> ## Documentation Index
> Fetch the complete documentation index at: https://docs.pulsmarket.tech/llms.txt
> Use this file to discover all available pages before exploring further.

# Kelly Criterion Sizing

> Dynamic mathematical bankroll management and position sizing for autonomous prediction market agents

# 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.

***

## 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.

### Edge Calculation

For a **YES** position:

$$
\text{Edge} = p - P
$$

For a **NO** position:

$$
\text{Edge} = (1 - p) - (1 - P) = P - p
$$

### Optimal Fraction ($f^*$)

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

### 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):

$$
f_{\text{adj}} = \min\left(0.5 \times f^*, 0.08\right)
$$

$$
\text{Stake (USDC)} = \text{clamp}\left(B \times f_{\text{adj}}, \$0.05, \$5.00\right)
$$

***

## 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.