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Overview

The Kelly criterion is a mathematical formula for optimal bet sizing in scenarios with known edge. It maximizes long-term logarithmic growth of capital while minimizing risk of ruin. The Polymarket Bot implements a fractional Kelly strategy to reduce variance and account for model uncertainty.

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Mathematical Foundation

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Full Kelly Formula

For a binary outcome (win/lose) with:
  • p = probability of winning
  • q = probability of losing = 1 - p
  • b = net odds received on a win (profit / wager)
The optimal fraction of bankroll to wager is: 
f* = (p × b - q) / b

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Simplified Form for Prediction Markets

Polymarket markets use decimal odds where:
  • You pay q per share
  • Winning shares pay $1.00
  • Net profit per share = 1 - q
This gives b = (1 - q) / q, and substituting into the Kelly formula: 
f* = (p - q) / (1 - q)
This is the core formula used in the PositionSizer implementation. Source: src/risk/position-sizer.js:74
The formula f* = (p - q) / (1 - q) assumes you’re betting on the outcome priced at q. For NO bets (betting against the outcome), both p and q are inverted to 1 - p and 1 - q respectively.

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Why Fractional Kelly?

Full Kelly (using f* directly) maximizes long-term growth rate but has significant drawbacks:
  1. High variance: Full Kelly can recommend very large bets (up to 100% of bankroll)
  2. Model error sensitivity: Small errors in p estimation lead to substantial overbet
  3. Drawdown magnitude: Aggressive sizing causes deeper and longer drawdowns
Fractional Kelly (α × f* where α < 1) trades some growth rate for:
  • Lower variance
  • Reduced sensitivity to model error
  • Shallower drawdowns
  • More psychologically tolerable equity curves

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Growth Rate Comparison

For α = 0.5 (half Kelly), the growth rate is approximately 75% of full Kelly, but the variance is only 25% of full Kelly. This is a favorable tradeoff for most scenarios.
The Polymarket Bot uses α values ranging from 0.10 to 0.40 depending on model accuracy (Brier score). Even at maximum aggression, the system never exceeds 40% of full Kelly.

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Implementation Details

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Side Determination

The bot determines which side of the market to bet based on the model’s prediction: 
const side = p >= 0.5 ? 'YES' : 'NO'
  • If p ≥ 0.5: Bet YES (the outcome will occur)
  • If p < 0.5: Bet NO (the outcome will not occur)
Source: src/risk/position-sizer.js:68

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Probability Inversion for NO Bets

When betting NO, the effective probabilities are inverted: 
const pEff = side === 'YES' ? p : 1 - p
const qEff = side === 'YES' ? q : 1 - q
This ensures the Kelly formula is applied correctly from the perspective of the bet being placed. Source: src/risk/position-sizer.js:69-70

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Edge Calculation

The Kelly fraction is only positive when an edge exists: 
f* = (pEff - qEff) / (1 - qEff)
If f* ≤ 0, there is no edge, and the system returns a zero bet: 
if (fullKelly <= 0) {
  return { bet: 0, fullKelly, alpha: 0, fractionalKelly: 0, capped: false, side }
}
Source: src/risk/position-sizer.js:77-79
Negative Kelly fractions (pEff < qEff) indicate the market price is better than your model’s estimate. The system correctly avoids betting in this scenario, as there is no perceived edge.

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Fractional Alpha Selection

The bot uses a Brier-tiered alpha system that maps model accuracy to fractional Kelly values:
Brier ScoreAlphaKelly Fraction
< 0.18 (excellent)0.4040% of full Kelly
0.18 - 0.22 (good)0.2525% of full Kelly
0.22 - 0.26 (moderate)0.2020% of full Kelly
> 0.26 (low accuracy)0.1010% of full Kelly
< 100 predictions0.00No trading
See Position Sizing for complete tier definitions.

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Rationale

Lower Brier scores indicate better calibration and accuracy. The system uses higher alpha values when the model has proven accuracy, and conservative values when accuracy is unproven or degraded.

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Example Calculations

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Example 1: YES Bet with Edge

Given:
  • Model prediction: p = 0.65
  • Market price: q = 0.52
  • Bankroll: $10,000
  • Brier score: 0.20 → α = 0.25 (Tier 3)
Calculation: 
Side:            YES (p ≥ 0.5)
pEff:            0.65
qEff:            0.52
f*:              (0.65 - 0.52) / (1 - 0.52) = 0.13 / 0.48 = 0.2708
α:               0.25
f_real:          0.25 × 0.2708 = 0.0677
Bet:             0.0677 × $10,000 = $677
Result: Bet $677 on YES (6.77% of bankroll)

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Example 2: NO Bet with Edge

Given:
  • Model prediction: p = 0.30 (70% chance outcome does NOT occur)
  • Market price: q = 0.45
  • Bankroll: $10,000
  • Brier score: 0.19 → α = 0.25 (Tier 3)
Calculation: 
Side:            NO (p < 0.5)
pEff:            1 - 0.30 = 0.70  (effective prob of NO winning)
qEff:            1 - 0.45 = 0.55  (price of NO)
f*:              (0.70 - 0.55) / (1 - 0.55) = 0.15 / 0.45 = 0.3333
α:               0.25
f_real:          0.25 × 0.3333 = 0.0833
Bet:             0.0833 × $10,000 = $833
Result: Bet $833 on NO (8.33% of bankroll)

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Example 3: No Edge (Market Efficient)

Given:
  • Model prediction: p = 0.55
  • Market price: q = 0.56 (market implies higher probability)
  • Bankroll: $10,000
  • Brier score: 0.18 → α = 0.40 (Tier 4)
Calculation: 
Side:            YES (p ≥ 0.5)
pEff:            0.55
qEff:            0.56
f*:              (0.55 - 0.56) / (1 - 0.56) = -0.01 / 0.44 = -0.0227
Result: f* < 0 → No edge → Bet $0 (no trade)

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Example 4: Drawdown Adjustment (Yellow Mode)

Given:
  • Model prediction: p = 0.68
  • Market price: q = 0.50
  • Bankroll: $9,200 (down from HWM of $10,500)
  • Drawdown: (10,500 - 9,200) / 10,500 = 12.38% → Yellow mode
  • Brier score: 0.17 → α_base = 0.40 (Tier 4)
  • Yellow adjustment: alphaMultiplier = 0.5
Calculation: 
Side:            YES
pEff:            0.68
qEff:            0.50
f*:              (0.68 - 0.50) / (1 - 0.50) = 0.18 / 0.50 = 0.36
α_base:          0.40
α_adjusted:      0.40 × 0.5 = 0.20  (yellow mode cuts alpha in half)
f_real:          0.20 × 0.36 = 0.072
Bet:             0.072 × $9,200 = $662
Result: Bet 662onYES(7.2662 on YES (7.2% of bankroll), reduced from 1,324 if in green mode

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Risk of Ruin

One of the Kelly criterion’s key properties is that it never risks total ruin (assuming f* < 1 and infinite divisibility of capital). The bankroll approaches zero asymptotically but never reaches it. However, practical constraints introduce ruin risk:
  1. Minimum bet size: minBetUsd creates a floor below which trading stops
  2. Model error: Incorrect p estimates can lead to overbetting
  3. Non-ergodic outcomes: Sequential betting can amplify losses during streaks
The fractional Kelly approach mitigates these risks by reducing bet sizes.

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Expected Growth Rate

For a single bet, the expected logarithmic growth is: 
E[log(bankroll)] = p × log(1 + b × f) + q × log(1 - f)
Full Kelly maximizes this quantity. Fractional Kelly (α × f*) reduces the growth rate but also reduces variance, making the actual realized growth more stable.

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Assumptions and Limitations

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Assumptions

  1. Independent outcomes: Each bet is independent (no correlation)
  2. Known probabilities: p is accurately estimated
  3. Binary payoffs: Win or lose, no partial outcomes
  4. Infinite divisibility: Can bet any fraction of bankroll
  5. No fees: Original Kelly assumes zero transaction costs

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Polymarket Adjustments

  1. 3% fee on wins: Accounted for in recordTrade() (see Drawdown Tracking)
  2. Minimum bet size: minBetUsd creates discrete rather than continuous sizing
  3. Maximum bet cap: maxBetPct prevents single bets from dominating bankroll
  4. Model uncertainty: Brier-tiered alpha adjusts for estimation error
The Kelly criterion assumes p is the true probability. In practice, p is a model estimate with its own uncertainty. The Brier-tiered system partially addresses this by reducing alpha when model accuracy is unproven.

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Practical Considerations

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When Kelly Fails

  1. Model miscalibration: If p systematically overestimates probabilities, Kelly will overbet
  2. Correlated outcomes: Betting on related markets violates independence assumption
  3. Non-stationary edge: If p accuracy degrades over time, historical Brier may lag reality
  4. Black swans: Extreme outcomes outside the model’s training distribution
The bot’s cold-streak circuit breaker partially addresses non-stationarity by detecting consecutive misses.

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Optimal vs. Practical Kelly

Theoretical optimal Kelly assumes:
  • Perfect knowledge of p
  • Zero fees
  • Continuous betting
  • No psychological constraints
Practical implementation uses:
  • Fractional Kelly (α = 0.10 to 0.40)
  • Fee adjustments in payout calculation
  • Minimum bet floors and maximum bet caps
  • Drawdown-based risk reduction
These modifications sacrifice some theoretical growth for robustness and psychological sustainability.

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Further Reading

  • Kelly, J. L. (1956). “A New Interpretation of Information Rate”. Bell System Technical Journal.
  • Thorp, E. O. (1969). “Optimal Gambling Systems for Favorable Games”. Review of the International Statistical Institute.
  • MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (2011). The Kelly Capital Growth Investment Criterion. World Scientific.

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