Logarithmic Market Scoring Rule

LMSR acts as a permanent Liquidity Provider that is always willing to buy or sell shares at algorithmically determined prices. When a trader buys shares in outcome i, the outstanding quantity q_i increases, which raises the softmax price of outcome i and lowers the prices of all other outcomes proportionally. The cost a trader pays for a bundle of shares is the difference in the cost function value before and after the trade: C(q_after) minus C(q_before). Because the cost function is strictly convex, each successive share purchased costs slightly more than the last, creating a natural Slippage curve that penalizes large single trades. This property is what bounds the operator loss: no sequence of trades can extract more than b * ln(n) from the initial subsidy.

LMSR was the foundational mechanism behind early prediction market platforms and remains the theoretical basis for most Automated Market Maker designs in the space. Platforms like Augur and Gnosis used LMSR-derived formulations. The liquidity parameter b is the key design decision: a small b means prices react sharply to each trade (good for price discovery with few traders), while a large b requires more capital but absorbs large orders with less impact on the Contract Price. LMSR differs from constant-product AMMs used in DeFi in that it provides tighter theoretical guarantees for prediction market contexts, specifically the property that prices always reflect a proper probability distribution summing to one. It also connects naturally to the Log Score proper scoring rule, giving it strong theoretical foundations in Information Aggregation.