2. Weighted Balancing

Weighted Balancing manages how the weights of two tokens change over time during a Liquidity Bootstrapping Pool (LBP) event. This adjustment influences both the pricing and availability of the tokens, ensuring a stable and predictable price discovery process.

Within this Litepaper, when describing LBPs, "Traded Token" ($Token_{Traded}$) refers to the token actively being exchanged or swapped within a transaction, while "Anchor Token" ($Token_{Anchor}$) refers to the token used as a stable measure or benchmark for determining the price of the traded token, such as a Toncoin (TON), a stablecoin, or a similarly stable asset.

2.1. Mathematical Foundations

At the start of the LBP, $Token_{Traded}$ and $Token_{Anchor}$ are assigned initial weights. Typically, $Token_{Traded}$ might start with a high weight, while $Token_{Anchor}$ starts with a low weight.

The weights of these tokens adjust linearly over the duration of the LBP. The formula for the current weight at any given time is calculated as follows:

$Weight_{Traded}^{Current}= Weight_{Traded}^{Start} + \frac{ElapsedTime}{TotalDuration} \times (Weight_{Traded}^{End}- Weight_{Traded}^{Start})$

$Weight_{Anchor}^{Current}= Weight_{Anchor}^{Start} + \frac{ElapsedTime}{TotalDuration} \times (Weight_{Anchor}^{End}- Weight_{Anchor}^{Start})$

$Weight_{Traded}​+Weight_{Anchor}​=1$

Where

• $Weight_{Traded}^{Current}$: Current weight of $Token_{Traded}$

• $Weight_{Anchor}^{Current}$ : Current weight of $Token_{Anchor}$

• $Weight_{Traded}^{Start}$ : Initial weight of $Token_{Traded}$

• $Weight_{Anchor}^{Start}$ : Initial weight of $Token_{Anchor}$

• $Weight_{Traded}^{End}$ : Final weight of $Token_{Traded}$

• $Weight_{Anchor}^{End}$ : Final weight of $Token_{Anchor}$

• $ElapsedTime$ : Time elapsed since the start of the LBP

• $TotalDuration$ : Total duration of the LBP event

2.2. Impact on Token Pricing

As $Weight_{Traded}^{Current}$ decreases and $Weight_{Anchor}^{Current}$ increases, $Token_{Traded}$ becomes cheaper in terms of $Token_{Anchor}$. This mechanism ensures that the price of $Token_{Traded}$ drops gradually, preventing early price spikes and allowing for a more stable price discovery process.

This weighted balancing mechanism is particularly important in preventing manipulative practices where early participants could drive up the price disproportionately. By gradually adjusting the weights, the LBP ensures that the price decreases in a controlled manner, fostering fairer market conditions.

For more details, please refer to the Spot Price and Effective Price sections.

Figure below demonstrates the evolution of the price over time within an LBP under varying $Weight_{Starting}$ conditions.