3.1 Phased Dynamic LMSR Market Maker

Path Protocol implements a Phased Dynamic LMSR system, a mathematically elegant automated market maker that provides instant liquidity without requiring liquidity providers or protocol reserves. The system uses a phased architecture that transitions from capital accumulation to full LMSR operation with dynamic reserves.

Phased Market Architecture

Path Protocol's LMSR implementation uses a phased architecture with five distinct market states:

┌─────────────────────────────────────────────────────────┐
            Phased Dynamic LMSR Market States
└─────────────────────────────────────────────────────────┘


BOUNDING Phase

   ├─> Market starts with target_b (e.g., 1000 USDC)
   ├─> Only BUY orders allowed
   ├─> Pool accumulates from trader deposits
   ├─> virtual_b = target_b (fixed, not dynamic yet)
   ├─> Transition: When real_pool >= target_b → ACTIVE
   └─> Timeout: If end_time reached → REFUNDED (full refund)


ACTIVE Phase

   ├─> Both BUY and SELL orders allowed
   ├─> CRITICAL: virtual_b = real_pool (dynamic, updates after each trade)
   ├─> Prices calculated using virtual_b
   ├─> True LMSR behavior with accurate pricing
   └─> Timeout: When end_time reached → COMPLETED


COMPLETED Phase

   ├─> Trading stopped
   └─> Awaiting oracle resolution


RESOLVED Phase

   ├─> Oracle has resolved winning outcome
   ├─> Winners redeem shares (proportional to pool)
   ├─> Losers receive nothing
   └─> Zero-sum settlement (winners from losers)


REFUNDED Phase (BOUNDING timeout)

   ├─> Market never reached ACTIVE
   ├─> All traders refunded their deposits
   └─> No winners/losers, clean closure

How LMSR Works

LMSR uses a cost function to price outcome shares based on outstanding quantities:

Cost Function: C(q) = virtual_b × ln(e^(q₁/virtual_b) + e^(q₂/virtual_b) + ... + e^(qₙ/virtual_b))

Where:

q = vector of outstanding shares for each outcome [q₁, q₂, ..., qₙ]
virtual_b = liquidity parameter (dynamic in ACTIVE phase, equals real_pool)
n = number of outcomes

The marginal price for any outcome i is calculated as:

Price: Pᵢ = e^(qᵢ/virtual_b) / Σⱼ e^(qⱼ/virtual_b)

Properties:

Σ Pᵢ = 1 (prices always sum to 100%)
0 < Pᵢ < 1 (prices bounded between 0 and 100%)
Prices respond smoothly to trades
In ACTIVE phase: Prices reflect true liquidity (virtual_b = real_pool)

Dynamic Virtual Reserves

Key Innovation: In the ACTIVE phase, virtual_b dynamically adjusts to always equal real_pool:

After every trade:
├─> Buy: real_pool += cost → virtual_b = real_pool
├─> Sell: real_pool -= payout → virtual_b = real_pool

This ensures:
├─> Prices always reflect true liquidity
├─> No price-reserve mismatch
├─> Accurate price discovery
├─> Zero settlement risk (pool matches prices)

Settlement Risk:

With dynamic virtual_b, settlement risk is zero:

• Pool always matches prices (virtual_b = real_pool)

• Winners paid from pool (trader money)

• Losers receive nothing

• Protocol never adds money

Why LMSR is Superior

Compared to CPMM (Constant Product Market Makers):

Feature	LMSR	CPMM (e.g., Uniswap)
Capital Required	$0 (phased dynamic LMSR)	$10k-$100k+ (LP provided)
Liquidity Available	Instant at creation	Depends on LP deposits
Provider Risk	None (no LPs)	Impermanent loss risk
Price Bounds	Guaranteed [0,1]	Can break invariants
Protocol Risk	Zero (dynamic virtual_b)	Unbounded (IL unlimited)
Multi-outcome	Native support	Requires N² pairs

Compared to Order Books:

Feature	LMSR	Order Book (CLOB)
Liquidity Depth	Smooth curve	Depends on limit orders
Price Discovery	Instantaneous	Requires market makers
Thin Markets	Excellent performance	Poor spreads
Gas Efficiency	High (single tx)	Low (order matching)

Trade Execution Example

Scenario: User wants to buy 100 YES shares in a binary market

Current State:

• b = 1000 USDC

• q_yes = 500 shares outstanding

• q_no = 500 shares outstanding

Step 1: Calculate current cost

C_before = 1000 × ln(e^(500/1000) + e^(500/1000))
C_before = 1000 × ln(1.649 + 1.649)
C_before = 1000 × ln(3.297)
C_before = 1000 × 1.193
C_before = 1193 USDC

Step 2: Calculate cost after purchase

C_after = 1000 × ln(e^(600/1000) + e^(500/1000))
C_after = 1000 × ln(1.822 + 1.649)
C_after = 1000 × ln(3.471)
C_after = 1000 × 1.245
C_after = 1245 USDC

Step 3: Purchase price

Cost = C_after - C_before = 1245 - 1193 = 52 USDC

The user pays 52 USDC (plus fees) to buy 100 YES shares. If the YES outcome occurs, they redeem shares proportionally from the pool (parimutuel settlement), with winners paid by losers and zero protocol risk.

Note: In Path Protocol's phased dynamic LMSR, the market would be in ACTIVE phase for this trade, with virtual_b dynamically equal to real_pool, ensuring prices accurately reflect true liquidity.