LMSR Market Maker

Settled uses the Logarithmic Market Scoring Rule (LMSR) as its automated market maker. There is no order book — prices are computed mathematically based on the current share quantities in the market.

How LMSR Works

LMSR is a cost-function-based AMM designed for prediction markets. It was introduced by Robin Hanson and has strong theoretical properties: bounded loss for the market maker, continuous pricing, and guaranteed liquidity at all price levels.

The Cost Function

The total cost of all shares in the market is:

C(q_yes, q_no) = b · ln(e^(q_yes/b) + e^(q_no/b))

Where:

• q_yes — total YES shares outstanding
• q_no — total NO shares outstanding
• b — the liquidity parameter (controls price sensitivity)

Price Functions

The current price of a YES share is:

p_yes = e^(q_yes/b) / (e^(q_yes/b) + e^(q_no/b))

And symmetrically for NO:

p_no = e^(q_no/b) / (e^(q_yes/b) + e^(q_no/b))

Prices always satisfy p_yes + p_no = 1.

Cost to Buy Shares

The cost to buy Δq additional YES shares is the difference in the cost function:

cost = C(q_yes + Δq, q_no) - C(q_yes, q_no)

This means buying shares gets progressively more expensive as the price moves toward 1 — providing natural slippage protection.

The Liquidity Parameter (b)

The parameter b controls how sensitive the price is to trades:

b value	Effect
Small b	Prices move quickly with small trades. Higher slippage.
Large b	Prices move slowly. Lower slippage but requires more capital.

The maximum possible loss for the market maker is bounded by b · ln(2) — this is the worst case where all shares end up on one side.

Settled sets b per market based on the expected volume tier. Higher-volume pairs (BTC, ETH) get a larger b for tighter spreads.

OU Model Prior

Each market starts with an initial price derived from the Ornstein-Uhlenbeck (OU) model — a mean-reverting stochastic process fitted to historical funding rate data.

The OU prior represents the probability that the funding rate will be positive at settlement time, based on:

• Current funding rate
• Historical mean reversion speed
• Volatility of the funding rate for that exchange/symbol pair

This prior is used as the initial q_yes / q_no ratio when the market opens, so the starting price reflects the statistically expected outcome rather than a naive 50/50.

The OU confidence score (0–1) indicates how reliable the prior is. Higher confidence means more historical data and a better-fit model.

On-Chain Implementation

The LMSR math runs entirely on-chain using fixed-point i64 arithmetic (6 decimal places, matching USDC). The Solana program computes:

1. Current prices from share quantities
2. Cost for a given trade size
3. New share quantities after a trade
4. Fee deduction

No floating-point operations. All math uses integer multiplication and division with overflow checks (checked_mul, checked_div).

Price Impact

Before placing a trade, you can query the price impact via the API:

GET /v1/markets/{id}/quote?side=yes&amount=100

Response includes:

• shares — number of shares you'd receive
• avg_price — average price per share
• price_after — YES price after the trade
• price_impact — how much your order moves the price

Always check the quote before large trades to understand slippage.

Properties

Property	Value
Liquidity	Guaranteed at all price levels — no empty order book
Price range	(0, 1) — approaches but never reaches 0 or 1
Bounded loss	Maximum market maker loss = b · ln(2)
Path independence	Final cost depends only on final quantities, not order of trades
Arbitrage-free	YES + NO prices always sum to 1