket’s documentation establishes a single fraction of full Kelly as best for Polymarket markets. Full, half, quarter, or a hard cap is a choice about how much model error you accept, and it sits outside the formula.
How a Polymarket share becomes a payoff model
Polymarket’s FAQ defines the contract mechanics. Outcome shares are priced between 0.00 and 1.00 USDC. Each paired YES and NO outcome is fully collateralized by 1.00 USDC. Shares for the correct final outcome pay out on resolution:
“The shares representing the correct, final outcome are paid out $1.00 USDC each upon market resolution.”
Because each share pays a fixed amount, the bot’s payoff model reduces to three numbers: the price paid, the probability of the outcome that pays, and the $1.00 payout. The table uses an illustrative YES price of 0.40 and ignores fees and spread.
| Position at an illustrative YES price of 0.40 | Cost per share | Profit per share if correct | Loss per share if wrong | Break-even probability (before fees and spread) |
|---|---|---|---|---|
| Buy YES at 0.40 | 0.40 | 0.60 | 0.40 | Probability of YES of 0.40 |
| Buy NO at 0.60 (complement of YES at 0.40) | 0.60 | 0.40 | 0.60 | Probability of NO of 0.60 |
The FAQ also says shares can be sold before the outcome is known. That exit price is a market price that can move between entry and resolution, and it is not the $1.00 resolution payoff. A bot that plans to hold to resolution and a bot that plans to exit early are running different payoff models. Only the first matches the Kelly formulas below.
Kelly formulas for YES and NO positions
Define the terms before using the formulas. p is the price you pay per share, which should be the ask you can execute against, not the midpoint. q is your estimated probability that the outcome you are buying resolves correct. f is the fraction of bankroll committed to the position’s cost. B is bankroll.
Buying YES
Net odds per dollar at risk are b = (1 − p) / p. The general Kelly form is f = (b·q − (1 − q)) / b. Substituting b and simplifying gives:
f = (q − p) / (1 − p)
If q is not greater than p, the result is zero or negative, and the bot should not buy YES. The dollar stake is f × B, and the share count is (f × B) / p.
Buying NO
Treat NO as its own contract with its own ask. Let pNO be the NO ask and qNO = 1 − q, the probability that NO is correct. Then:
f = (qNO − pNO) / (1 − pNO)
When the two sides sum exactly to 1.00, this is the same decision as the YES formula. Quoted prices on the two sides are not guaranteed to sum that way, so compute NO from its own ask.
Choosing stake units
A bot can express a position as a capital fraction, a share count or a dollar amount. Pick one internal unit, convert the others at the moment of the order, and round to the market’s tick size and minimum order size. Read those values from current documentation rather than assuming them.
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Worked example with illustrative numbers
The numbers below are hypothetical. They assume a $1,000 bankroll, a bot probability of 0.60 that YES resolves, and a YES ask of 0.50.
Full Kelly: f = (0.60 − 0.50) / (1 − 0.50) = 0.20. That is 20% of bankroll, or $200, which buys 400 shares at 0.50.
| Sizing rule | Fraction of bankroll | Stake (cost) | Shares at 0.50 |
|---|---|---|---|
| Full Kelly | 20% | $200 | 400 |
| Half Kelly | 10% | $100 | 200 |
| Quarter Kelly | 5% | $50 | 100 |
The table shows scaling under one model, not a recommendation. Full Kelly is not an all-in bet, but if this position loses, the full $200 is gone.
Why the output is only as sound as its inputs
Probability estimate
The stake is sensitive to q. The table holds the bankroll at $1,000 and varies the estimate and the ask:
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|---|---|---|---|
| 0.60 | 0.50 | 20.0% | $200 |
| 0.55 | 0.50 | 10.0% | $100 |
| 0.50 | 0.50 | 0% | $0 |
| 0.60 | 0.55 | 11.1% | $111 |
If the true probability is 0.55 but the bot believes 0.60, the full Kelly stake doubles. When the estimate equals the ask, the formula says take no position. A bot that cannot justify its probability to within a few points should size far below full Kelly, and a stale or noisy estimate should shrink the stake rather than pass through unchanged. Neither the academic work nor Polymarket’s documentation prescribes a calibration method, so the method and the evidence for it are the builder’s responsibility.
Payoff assumptions
The payoff model is only as correct as the bot’s reading of what the market pays on. Polymarket’s wording pays shares for the correct, final outcome, so the bot needs each market’s resolution criteria encoded explicitly. A question whose wording leaves room for dispute creates payoff risk even when the probability estimate is right. Where resolution is unclear, reduce the stake or skip the market.
Execution price: midpoint, last trade and executable price
A community-maintained Polymarket CLOB API guide (not an official Polymarket document) distinguishes the midpoint, the last trade and the executable price. The midpoint is a reference point. A buy order fills against the ask side of the book, and a larger order can fill across several levels.
Consider a hypothetical book with 150 shares offered at 0.50 and the next 250 at 0.56. Buying 400 shares costs $215, an average of 0.5375 rather than 0.50. Recomputing full Kelly at that average gives (0.60 − 0.5375) / (1 − 0.5375), about 13.5%, not 20%. Evaluate each candidate size against the average fill it would actually receive, and do not commit to a size the book cannot fill at a price the formula still supports.
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Fees, tick sizes, spreads and minimum order requirements all change the cost per share. Polymarket’s Institute data page directs readers to its pricing documentation for these details, and that is where current values should come from. This article does not state a current fee rate or tick size, because those are market-specific and can change.
Model fees in the payoff rather than assuming they are zero. A fee charged on winnings changes the profit column of the payoff table, not just the price paid, so it needs its own term in the calculation.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Risk controls the formula does not provide
Kelly sizes a single bet against a bankroll. The controls below sit around it. Polymarket’s documentation does not define them as platform rules, and their parameters are the builder’s decision.
- Per-market cap: a hard maximum stake in dollars or shares, applied after the Kelly fraction and any fractional multiplier.
- Portfolio cap that includes open orders: resting orders and filled positions counted together against one exposure limit.
- Stale-data guard: no sizing on a probability or order-book snapshot older than an age limit you set.
- Drawdown stop: no new entries once equity falls below a threshold set in advance, until the position is reviewed.
- Kill switch: one control that cancels resting orders and halts the trading loop.
Correlated positions
Kelly applied market by market treats each position as independent. Two markets that resolve on the same underlying event, or that respond to the same news, can lose together. Summing the per-market stakes can therefore oversize the combined exposure. Group correlated positions and treat them as one exposure, and count simultaneous open orders the same way. No Polymarket-specific correlation model is documented, so the grouping rule is a design choice the builder must define.
Monitoring and reconciliation
Polymarket’s Data API reference documents wallet portfolios, trade and activity data, market state and ranked boards. It also identifies HTTP 429 responses for rate limits. A bot should back off on 429, avoid polling faster than it needs, and treat every poll as fallible. Before updating bankroll or exposure, reconcile the bot’s own order records against the trade and activity data, because a missed fill or a duplicated poll changes the next Kelly calculation. Endpoints and response schemas are operational dependencies that can change, so pin the API version the bot is built against and check the reference whenever a response fails to parse.
Validating a sizing rule before committing capital
This article does not present backtest results or live performance for any bot, and nothing here shows a Kelly-based strategy to be profitable. Before committing capital, a builder should be able to answer these questions from logged evidence:
- How often did resolved outcomes match the probabilities the bot assigned, grouped by estimated probability?
- What executable price, at what size, did each order face, compared with the midpoint the bot used for its decision?
- How many positions were open at once, and which of them resolved on the same event?
- Did any stake exceed the per-market or portfolio cap, and how did the bot behave during a 429 response or a stale snapshot?
Start with stakes small enough that a full run of losses is survivable under your own risk limits, and increase size only after the log answers those questions.
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