Bonding curves are mathematical functions used to price and allocate digital assets in a decentralized manner. There are several ways to classify bonding curves, including:

Based on the shape of the graph as determined by price function-

Visually and through the Mathematical Price Function - Shape & Price

• Shape: Bonding curves can have different shapes, including linear, concave, convex, or a combination of these shapes. The shape of the bonding curve determines how the price of the asset changes as the supply of the asset increases or decreases.

  1. Linear Bonding Curve: The price of the token in a linear bonding curve increases linearly as the supply of the token increases. Linear bonding curves are the simplest type of bonding curves and have a constant slope.

  2. Concave Bonding Curve: In a concave bonding curve, the price of the token increases at a decreasing rate as the supply of the token increases. Concave bonding curves are also known as sublinear bonding curves, and they are commonly used for platforms that require a high degree of liquidity.

  3. Convex Bonding Curve: In a convex bonding curve, the price of the token increases at an increasing rate as the supply of the token increases. Convex bonding curves are also known as super-linear bonding curves, and they are commonly used for platforms that require a high degree of scarcity.

  4. Sigmoid Bonding Curve: A sigmoid bonding curve is a combination of a concave and convex curve, where the price of the token increases at a decreasing rate at first, then at an increasing rate, and then at a decreasing rate again. Sigmoid bonding curves are often used for platforms that require a balance between liquidity and scarcity.

  5. Stepped Bonding Curve: A stepped bonding curve is a piecewise-linear curve with flat regions, where the price of the token remains constant for a certain range of the supply and then increases sharply. Stepped bonding curves are often used for platforms that require a high degree of liquidity at certain points in the supply curve.

• Function: Bonding curves can be based on different mathematical functions, including quadratic, exponential, logarithmic, and more. The function used in the bonding curve determines how the price of the asset changes as the supply of the asset increases or decreases.

  1. Linear Bonding Curve: The formula for a linear bonding curve is simply $y = mx + b,$ where

     1. $"y"$ is the price of the token,
     2. $"x"$ is the token supply,
     3. $"m"$ is the slope of the curve, and
     4. $"b"$ is the y-intercept.
     5. The slope $"m"$ is constant for linear bonding curves, meaning that the price of the token increases linearly as the token supply increases.

     Super Linear Curves

  2. Exponential Bonding Curve: The formula for an exponential bonding curve is $y =$ $ab^x,$ where

     1. $"y"$ is the price of the token,
     2. $"x"$ is the token supply,
     3. $"a"$ is the initial price of the token, and
     4. $"b"$ is the constant base of the exponential function. .

  3. Power Bonding Curve: The formula for a power bonding curve is

     $y = ax^b$, where

     1. $"y"$ is the price of the token,
     2. $"x"$ is the token supply,
     3. $"a"$ is the initial price of the token, and
     4. $"b"$ is the constant exponent. In power bonding curves, the price of the token increases at an increasing rate as the token supply increases.

  4. Sigmoid Bonding Curve: The formula for a sigmoid bonding curve is $y = a / (1 + e^(-bx+c)),$ where

     1. $"y"$ is the price of the token,
     2. $"x"$ is the token supply,
     3. $"a"$ is the initial price of the token,
     4. $"b"$ is a constant that determines the steepness of the curve, and
     5. $"c"$ is the point where the curve crosses the y-axis. In sigmoid bonding curves, the price of the token increases at a decreasing rate at first, then at an increasing rate, and then at a decreasing rate again.

  5. Hybrid Bonding Curve: The formula for a hybrid bonding curve combines multiple functions to create a more complex curve. For example, a hybrid bonding curve might use a linear function for the first portion of the curve and then switch to an exponential or power function for the later portions of the curve. The specific formula for a hybrid bonding curve depends on the combination of functions used.

  $y = 0.5x + 1, x <= 1000$

  $;1.5 * (1.2^(x-1000)), x > 1000$

  Linear for first 1000 tokens and then exponential for all values >1000

• Price determination: Price determined bonding curves are a type of bonding curve where the price of the asset is determined by the bonding curve itself. In other words, the bonding curve determines the price at which the asset can be bought or sold, based on. Bonding curves can be used to determine the price of an asset in different ways. For example, some bonding curves use a constant price function, while others use a supply-and-demand-based pricing mechanism.

  1. Constant function bonding curve: In this type of bonding curve, the **price of the asset remains constant (**Jeff’s Carrot farm example- x tokens redeemable for “y” tokens - always ) throughout the trading period. This means that the price of the asset does not change even when there are changes in the demand and supply of the asset.

  2. Supply and demand-based pricing bonding curve: In this type of bonding curve, the price of the asset is determined based on the supply and demand of the asset. When the demand for the asset is high, the price increases and when the demand is low, the price decreases.

     Supply and demand-based pricing

     $$ Price of energy = (Demand - for- energy / Supply- of- energy) + Price premium $$

  3. Time-based pricing bonding curve: In this type of bonding curve, the price of the asset is determined based on time . The longer the asset is held, the higher the price becomes. This type of bonding curve is often used in projects that incentivize long-term holding of the asset.

     Time-based pricing in Energy markets e.g Time based adjustment can be updated based on

     $$ Price of energy = (Base price) + (Time-based-adjustment) $$

     1. Would be interested to find out how staking and especially yield farming would tie into this
     2. Also can find applications in dynamic bond yields, incentivizing long term holding through some kind of yield on top of yield till maturity.
     3. Examples include Curve Boosts, Gauge Weights etc

  4. Weather-based pricing bonding curve: In this type of bonding curve, the price of the asset is determined based on weather conditions . For example, the price of a renewable energy token may be higher during periods of low wind or solar production, as there is a greater demand for the token during these times.

     • Weather-based pricing

     $$ Price of energy = Base price + (Weather-based- adjustment) $$

  5. Hybrid pricing bonding curve: In this type of bonding curve, a combination of different price determination mechanisms is used to determine the price of the asset. For example, a bonding curve may use a constant price function for the first few tokens sold and then switch to a supply and demand-based pricing mechanism as more tokens are sold.

• Purpose: Bonding curves can be used for different purposes, including fundraising, governance, and incentivizing behavior. For example, bonding curves can be used to raise funds for a project by selling tokens at a fixed price, or they can be used to incentivize users to participate in a decentralized platform by rewarding them with tokens.

  1. Fundraising bonding curves
     1. Augmented Bonding Curve
     2. The Gnosis Dutch Auction
     3. Juicebox.eth
  2. Liquidity provisioning bonding curves
     1. The Bancor Protocol
     2. The Balancer Protocol
     3. The Curve Protocol
     4. Uniswap
  3. Incentive-based bonding curves
     1. Ocean Protocol
     2. Augur
     3. Kleros
     4. Livepeer
  4. Governance bonding curves
     1. Compound Governance
     2. Giveth (Gurves - WIP)

• Implementation: Bonding curves can be implemented in different ways, including on-chain and off-chain implementations. On-chain bonding curves are implemented directly on a blockchain, while off-chain bonding curves are implemented using a separate layer or protocol.

  • Smart contract-based implementation
    • Bancor
  • Off-chain oracle-based implementation
    • Augur is an example of an off-chain oracle-based implementation of a bonding curve. Augur is a decentralized prediction market platform that uses a bonding curve to determine the price of its REP token. The bonding curve is implemented on-chain, but it relies on external price information from oracles to determine the token price.
  • Federated implementation
    • OmiseGO is an example of a federated implementation of a bonding curve. OmiseGO is a decentralized exchange that uses a plasma chain to enable fast and cheap transactions. The bonding curve in OmiseGO is determined by a group of validators, who are responsible for executing transactions and determining the token price.
  • On-chain implementations:
    • Bancor
    • Augur
  • Off-Chain Implementation
    • Gnosis
    • Ocean Protocol
    • dxDAO: The dxDAO is a decentralized autonomous organization that governs the DutchX protocol. The dxDAO uses a bonding curve to distribute its DXD token. The bonding curve is implemented off-chain using the Gnosis Safe
  • Hybrid Implementation
    • Uniswap is an example of a hybrid implementation of a bonding curve. Uniswap is a decentralized exchange that uses an automated market maker (AMM) model to provide liquidity for its tokens. The AMM model uses a constant product formula to determine the token price, and the buying and selling of tokens is implemented on-chain. However, the AMM price feed is updated off-chain using a decentralized oracle service called Chainlink.

• Time-based incentives: Bonding curves can be used to incentivize certain behaviors or activities over time. For example, some bonding curves offer discounts or bonuses for early adopters, while others offer rewards for long-term holders.

• Scaling mechanisms: Bonding curves can be used to scale a platform or protocol in different ways. For example, some bonding curves use multi-level bonding curves to allow for greater scalability, while others use bonding curves in combination with other scaling mechanisms such as sharding.

Needs further exploration

• Governance mechanisms: Bonding curves can be used to implement different types of governance mechanisms in a decentralized system. For example, some bonding curves allocate governance tokens to users based on the amount of tokens they hold, while others allocate governance tokens based on the amount of contributions made to the system.