Mathematical Formulations and Models

Mathematical Formulations and Models (From the AI Agents’ Perspective)

This section dives into the mathematical underpinnings that guide each AI agent’s proposal generation and the subsequent orchestration process. By detailing the analytical foundations, we clarify how the agents transform data inputs into parameter recommendations (e.g., bond price adjustments, APY changes, buyback decisions) and how the final agreed-upon parameters emerge from a formalized, data-driven model.

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Agent-Level Proposal Generation

Each AI agent ( A_i ) specializes in different data domains and computational models. Despite their uniqueness, all agents follow a similar conceptual approach:

1. Data Inputs: Each agent retrieves a subset of data sources (market conditions, treasury composition, user behavior, risk signals, user growth metrics).

   Formally, let each agent ( A_i ) have a data vector:

   $$D_i = (d_{i1}, d_{i2}, \ldots, d_{in})$$

   where ( d_{ij} ) represents a specific input metric (e.g., current price, volatility index, treasury balance, staking participation rate).

2. Agent-Specific Transformation Functions: Each agent applies a model Fi(Di) to its data vector. These functions can be nonlinear and are often based on machine learning models, econometric methods, or heuristic strategies refined over time.

   Example:

   • Market Analysis Agent (A1) might compute an optimal bond price ( BA1} ) as:

     $$B_{A1} = B_{base} + \alpha_1 (P_{target} - P_{current}) + \alpha_2 (\text{Volatility}) + \cdots$$

     and propose APY adjustments or small buybacks based on secondary indicators.

   • Treasury Management Agent (A2) might focus on:

     $$\Delta N_{A2} = \beta_1 (C_{treasury} - N_{MID} \cdot P_{target}) + \beta_2 (\text{Premium Accumulated})$$

     suggesting buybacks ((\Delta N < 0)) or controlled expansions ((\Delta N > 0)).

   In general, each agent produces a proposal vector:

   $$\Theta_{A_i} = (B_{A_i}, R_{A_i}, I_{A_i})$$

   where ( B_{A_i} ) = Bond price suggestion, ( R_{A_i} ) = APY (staking reward) adjustment, ( I_{A_i} ) = buyback or issuance amount.

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Orchestration Model

Once each agent ( A_i ) provides a proposal vector (\Theta_{A_i}), the orchestration layer merges these into a single parameter set (\Theta_{final}).

Weighted Aggregation

1. Assigning Weights: Each agent’s reliability, historical performance, and alignment with desired outcomes determine a weight ( w_i \geq 0 ). Agents that have historically produced stable results or quickly corrected price deviations may receive higher weights.

2. Parameter-by-Parameter Aggregation: Suppose we have three main parameters to finalize: Bond Price ((B)), Staking Reward Rate ((R)), and Buyback Amount ((I)). For each parameter, we compute a weighted average:

   • Bond Price:

     $$B_{final} = \frac{\sum_{i} w_i \cdot B_{A_i}}{\sum_i w_i}$$
   • APY (Staking Reward):

     $$R_{final} = \frac{\sum_{i} w_i \cdot R_{A_i}}{\sum_i w_i}$$
   • Buyback/Issuance:

     $$I_{final} = \frac{\sum_{i} w_i \cdot I_{A_i}}{\sum_i w_i}$$

   This linear aggregation ensures no single agent dominates. The final results reflect a consensus influenced by each agent’s confidence and track record.

Conflict Resolution and Constraints

While weighted averaging is a start, the orchestration may incorporate additional constraints or optimization steps:

• Nonlinearity and Constraints: In practice, ( \Theta_{final} ) might need to respect certain inequalities (e.g., buyback cannot exceed treasury liquid reserves, APY cannot exceed a given maximum to prevent hyperinflation).

  We can model these constraints as:

  $$g_j(B_{final}, R_{final}, I_{final}) \leq 0, \quad j=1,\ldots,m$$

  If initial weighted results violate any constraint, the orchestration layer solves a constrained optimization problem:

  $$\min_{B,R,I} \sum_i w_i \cdot \| \Theta_{A_i} - (B, R, I) \|$$

  subject to

  $$g_j(B,R,I) \leq 0.$$

  This might require applying methods like quadratic programming, Lagrangian multipliers, or iterative approximation. The AI orchestration layer can run these optimizations off-chain, then propose final parameters on-chain once a feasible solution is found.

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Accounting for Risk and User Growth

With the introduction of the Risk Management Agent (A4) and User Flow Agent (A5):

1. Risk Management Adjustments: If A4 detects risk scenarios, it might impose a penalty term if the final parameters stray too far into risky territory. For example, if a large APY increase is proposed while A4 signals potential oracle manipulation, the orchestration might add a penalty term:

   $$\text{Penalty}_{risk} = \gamma \cdot (\text{Risk Score}) \cdot \max(0, R_{final} - R_{safe})$$

   This ensures that high APY values are discouraged under risky conditions, shifting final parameters towards safety.

2. User Flow Considerations: The User Flow Agent (A5) might suggest moderate increases in APY or small bond discounts to attract new participants if user growth is lagging. The orchestration includes these user growth signals as additional weights or as terms that pull final parameters slightly towards more user-friendly settings.

   For instance, if user growth is slow, we might have:

   $$R_{final} := R_{final} + \delta \times (\text{User Growth Deficit})$$

   The orchestration layer integrates this increment into the final calculation, again possibly constrained by risk and treasury logic.

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Dynamic Calibration of Weights and Coefficients

Over time, the system updates:

• Weights ( w_i ) based on each agent’s performance:

  $$w_i(t+1) = w_i(t) + \eta \cdot (\text{Accuracy Improvement})$$

  where (\eta) is a learning rate. Agents that repeatedly yield stable outcomes gain influence; those that lead to volatility lose it.

• Coefficients ( k_1, k_2, k_3, \ldots ) can be periodically recalibrated: AI agents run simulations, measure outcome stability, and adjust these coefficients to minimize price variance, inflation, or user churn.

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Example Calculation

• Given proposals:

  • A1: $B_{A1}=1.02$, $R_{A1}=+0.5%$, $I_{A1}=0$ (no buyback)

  • A2: $B_{A2}=1.00$ (neutral), $R_{A2}=+0.3%$, $I_{A2}=-1000$ (buyback 1000 MID if price < $1)

  • A3: $B_{A3}=0.99$ (slight discount), $R_{A3}=+0.1%$, $I_{A3}=0$

  • A4: Proposes $R_{A4}=0.0%$ change due to risk alert, $B_{A4}=1.01$, $I_{A4}=0$

  • A5: $B_{A5}=1.01$, $R_{A5}=+0.4%$, $I_{A5}=0$

  Assigning weights (example): $w_1=2, w_2=3, w_3=1, w_4=4, w_5=2$.

  • Bond Price:

    $$B_{final} = \frac{2\cdot1.02 + 3\cdot1.00 + 1\cdot0.99 + 4\cdot1.01 + 2\cdot1.01}{2+3+1+4+2}$$

    Calculate and find a balanced value around ~1.01.

  • APY:

    $$R_{final} = \frac{2\cdot0.5\% + 3\cdot0.3\% + 1\cdot0.1\% + 4\cdot0.0\% + 2\cdot0.4\%}{2+3+1+4+2}$$

    Weighted APY might end near ~0.2–0.3%.

  If risk constraints apply, the orchestration might slightly reduce APY if it’s above a certain threshold or not meet certain conditions.

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Conclusion

Mathematically, the multi-AI-agent orchestration uses:

• Weighted averages,

• Constrained optimization,

• Penalty functions for risk,

• Incremental adjustments based on user growth deficits,

to produce a stable, balanced set of parameters (Bond price, APY, Buyback decisions) that uphold MID’s value proposition. By formalizing each agent’s proposals and merging them through a consistent mathematical framework, the system ensures rational, transparent, and adaptive economic governance aligned with long-term stability and user-friendly growth.