Mathematical Foundations for AI in MEV

Integrating AI into MEV strategies involves the application of several mathematical frameworks to enhance predictive accuracy, optimize gas costs, and maximize profitability. Below are key mathematical equations and models that support EOALabs’ approach:

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1. Predictive Algorithms for Transaction Forecasting

AI models use time-series analysis to predict transaction behavior in the mempool. The Autoregressive Integrated Moving Average (ARIMA) model is applied as follows:

$$y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} + \epsilon_t + \theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q}$$

Where:

• $y_t$: Predicted value (e.g., gas price or transaction volume).

• $c$: Constant term.

• $\phi$: Autoregressive coefficients.

• $\epsilon_t$: Error term.

• $\theta$: Moving average coefficients.

• $p, q$: Orders of the AR and MA terms, respectively.

Application:

• MEV Use Case: Forecast transaction spikes in the mempool to preemptively identify arbitrage opportunities or liquidation risks.

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2. Dynamic Gas Adjustments Using Queueing Theory

Gas fee optimization is modeled using M/M/1 queueing systems in queueing theory:

$$L = \frac{\lambda}{\mu - \lambda}$$

Where:

• $L$: Average number of transactions in the queue (pending in the mempool).

• $\lambda$: Arrival rate of transactions (TPS, transactions per second).

• $\mu$: Service rate of the blockchain (TPS capacity).

Optimal Gas Price Calculation: The total cost function includes gas fees G and delay penalties D :

$$C = G + D$$

To minimize costs:

$$G^* = \arg\min_{G} \left( G + \frac{\lambda}{\mu - \lambda} \cdot P \right)$$

Where:

• $P$: Penalty cost per unit delay.

Application:

• MEV Use Case: Dynamically adjust gas fees to minimize the total cost while maintaining priority inclusion in the next block.

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3. Smart Bidding with Game Theory

Private auctions for transaction bundles are analyzed using Nash Equilibrium in game theory. For ( n ) bidders, each bidder’s strategy is to maximize utility:

$$U_i(b_i, b_{-i}) = V_i - b_i \quad \text{if } b_i > b_{-i}, \; U_i = 0 \text{ otherwise.}$$

Where:

• $U_i$: Utility of bidder $i$.

• $b_i$: Bid of bidder $i$.

• $b_{-i}$: Bids of other participants.

• $V_i$: Value of winning (expected profit from transaction inclusion).

The optimal bid is derived using reinforcement learning:

$$b_i^* = \frac{1}{n} \left( V_i - \frac{\partial U_i}{\partial b_i} \right)$$

Application:

• MEV Use Case: Optimize bids in private auctions, ensuring high profitability and transaction inclusion while minimizing overpayment.

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4. Risk Management with AI-Driven Optimization

AI models use Markov Decision Processes (MDPs) to manage risk by modeling MEV scenarios as states, actions, and rewards:

$$V(s) = \max_a \left[ R(s, a) + \gamma \sum_{s'} P(s' | s, a) V(s') \right]$$

Where:

• $V(s)$: Value of state $s$.

• $a$: Action taken in state $s$ (e.g., gas bid adjustment or transaction sequencing).

• $R(s, a)$: Immediate reward for taking action $a$.

• $P(s' | s, a)$: Transition probability to the next state $s'$.

• $\gamma$: Discount factor (importance of future rewards).

Application:

• MEV Use Case: AI dynamically adjusts strategies (e.g., frontrunning vs. arbitrage) based on real-time risk and reward evaluations.

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5. Real-Time Optimization Using Neural Networks

AI systems often use Convolutional Neural Networks (CNNs) or Recurrent Neural Networks (RNNs) to analyze mempool data. The loss function ensures predictions align with observed outcomes:

$$\mathcal{L}(\theta) = \frac{1}{n} \sum_{i=1}^n \left[ y_i - f(x_i; \theta) \right]^2$$

Where:

• $\mathcal{L}$: Loss function.

• $y_i$: Actual outcome (e.g., transaction profitability).

• $f(x_i; \theta)$: Model prediction.

• $\theta$: Model parameters.

Application:

• MEV Use Case: Train models to predict profitable transactions, reducing errors in execution and improving success rates.

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