AI Coin Tokenomics
How does price discovery happen ?
The Step Bonding Curve Price Mechanism: Theory and Practice
The pricing of AI Coins is central to the economic integrity of the Data3 Network. Our approach employs a step bonding curve mechanism, which combines the benefits of traditional bonding curves with a discrete, stepwise pricing model.
Introduction to Bonding Curves
Bonding curves are mathematical constructs that define the relationship between a token’s supply and its price. In a conventional bonding curve, the price P(s)P(s)P(s) of a token is a continuous function of the total supply sss, such that each additional token purchased increases the overall price. The core principles are:
Dynamic Pricing: More tokens purchased lead to a higher price, while selling tokens reduces the price.
Reserve Calculation: A reserve pool holds the collateral (in our case, ETH), ensuring that tokens can be redeemed at the prevailing price.
Incentive Structure: Early adopters benefit from lower prices, encouraging early investment and long-term holding.
The Mathematics Behind Bonding Curves
In continuous bonding curve models, the price function is often expressed as:
P(s)=k⋅sn,P(s) = k \cdot s^n,P(s)=k⋅sn,
where:
sss is the total supply of tokens,
kkk is a constant scaling factor, and
nnn determines the curvature (with n>0n > 0n>0).
The total cost R(s)R(s)R(s) to purchase sss tokens is the integral of the price function:
R(s)=∫0sP(x) dx=∫0sk⋅xn dx=kn+1⋅sn+1.R(s) = \int_0^s P(x) \, dx = \int_0^s k \cdot x^n \, dx = \frac{k}{n+1} \cdot s^{n+1}.R(s)=∫0sP(x)dx=∫0sk⋅xndx=n+1k⋅sn+1.
For a buyer wanting to purchase an incremental amount Δs\Delta sΔs starting from a current supply sss, the cost is:
ΔR=∫ss+ΔsP(x) dx.\Delta R = \int_s^{s+\Delta s} P(x) \, dx.ΔR=∫ss+ΔsP(x)dx.
Step Bonding Curve: Definition and Formulae
Unlike a continuous bonding curve, a step bonding curve increases the token price in discrete increments after a set threshold. This mechanism makes the pricing process more intuitive and easier to audit on-chain.
Defining the Step Function
Let:
P0P_0P0 be the initial token price.
ΔP\Delta PΔP be the fixed price increment applied at each step.
TTT be the token threshold (i.e., the number of tokens that can be purchased before a price increase is triggered).
The token price P(s)P(s)P(s) as a function of total supply sss is defined by:
P(s)=P0+ΔP⋅⌊sT⌋,P(s) = P_0 + \Delta P \cdot \left\lfloor \frac{s}{T} \right\rfloor,P(s)=P0+ΔP⋅⌊Ts⌋,
where ⌊⋅⌋\lfloor \cdot \rfloor⌊⋅⌋ represents the floor function, indicating the number of complete steps achieved.
Calculating the Total Cost
When a buyer intends to purchase Δs\Delta sΔs tokens, the total cost must account for the possibility that the purchase spans multiple steps. Consider:
sss is the current total supply.
Δs\Delta sΔs is the number of tokens to be bought.
sfinal=s+Δss_{\text{final}} = s + \Delta ssfinal=s+Δs is the new supply after purchase.
The total cost CCC can be expressed as the sum of the costs across each step interval:
C=∑i=⌊s/T⌋⌊sfinal/T⌋Ci,C = \sum_{i=\lfloor s/T \rfloor}^{\lfloor s_{\text{final}}/T \rfloor} C_i,C=i=⌊s/T⌋∑⌊sfinal/T⌋Ci,
where CiC_iCi is the cost incurred in the ithi^\text{th}ith step. Each CiC_iCi is computed based on the number of tokens purchased in that step multiplied by the step’s price.
For a step where the token price remains constant at:
Pi=P0+ΔP⋅i,P_i = P_0 + \Delta P \cdot i,Pi=P0+ΔP⋅i,
if nin_ini tokens are purchased in that step, then:
Ci=Pi⋅ni.C_i = P_i \cdot n_i.Ci=Pi⋅ni.
Piecewise Calculation
If the buyer’s purchase does not perfectly align with the step thresholds, the computation must be done piecewise:
Remaining Tokens in Current Step: Let r=T−(smod T)r = T - (s \mod T)r=T−(smodT) denote the remaining tokens available at the current price level.
If Δs≤r\Delta s \leq rΔs≤r, then the entire purchase occurs at the current step price P(s)P(s)P(s): C=P(s)⋅Δs.C = P(s) \cdot \Delta s.C=P(s)⋅Δs.
Crossing Multiple Steps: If Δs>r\Delta s > rΔs>r, then:
Purchase rrr tokens at price P(s)P(s)P(s).
The remaining Δs−r\Delta s - rΔs−r tokens are purchased in subsequent steps. The overall cost becomes:
C=P(s)⋅r+∑j=1k(P0+ΔP⋅(i+j))⋅nj,C = P(s) \cdot r + \sum_{j=1}^{k} \left( P_0 + \Delta P \cdot \left(i + j\right) \right) \cdot n_j,C=P(s)⋅r+j=1∑k(P0+ΔP⋅(i+j))⋅nj, where kkk is the number of additional steps crossed, and njn_jnj is the number of tokens purchased in each step jjj.
This discrete, stepwise mechanism not only simplifies the pricing logic but also enhances transparency. Each step is clearly defined and can be audited by any participant on the blockchain.
Practical Calculation Examples
Let’s illustrate the step bonding curve with a practical example.
Example Scenario
Suppose the following parameters are set:
Initial Price: P0=0.01P_0 = 0.01P0=0.01 ETH per token.
Price Increment: ΔP=0.005\Delta P = 0.005ΔP=0.005 ETH.
Step Threshold: T=100T = 100T=100 tokens.
Scenario 1: Purchase Within a Single Step
Current Supply: s=50s = 50s=50 tokens.
Purchase Amount: Δs=30\Delta s = 30Δs=30 tokens.
Since smod T=50s \mod T = 50smodT=50 and r=T−50=50r = T - 50 = 50r=T−50=50, the entire purchase of 30 tokens falls within the current step. Therefore, the cost is:
C=P(s)×Δs=(0.01+0.005⋅⌊50100⌋)×30=0.01×30=0.3 ETH.C = P(s) \times \Delta s = \left(0.01 + 0.005 \cdot \left\lfloor\frac{50}{100}\right\rfloor \right) \times 30 = 0.01 \times 30 = 0.3 \, \text{ETH}.C=P(s)×Δs=(0.01+0.005⋅⌊10050⌋)×30=0.01×30=0.3ETH.
Scenario 2: Purchase Spanning Two Steps
Current Supply: s=90s = 90s=90 tokens.
Purchase Amount: Δs=30\Delta s = 30Δs=30 tokens.
In this case:
Tokens remaining in the current step: r=T−(smod T)=100−90=10r = T - (s \mod T) = 100 - 90 = 10r=T−(smodT)=100−90=10 tokens.
Cost for the first 10 tokens: The current price is: P(90)=0.01+0.005⋅⌊90100⌋=0.01 ETH.P(90) = 0.01 + 0.005 \cdot \left\lfloor\frac{90}{100}\right\rfloor = 0.01 \, \text{ETH}.P(90)=0.01+0.005⋅⌊10090⌋=0.01ETH. Thus, cost for 10 tokens: C1=10×0.01=0.1 ETH.C_1 = 10 \times 0.01 = 0.1 \, \text{ETH}.C1=10×0.01=0.1ETH.
Remaining tokens to be purchased: 30−10=2030 - 10 = 2030−10=20 tokens at the next step. For the next step: Pnext=0.01+0.005⋅⌊100100⌋=0.01+0.005×1=0.015 ETH.P_{\text{next}} = 0.01 + 0.005 \cdot \left\lfloor\frac{100}{100}\right\rfloor = 0.01 + 0.005 \times 1 = 0.015 \, \text{ETH}.Pnext=0.01+0.005⋅⌊100100⌋=0.01+0.005×1=0.015ETH. Cost for these 20 tokens: C2=20×0.015=0.3 ETH.C_2 = 20 \times 0.015 = 0.3 \, \text{ETH}.C2=20×0.015=0.3ETH.
Total cost: C=C1+C2=0.1 ETH+0.3 ETH=0.4 ETH.C = C_1 + C_2 = 0.1 \, \text{ETH} + 0.3 \, \text{ETH} = 0.4 \, \text{ETH}.C=C1+C2=0.1ETH+0.3ETH=0.4ETH.
These examples clearly illustrate how the step bonding curve mechanism works, with price adjustments occurring discretely as token supply crosses predefined thresholds.
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