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LMSR Automated Market Maker: Logarithmic Market Scoring Rule

LMSR (Hanson 2003): cost function C(q) = b·ln(Σ exp(qᵢ/b)). Price of outcome i: pᵢ = exp(qᵢ/b)/Σexp(qⱼ/b). Liquidity parameter b controls price sensitivity. Market maker always willing to buy/sell at LMSR price. Bounded loss: max loss = b·ln(n) where n=outcomes. Used in: Augur v1, Gnosis, Manifold Markets, Polymarket (AMM variant). Properties: proper scoring rule, myopic loss-minimization, subsidized liquidity. Limitations: creator must fund initial liquidity, bounded but non-zero loss. Categorical markets: k outcomes, LMSR price sums to 1. Binary special case: reduces to logit.