Token Economy Simulation Suite - Mathematical Foundations
Version 1.0
The unified mathematical framework employs consistent notation to represent economic concepts across all simulation subsystems (Affiliate, Airdrop, Bonding Curve, and MCP). The notation convention follows a hierarchical structure:
Entity.Index[].Time[Parameter]
Convention Rules: - Entity Type: Capital letters for system-level concepts (Tokens, Resources, Agents) - Individual Elements: Lower case with subscripts (agent $i$, resource $j$, token $k$) - Time Dimension: Parenthetical time notation $(\cdot(t))$ for temporal evolution - Parameter Variants: Bracket notation $(\cdot[param])$ for parameter-dependent variants - Vector/Matrix Quantity: Bold font $(\mathbf{V})$ for multi-dimensional quantities
| Symbol | Meaning | Context | Status |
|---|---|---|---|
| $t$ | Discrete time step | All systems | Universal |
| $i$ | Individual entity index (Agent $i$) | All systems | Universal |
| $j$ | Resource/Asset index | MCP, Affiliate | Adapter |
| $k$ | Token type index | Affiliate, Bonding | Adapter |
| $\mathbf{A}$ | Set of agents | All systems | Universal |
| $\mathcal{T}$ | Time horizon | All systems | Universal |
| $\mathbb{I}(\cdot)$ | Indicator function | All systems | Universal |
| $\mathbf{P}(X; \theta)$ | Probability of event $X$ given parameter $\theta$ | All systems | Universal |
Affiliate System Adaptation:
S_i(t) → Token i supply at time t
B_j(t) → Base currency balance of affiliate j
C_j(t) → Commission rate of affiliate j
W_{j,k}(t) → Token k holdings of affiliate j
Airdrop System Adaptation:
T_t → Total token supply (scalar)
H_i(t) → Holdings of agent i
α_i(t) → Demand multiplier of agent i
M(t) → Market sentiment factor
Bonding Curve Laboratory Adaptation:
S(t) → System supply (scalar focus)
C_i(t) → Capital of agent i
T_i(t) → Token holdings of agent i
M_i(t) → Price memory array of agent i
MCP System Adaptation:
C_j(t) → Capacity of resource j
L_j(t) → Load/current usage of resource j
Pr_j(t) → Price of resource j
B_i(t) → Balance of agent i
Each simulation subsystem models autonomous agents that interact within an economic environment. The unified framework abstracts these agents into a universal representation characterized by behavioral parameters, decision-making mechanisms, and resource constraints.
Formal Agent Definition:
Let $\mathcal{A}$ be the set of all agents in a simulation system. Each agent $a \in \mathcal{A}$ is characterized by a tuple:
$$a = (\mathbf{State}(t), \mathbf{Params}, \mathbf{Behavior}, \mathbf{Constraints})$$
Where:
All subsystems follow a common discrete-time simulation paradigm that can be abstracted as Sequential Decision Process:
Initialization: $t = 0,\; \mathbf{State}(0) \sim \mathbf{p}_0$
For $t = 0, 1, \dots, T-1$:
Termination Criteria: - Time horizon reached: $t = T$ - Convergence conditions met - Bankruptcy thresholds exceeded
Communication Layer: $\gamma: \mathcal{A} \times \mathcal{A} \to {0,1}$ indicating interaction possibility
Trade Function: $\tau: \mathbf{State}^2 \to \mathbf{State}^2$ atomic exchange mechanism
Market Clearing: $\phi: \mathbf{Actions}, \mathcal{E} \to \mathbf{Allocations}$ resource distribution algorithm
Theorem 1 (Asset Equivalence): Token-based economic systems (Affine) and Resource-based systems (MCP) are mathematically isomorphic when considering equivalent economic observables.
Formal Statement: Let $\mathcal{T} = (T, P)$ be a token system and $\mathcal{R} = (C, Pr)$ be a resource system. Then:
$$ \begin{pmatrix} S_i \ P_i \end{pmatrix} \cong \begin{pmatrix} C_j(1-L_j/C_j) \ Pr_j \end{pmatrix} $$
Under the transformation mapping supply $S_i \mapsto$ available capacity $C_j\,(1-L_j/C_j)$ and price $P_i \mapsto$ resource price $Pr_j$.
Implications: - Price elasticity relationships equivalent across systems - Supply-demand dynamics follow identical mathematical forms - Optimization problems can be translated between systems - Cross-validation possible between token and resource models
Theorem 2 (Decision Unification): All agent decision probabilities across subsystems can be expressed in terms of unified sigmoid functions with equivalent parameter mappings.
Mathematical Form: For any agent decision probability $\mathbf{P}(action|\input)$:
$$\mathbf{P}(action) = \sigma(\mathbf{w} \cdot \mathbf{z} + b)$$
Where: - $\sigma$ is the logistic function - $\mathbf{z}$ is the normalized input vector - $\mathbf{w}$ is the influence vector - $b$ is the bias term
Subsystem Mappings: - Affiliate: $action \in {buy, sell}$, $input = (price_{\text{target}} - price_{\text{current}})/price_{\text{base}}, agent_{\text{balance}}$ - Airdrop: $action \in {buy, sell}$, $input = price_{\text{variation}}, market_{\text{sentiment}}, holdings$ - Bonding: $action \in {hold, trade}$, $input = price_{\text{trend}}, memory_{\text{analysis}}$ - MCP: $action \in {request, hold}$, $input = demand_{\text{preference}}, resource_{\text{availability}}, balance$
Theorem 3 (Gini Generalization): All subsystems employ equivalent wealth inequality metrics that can be unified under a generalized Gini coefficient framework.
Unified Gini Formulation: For any wealth distribution $W = {w_1, \dots, w_n}$, the Gini coefficient can be computed as:
$$G(W) = \frac{\sum_{i}\sum_{j}|w_i - w_j|}{2n\sum_{i}w_i}$$
All simulation subsystems model economic value through assets that can be traded, held, and accumulated. The unified framework represents these assets as stateful objects with supply-demand dynamics.
Universal Asset Definition:
An economic asset $\mathcal{A}$ is characterized by the tuple:
$$\mathcal{A} = (S(t), P(t), \theta, \mathcal{D}(t), \Phi)$$
Where: - $S(t)$: Supply/availability at time $t$ - $P(t)$: Market price at time $t$ - $\theta$: Static structural parameters (e.g., curve types, maxima) - $\mathcal{D}(t)$: Demand profile at time $t$ - $\Phi$: Transformation rules (e.g., regeneration, allocation)
The affiliate system employs parallel token economies where each token constitutes an independent but interconnected economic asset.
Token Definition: For each token $i$ in ${1, \dots, N_T}$:
$$\mathcal{T}i = (S_i(t), P_i(t), f_i, \mathcal{D}_i(t), \Phi{\text{trading}})$$
Supply Dynamics: $$S_i(t+1) = S_i(t) + \sum_{j \in \mathcal{F}} \Delta T_{j,i}(t)$$
Where $\mathcal{F}$ is the set of affiliates actively trading token $i$.
Price Determination: $$P_i(t) = f_i(S_i(t), \theta_i(t))$$
Tokens interact through correlation of affiliate behavior and economic disturbances.
Cross-Token Influence: $$\rho_{i,j}(t) = \text{corr}(P_i(t), P_j(t))$$
Where correlation is measured over a sliding window of historical prices.
The MCP system models assets as depletable but regenerable resources with capacity constraints.
Resource Definition: For each resource $j$ in ${1, \dots, R}$:
$$\mathcal{R}j = (C_j(t), L_j(t), Pr_j(t), \mathcal{U}_j(t), \Phi{\text{maintenance}})$$
Capacity Evolution: $$C_j(t+1) = \min(C_{\max}, C_j(t) \cdot (1 + \gamma_j + \eta_j \cdot \bar{B}(t)))$$
Demand Relationship: $$L_j(t) = \sum_{i \in \mathcal{A}} A_{i,j}(t)$$
Utilization Ratio: $$u_j(t) = \frac{L_j(t)}{C_j(t)}$$
Efficiency Metrics: $$\eta_j(t) = \frac{\sum_{i} U_{i,j}(t)}{L_j(t)}$$
The bonding curve system views assets through the lens of immediate market interactions.
Asset State Representation: $$\mathcal{B} = (S(t), P(t), \mathbf{f}{\mathcal{C}}, V(t), \Phi{\text{curve}})$$
Instantaneous Price Response: $$P(t) = \mathbf{f}_{\mathcal{C}}(S(t))$$
Volatility Measures: $$\sigma_P(t) = \sqrt{\frac{1}{M-1} \sum_{k=1}^{M} (P(t-k+1) - \bar{P}(t))^2}$$
Where $\bar{P}(t)$ is the mean price over the memory window of size M.
The airdrop system integrates asset distribution as a fundamental property.
Distribution-Enhanced Asset: $$\mathcal{A} = (\mathbf{abundance}, P(t), \mathbf{v}, \mathbf{h}, \Phi_{\text{distribution}})$$
Vested Asset Accumulation: $$S^v_i(t) = \min(S^{\text{allocated}}_i, \mathbf{v}(t, \text{threshold}))$$
Perceived Asset Value: $$V_i(t) = P(t) \cdot (1 + \alpha_i(t) \cdot u_i(t))$$
Where $u_i(t)$ represents utility modifier based on holdings.
Theorem 4 (Resource Token Equivalence): Token supply $S_i$ and resource available capacity $C_j(1-L_j/C_j)$ are economically equivalent under proper scaling.
Transformation Mapping: $$\varphi: S_i \mapsto C_j \cdot (1 - u_j) : \mathbb{R}^+ \to \mathbb{R}^+)$$
Price Relationship: $$P_i \cong Pr_j \cdot \frac{S_i}{C_j \cdot (1 - u_j)}$$
For equivalent economic value per unit across systems.
General Supply Dynamics: $$S(t+1) = S(t) + \Delta S^{\text{inflow}}(t) - \Delta S^{\text{outflow}}(t) + \Delta S^{\text{internal}}(t)$$
Where: - $\Delta S^{\text{inflow}}$: External additions (airdrops, regeneration) - $\Delta S^{\text{outflow}}$: External removals (burns, depreciation) - $\Delta S^{\text{internal}}$: Internal reallocations (trading, redistribution)
Universal Asset Parameters Table:
| Parameter | Affiliate System | MCP System | Bonding System | Airdrop System |
|---|---|---|---|---|
| Supply | $S_i(t)$ | $C_j(t)$ | $S(t)$ | $T(t)$ |
| Price | $P_i(t)$ | $Pr_j(t)$ | $P(t)$ | $P(t)$ |
| Demand | $\mathcal{D}_i(t)$ | Aggregation | Instant | $\mathcal{D}(t)$ |
| Allocation | Commissions | Capacity | Curve | Vesting |
This chapter establishes the common ground for understanding how each subsystem represents economic value, enabling cross-system comparisons and theoretical developments in subsequent chapters.
Definition: A bonding curve is a functional relationship $f: \mathbb{R}^+ \to \mathbb{R}^+$ between asset supply and price:
$$P(S) = f(S; \theta)$$
Where $\theta \in \Theta$ is the parameter set characterizing the curve's shape and characteristics.
The framework supports a generalized bonding curve family that encompasses the specific implementations from all subsystems:
Linear Curves: $$f_L(S) = m S + b$$
Exponential Curves: $$f_E(S) = a e^{k S}$$
Sigmoid Curves: $$f_S(S) = K / (1 + e^{-k(S - S_0)}) + b$$
Multi-Segment Curves: $$f_M(S) = \begin{cases} m_1 S & \text{if } S \le S_b \ m_1 S_b + f_E(S - S_b) & \text{if } S > S_b \end{cases}$$
Root Curves: $$f_R(S) = k \sqrt{S} + b$$
Inverse Curves: $$f_I(S) = k / (S + \epsilon) + b$$
Where $\epsilon > 0$ prevents singularity.
Elasticity Analysis: $$\varepsilon(S) = \frac{dP/dS}{P/S} = \frac{f'(S) \cdot S}{f(S)}$$
Inflection Points: Points where curve curvature changes, determining price stability regions: $$f''(S) = 0$$
Asymptotic Behavior: $$\lim_{S \to \infty} P(S) = asymptote$$
General Equilibrium Condition: $$\mathcal{S}(t) = \mathcal{D}(t)$$
Where supply and demand may be implicit functions of price.
Multiple System Approaches:
Direct Price Setting (Bonding Curves): $$P(t+1) = f(S(t+1))$$
Market Clearing (Airdrop/MCP): $$P(t+1) = P_0 \cdot (1 + \alpha \cdot (D(t) - S(t))/V_0 + \noise)$$
Agent-Based Equilibrium (Affiliate): $$P(t+1) = f(S(t+1)) + \beta \cdot \mathcal{M}(t)$$
Where $\mathcal{M}(t)$ represents market sentiment or demand factors.
Fixed Point Analysis: Solutions to $P^ = f(S^(P^*))$
Stability Condition: $$|\partial P/\partial S| < 1$$ for stable price-supply relationship.
Standard Volatility: $$\sigma_P = \sqrt{\frac{1}{T-1} \sum_t (P_t - \bar{P})^2}$$
Logarithmic Volatility: $$\sigma_{\ln P} = \sqrt{\frac{1}{T-1} \sum_t (\ln P_t - \ln P_{t-1})^2}$$
Lyapunov Function for Stability: $$V(S, P) = (P - P^)^2 + \delta (S - S^)^2 + \gamma \dot{P}^2$$
Stability when $\dot{V} \le 0$.
Market Sentiment Model: $$M(t+1) = \rho M(t) + (1-\rho) \mu + \sigma \epsilon_t$$ $$\epsilon_t \sim \mathcal{N}(0,1)$$
Price Impact Integration: $$P(t+1) = f(S(t+1)) \cdot (1 + \eta M(t) + \epsilon_t)$$
Price Cross-Correlation: $$\rho_{i,j} = \frac{\text{Cov}(P_i, P_j)}{\sqrt{\text{Var}(P_i)\text{Var}(P_j)}}$$
This completes the foundational chapters establishing the unified theoretical framework. The subsequent chapters will integrate these foundations with specific analytical methods and optimization approaches.
Continued in subsequent sections...
| Subsystem | Wealth Measurement $w_i$ |
|---|---|
| Affiliate | Base currency balance $B_j$ |
| Airdrop | Token holdings $H_i \cdot P(t)$ |
| Bonding | Portfolio value $C_i + T_i \cdot P(t)$ |
| MCP | Balance $B_i$ |
Properties: - Scale invariant: $G(cW) = G(W)$ for constant $c > 0$ - Population invariant: Independent of agent count - Lorenz curve representation possible in all systems
These foundational theorems establish the mathematical basis for cross-subsystem analysis and parameter mapping, enabling unified theoretical treatment of the diverse economic simulation approaches.
This document provides the mathematical foundations for unifying the four simulation subsystems. Subsequent chapters will expand on the specific models while maintaining consistency with these foundational principles.