Bonding Curve Mechanics

Overview

The bonding curve phase implements the LMSR (Logarithmic Market Scoring Rule) pricing model, a mathematically rigorous market maker designed specifically for prediction markets. This mechanism enables permissionless market creation without seed liquidity while providing bounded losses and smooth price discovery.

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Pricing Model

LMSR Formula

The protocol uses the Logarithmic Market Scoring Rule with liquidity parameter b:

Cost Function:

C(q) = b × ln(e^(q_yes/b) + e^(q_no/b))

Price Formulas:

Price(YES) = e^(q_yes/b) / (e^(q_yes/b) + e^(q_no/b))
Price(NO)  = e^(q_no/b) / (e^(q_yes/b) + e^(q_no/b))

[!IMPORTANT] Key Property: Price(YES) + Price(NO) = 1.0 ALWAYS. This fundamental invariant ensures market coherence.

Liquidity Parameter (b)

The b parameter controls price sensitivity:

• Higher b = More liquidity, less price impact per trade

• Lower b = Less liquidity, more volatile prices

• Default: b = bondingTarget / 7 (approximately $143 for a $1,000 target)

Price Impact Examples

Starting from 50/50 prices with b ≈ $143:

Action	Shares Bought	YES Price After	NO Price After
$100 YES buy	~97 shares	55.2%	44.8%
$200 YES buy	~189 shares	60.1%	39.9%
$500 YES buy	~420 shares	73.5%	26.5%

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Trade Execution

Buy Order Processing (LMSR)

For a YES purchase of USDC amount A:

1. Calculate current cost: C_before = b × ln(e^(qYes/b) + e^(qNo/b))
2. Binary search for shares where: C(qYes + shares, qNo) - C_before = A
3. shares = MAX(calculated_shares, A)  ← $1 CEILING
4. qYes += shares
5. vault += A
6. Mint shares to user

[!TIP] $1 Ceiling Guarantee: You always receive at least 1 share per $1 spent. If LMSR price exceeds $1, you still get 1:1. This ensures winners always profit.

Sell Order Processing (LMSR)

For selling S YES shares:

1. Calculate current cost: C_before = b × ln(e^(qYes/b) + e^(qNo/b))
2. Calculate new cost: C_after = b × ln(e^((qYes-S)/b) + e^(qNo/b))
3. payout = C_before - C_after (capped at vault balance)
4. qYes -= S
5. vault -= payout
6. Burn shares from user

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LMSR vs CPMM

Property	LMSR	CPMM
Price Sum	Always = 1.0	May drift
Bounded Loss	Yes, for market maker	No inherent bound
Price Sensitivity	Adjustable via b	Fixed by pool ratio
Mathematical Basis	Information theory	Uniswap-style
Prediction Markets	Designed for	Adapted from DeFi

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Fee Structure

Operation	Fee Rate	Notes
Buy	0%	No trading fees
Sell	0%	No trading fees

[!NOTE] The bonding phase has zero trading fees. Solvency is maintained through the natural dynamics of the LMSR curve. Fees are only collected at graduation (2%).

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Early Exit Mechanism

Users may liquidate positions at any time during the bonding phase:

Exit Dynamics

• LMSR Payout: Calculated as decrease in cost function when removing shares

• First-mover advantage: Earlier sellers when prices are extreme receive better payouts

• Vault protection: Payout limited by available reserves

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Bonding Phase Termination

The bonding phase ends under one of two conditions:

Condition 1: Graduation (Success)

VaultBalance >= bondingTarget AND Solvency >= 100% AND each side >= 20%

Market transitions to Order Book trading with real transferable ERC-20 tokens.

Condition 2: Deadline Expiry (Failure)

block.timestamp > bondingDeadline AND (Solvency < 100% OR Vault < bondingTarget)

Refund mode activates; users claim pro-rata share of vault.

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Economic Incentives

Actor	Incentive	Mechanism
Early buyers	Lower prices	LMSR starting at 50%
Large buyers	Price discovery	Exponential price movement
Arbitrageurs	Balance markets	Cheap side offers profit
OG Participants	Bonus pool	Early depositors share excess vault
Protocol	Graduation fees	2% collected at graduation

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Mathematical Properties

Solvency Guarantee

LMSR ensures the market maker's maximum loss is bounded:

Max Loss = b × ln(2)

This property, combined with the $1 ceiling on share prices, ensures the vault can always cover winning payouts.

Price Smoothness

Unlike CPMM with its hyperbolic curve, LMSR provides smoother price transitions through the entire probability range (0% to 100%), making it ideal for prediction markets where extreme probabilities are meaningful.