math.md

Math Example: Adding Liquidity

Let's trace a simplified example of a user adding liquidity to a single bin.

Scenario

• Bin 100 (Price = 1.00 USDC per SOL)
• User wants to add: 100 USDC and 100 SOL
• Initial State: Bin is empty (total_shares = 0)

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1. Input Distribution

User specifies:

• delta_id = 0 (Active Bin)
• dist_x = 5000 (50% of X)
• dist_y = 5000 (50% of Y)

2. Deposit Amount Calculation

Calculated from input amounts (e.g., total input 200 USDC, 200 SOL):

$$deposit_x = amount_x \times \frac{dist_x}{10000} = 200 \times \frac{5000}{10000} = 100$$ $$deposit_y = amount_y \times \frac{dist_y}{10000} = 200 \times \frac{5000}{10000} = 100$$

3. Shares Calculation (Minting)

Case A: First Deposit (Bin Empty)

When total_shares == 0, we use the geometric mean:

$$shares = \sqrt{deposit_x \times deposit_y}$$ $$shares = \sqrt{100 \times 100} = 100$$

User receives 100 shares.

Bin State After Case A:

• reserve_x = 100
• reserve_y = 100
• total_shares = 100

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Case B: Subsequent Deposit

Now User 2 comes along.

• Wants to add 50 USDC and 50 SOL.
• Bin already has liquidity (from Case A).

We calculate the share ratio for both X and Y and take the minimum:

$$shares_x = deposit_x \times \frac{total\_shares}{reserve\_x} = 50 \times \frac{100}{100} = 50$$ $$shares_y = deposit_y \times \frac{total\_shares}{reserve\_y} = 50 \times \frac{100}{100} = 50$$ $$shares = \min(50, 50) = 50$$

User 2 receives 50 shares.

Bin State After Case B:

• reserve_x = 100 + 50 = 150
• reserve_y = 100 + 50 = 150
• total_shares = 100 + 50 = 150

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Why usage of min? (Preventing Dilution)

Imagine User 3 tries to cheat. They add 1000 USDC but only 1 SOL.

• The bin ratio is 1:1 (150 X : 150 Y).
• User 3 adds 10:1 ratio.

$$shares_x = 1000 \times \frac{150}{150} = 1000$$ $$shares_y = 1 \times \frac{150}{150} = 1$$ $$shares = \min(1000, 1) = 1$$

They only get 1 share.

• Their extra 999 USDC is effectively "donated" to the pool (captured by existing LPs).
• This forces LPs to add liquidity matching the current bin ratio.