Mathematical Foundations in AI Integration

1. Optimization Algorithms

VirtualDaos utilizes optimization algorithms such as Gradient Descent to refine asset allocation and improve decision-making:

$$\theta^{t+1} = \theta^t - \eta \nabla f(\theta^t)$$

Where:

• θ : Model parameters

• η : Learning rate

• f(θ) : Objective function

2. Neural Network Modeling

Neural networks are applied to analyze on-chain and off-chain data:

$$y = \sigma(Wx + b)$$

Where:

• y : Output prediction

• σ : Activation function

• W : Weight matrix

• x : Input vector

• b : Bias term

3. Markov Decision Processes (MDP)

For decision-making, VirtualDaos applies MDP to model governance dynamics:

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

Where:

• V(s) : Value of state s

• P(s′∣s,a) : Transition probability

• R(s,a,s ′) : Reward function

• γ : Discount factor

4. Game Theory

Game theory underpins DAO collaboration and voting dynamics, applying Nash Equilibrium concepts:

$$\text{NE: } \forall i, u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*)$$

Where:

$$u_i : \text{Utility function of participant } i. \newline s_i^* : \text{Optimal strategy for participant } i.$$