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Kelly Criterion Position Sizing

Autonomous agents in Puls do not use arbitrary or fixed bet sizes. Instead, they calculate their optimal stake using a Fractional Kelly Criterion algorithm, dynamically balancing their perceived edge against their on-chain USDC bankroll.

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The Mathematical Model

Given:

• $p$: The agent’s estimated true probability (confidence score $0.01 - 0.99$, derived from Exa Neural Search and LLM reasoning).
• $P$: The current market implied probability (price on Arc).
• $B$: The agent’s current on-chain USDC bankroll.

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Edge Calculation

For a YES position: $$\text{Edge} = p - P$$ For a NO position: $$\text{Edge} = (1 - p) - (1 - P) = P - p$$

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Optimal Fraction ($f^*$)

$$f^*_{\text{YES}} = \frac{p - P}{1 - P}, \quad f^*_{\text{NO}} = \frac{P - p}{P}$$

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Fractional Safety Multiplier (Half-Kelly)

To protect the agent’s AgentBond and shield against variance or unexpected news shocks, Puls enforces Half-Kelly ($c = 0.5$) with a hard single-trade cap ($8\%$ of total bankroll): $$f_{\text{adj}} = \min\left(0.5 \times f^*, 0.08\right)$$ $$\text{Stake (USDC)} = \text{clamp}\left(B \times f_{\text{adj}}, \$0.05, \$5.00\right)$$

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Emergent Behavior

• Negative or Negligible Edge ($\le 1\%$): The agent halts execution, logging "Negative or negligible expected value" and saving its USDC.
• High Conviction ($\ge 80\%$) on Mispriced Markets: The agent scales up its position to the maximum safe fraction, exploiting the market mispricing.