airdrop

The user wants to transform the provided Python code into an unsolved mathematical problem statement and its solution in GitHub Markdown format.

Here's a breakdown of the steps:

1. Understand the Code: The Python code simulates a token economy with different airdrop strategies. The goal is to analyze the impact of these strategies on the token price.
2. Identify Key Variables and Relationships: Extract the core mathematical elements: token supply, price, user behavior (buy/sell probabilities), airdrop parameters, and the simulation dynamics.
3. Formulate the Problem Statement: Express the simulation as a mathematical model. Define the state variables, parameters, and the objective (e.g., predicting the token price under different airdrop schemes).
4. Develop the Solution: Translate the Python code's logic into mathematical equations. Represent user behavior, price dynamics, and airdrop mechanics mathematically.
5. Structure the Solution in Markdown: Use GitHub Markdown syntax, especially LaTeX for equations, to present the problem and solution clearly. of $P_0$. The economy consists of $N$ users, each characterized by a set of behavioral parameters influencing their buy and sell decisions. These parameters include a base buy probability ($b$), a base sell probability ($s$), a price sensitivity ($\sigma$), and a market influence ($\mu$). Users are categorized into archetypes, each with a distinct set of these parameters.

The token economy undergoes discrete time steps. At each step, users make buy or sell decisions based on probabilistic models influenced by the current token price ($P_t$), the initial price, a market sentiment factor ($M_t$), and a specific airdrop strategy implemented.

Several airdrop strategies can be employed, each defined by a set of parameters including:

• Type: (e.g., lottery, uniform, tiered, none)
• Percentage: The fraction of the initial token supply allocated for the airdrop.
• Vesting: The method by which airdropped tokens become available to users (e.g., none, linear, dynamic based on price or activity).
• Vesting Periods: The number of periods over which vesting occurs.
• Criteria: The metric used for eligibility or allocation (e.g., holdings, activity).
• Thresholds: Levels of the criteria used for tiered airdrops.
• Weights: Factors applied to different tiers in a tiered airdrop.
• Winners Fraction: The proportion of users selected in a lottery airdrop.
• Price Threshold: A price level used in dynamic price vesting.
• Activity Threshold: An activity level used in dynamic activity vesting.

The probability of a user $i$ buying at time $t$ ($P_{buy, i, t}$) and selling ($P_{sell, i, t}$) are modeled as sigmoid functions:

$P_{buy, i, t} = \frac{1}{1 + e^{-(\beta_{buy, i} + \sigma_i (P_0 - P_t) + \mu_i M_t - \Delta P_t \cdot \omega_{buy})}}$

$P_{sell, i, t} = \frac{1}{1 + e^{-(\beta_{sell, i} - \sigma_i (P_t - P_{airdrop}) + \mu_i M_t + \Delta P_t \cdot \omega_{sell})}}$

where:

• $\beta_{buy, i}$ and $\beta_{sell, i}$ are base probabilities for user $i$, potentially modified by the airdrop strategy.
• $P_{airdrop}$ is a reference price, potentially the initial price or a specific price target of the airdrop.
• $\Delta P_t = \frac{P_t - P_0}{P_0}$ represents the relative price change.
• $\omega_{buy}$ and $\omega_{sell}$ are weighting factors for the price change influence.

The quantity of tokens bought or sold by user $i$ is influenced by their decision and the available supply or their holdings. The token price at the next time step ($P_{t+1}$) is determined by the aggregate buy and sell orders, assuming a simplified market clearing mechanism where the price adjusts based on the difference between demand and supply relative to the total supply ($S_t$). A burn mechanism is also in place, reducing the total supply based on transaction volume.

The Problem:

Given the initial conditions, user archetypes and their parameters, and a defined set of airdrop strategies (characterized by the parameters listed above), mathematically model and predict the evolution of the token price ($P_t$) and total token supply ($S_t$) over a finite number of time steps ($T_{max}$) for each airdrop strategy. Analyze and compare the impact of different airdrop strategies on the final token price and supply.

Solution:

Let:

• $T_t$ be the total token supply at time $t$.
• $P_t$ be the token price at time $t$.
• $H_{i,t}$ be the holdings of user $i$ at time $t$.
• $\mathcal{A}$ be the set of user archetypes.
• $\theta_a = (b_a, s_a, \sigma_a, \mu_a)$ be the parameters for archetype $a \in \mathcal{A}$.
• $U$ be the set of users, $|U| = N$.
• $A_i \in \mathcal{A}$ be the archetype of user $i$.
• $\theta_i = \theta_{A_i}$ be the parameters of user $i$.
• $\mathcal{S}$ be the set of airdrop strategies.
• $s \in \mathcal{S}$ be a specific airdrop strategy with parameters $\Pi_s$.
• $E_{i,s}$ be the eligibility of user $i$ for airdrop strategy $s$.
• $\Gamma_{i,s}$ be the amount of tokens airdropped to user $i$ under strategy $s$.
• $V_{i,s,t}$ be the amount of vested tokens for user $i$ under strategy $s$ at time $t$.
• $\mathbb{I}(\cdot)$ be the indicator function.

1. User Behavior Model:

The probability of user $i$ buying at time $t$ under airdrop strategy $s$ is:

$P_{buy, i, t}^{(s)} = \frac{1}{1 + e^{-(\theta_{i,buy} + \sigma_i (P_0 - P_t) + \mu_i M_t - \frac{P_t - P_0}{P_0} \cdot 0.5)}}$

where $\theta_{i,buy}$ is the base buy probability of user $i$.

The probability of user $i$ selling at time $t$ under airdrop strategy $s$ is:

$P_{sell, i, t}^{(s)} = \frac{1}{1 + e^{-(\theta_{i,sell} - \sigma_i (P_t - P_{0}) + \mu_i M_t + \frac{P_t - P_0}{P_0} \cdot 0.3)}} \cdot (1 + \ln(H_{i,t} + 1)) \cdot \mathbb{I}(H_{i,t} > 0)$

where $\theta_{i,sell}$ is the base sell probability of user $i$, potentially modified by the airdrop strategy (e.g., halved for holding-based tiered airdrops).

2. Airdrop Mechanics:

The total amount of tokens allocated for airdrop strategy $s$ is $A^{(s)} = T_0 \cdot \text{percentage}^{(s)}$.

• Uniform Airdrop: $\Gamma_{i,s} = \frac{A^{(s)}}{N}$.
• Lottery Airdrop: Select $\lfloor N \cdot \text{winners_fraction}^{(s)} \rfloor$ winners randomly. Airdrop amount per winner is $\frac{A^{(s)}}{\lfloor N \cdot \text{winners_fraction}^{(s)} \rfloor}$.
• Tiered Airdrop: Eligibility is based on criteria (holdings or activity) and thresholds with corresponding weights. For holdings: $E_{i,s} = \sum_{j} \text{weight}j^{(s)} \cdot \mathbb{I}(H{i,0} \ge \text{threshold}j^{(s)})$. The airdrop amount is proportional to eligibility: $\Gamma{i,s} = A^{(s)} \cdot \frac{E_{i,s}}{\sum_{k \in U} E_{k,s}}$.
• No Airdrop: $\Gamma_{i,s} = 0$.

3. Vesting:

• No Vesting: $V_{i,s,t} = \Gamma_{i,s}$.
• Linear Vesting: $V_{i,s,t} = \Gamma_{i,s} \cdot \frac{\lfloor \frac{t}{\lfloor T_{max} / \text{vesting_periods}^{(s)} \rfloor} \rfloor}{\text{vesting_periods}^{(s)}}$.