What is LMSR (Logarithmic Market Scoring Rule)?
The Logarithmic Market Scoring Rule (LMSR) is an automated market-making formula designed specifically for prediction markets. It was introduced by economist Robin Hanson in 2003 and became the standard for play-money platforms and early on-chain venues.
What is LMSR (Logarithmic Market Scoring Rule)?
What problem it solves
Traditional orderbook markets need matched buyers and sellers — if no one wants to take the other side of your trade, nothing happens. LMSR solves this by having a programmatic market maker that will always quote a price for both YES and NO, no matter how lopsided the current state is. This means a market can exist even with just one trader on each side.
How the formula works
LMSR uses a logarithmic cost function, C(q) = b × ln(e^(q_YES/b) + e^(q_NO/b)), where q_YES and q_NO are the total quantities of each share outstanding and b is a liquidity parameter. Prices are the partial derivatives of that cost function — so every time someone buys shares, the cost curve moves and prices update smoothly. Crucially, the market maker's total loss is bounded by b × ln(n), where n is the number of outcomes. That bounded loss is what makes LMSR safe to deploy at scale.
Who still uses it
LMSR powered early prediction market pioneers like Inklings, Consensus Point, and Augur v1, and it's still used by Manifold Markets for many of its play-money markets. Modern real-money venues like Kalshi and Polymarket moved to orderbook models because LMSR's guaranteed quote comes at the cost of wider spreads in liquid markets — a worthwhile trade-off when liquidity is scarce, but not when sharp traders are already providing it.
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