Concept · mechanism-design

LMSR (Logarithmic Market Scoring Rule)

An automated market maker that prices trades using a logarithmic cost function, guaranteeing bounded loss to the subsidizer. Robin Hanson's LMSR is the canonical prediction-market AMM · but in practice it failed for binary contracts and Polymarket migrated to a CLOB in late 2022.

Key insights

LMSR failed for production binary markets. In a binary market that resolves to 0 or 1, impermanent loss becomes permanent: the pool inevitably holds worthless shares on the losing side, and trading fees cannot offset a guaranteed structural loss. Polymarket's late-2022 migration from an LMSR AMM to a central limit order book is the inflection point where the industry recognized this (Melee, Why AMMs Failed Prediction Markets).
Dalen / Daedalus Research formalizes LMSR's structural problem in a clean line: "For cost-function market makers such as LMSR, providing more liquidity increases worst-case loss; without external subsidies, they are expected to run at a deficit proportional to the liquidity they offer." This is why LMSR works in subsidized academic / forecasting-tournament contexts but bleeds capital in production.
LMSR is mathematically identical to the softmax function · this bridges quant finance and prediction-market pricing, and is why ML-derived methods (Bayesian regime detection, factor models, machine learning) transfer directly (gemchanger).
Chen & Pennock (Designing Markets for Prediction, Jan 2025) position LMSR as the canonical example among automated market makers designed for "(effectively infinite) liquidity for prediction markets." Their framing: "Many automated market maker mechanisms have been designed to provide (effectively infinite) liquidity for prediction markets; much effort has been put into understanding manipulation in prediction markets and designing prediction mechanisms to achieve incentive compatibility."
Shaw Dalen / Daedalus Research (Oct 2025) proposes a Black–Scholes for prediction markets: a unified stochastic kernel (logit jump-diffusion) that treats traded probability `pt = S(xt)` as a Q-martingale. The risk-neutral drift is pinned down by `µ(t,x) = -[½S''(x)σ_b²(t,x) + ∫(S(x+z)-S(x)-S'(x)χ(z))ν_t(dz)] / S'(x)`. What remains tradable: belief volatility `σb`, jump intensity, and cross-event correlation/co-jumps. Defines a derivative menu (belief variance/volatility swaps, correlation swaps, corridor variance, threshold/first-passage notes) on top of an LMSR-compatible state.
Dalen's central thesis on why LMSR alone isn't enough: "Today, venues execute via scoring rules or AMMs or CLOBs, but there is no shared stochastic model for how probabilities evolve across time, shocks, or related events. Without a kernel, makers cannot isolate 'belief risk' (level, volatility, jumps, co-movement) or lay it off in a standard way; spreads widen around news; inventory near the 0/1 boundaries becomes hard to manage."
Powell, Hanson, Laskey & Twardy (2013) ran the canonical experimental study on combinatorial prediction markets using the DAGGRE platform · a Bayesian-network-driven murder-mystery experiment where participants traded conditional and Boolean events. Their experimental result: "economic theory suggests that the greater expressivity of combinatorial prediction markets should improve accuracy by capturing dependencies among related questions" · empirically validated against flat structures. LMSR's log cost function decomposes naturally over the combinatorial state space.
Baheet's game-theory explainer: LMSR works because it's a proper scoring rule · truth-telling is the dominant strategy. Builders need economists, not just engineers.
Jo Lim's Strait of Hormuz piece argues LMSR/CLMSR is uniquely suited to events without underlying assets · the protocol-native liquidity property (you don't need a counterparty) is what makes scoring-rule markets viable for granular, multi-outcome risk pricing.
LMSR's "bounded loss" property is the killer feature for subsidized markets · a market designer puts up capital `b · ln(n)` and that's the max they can lose, irrespective of trade volume. This is what makes long-tail markets economically viable.
The liquidity parameter `b` is the key design knob: high `b` = tight prices, more capital risk, lower price sensitivity. Low `b` = volatile prices, less capital risk, more responsive to small trades.
Open problem: LMSR's structural-loss issue for binaries doesn't apply identically to continuous outcomes or to non-binary multi-outcome markets · there's an emerging design conversation (CLMSR, scoring-rule AMMs for continuous distributions) that Dekant's L2-norm CFAMM is part of.

In their words

In a binary market that resolves to 0 or 1, impermanent loss becomes permanent: the pool inevitably holds worthless shares on the losing side, and trading fees cannot offset a guaranteed structural loss.· Melee, *Why AMMs Failed Prediction Markets*

For cost-function market makers such as LMSR, providing more liquidity increases worst-case loss; without external subsidies, they are expected to run at a deficit proportional to the liquidity they offer.· Dalen et al., *Toward Black–Scholes for Prediction Markets*

Today, venues execute via scoring rules or AMMs or CLOBs, but there is no shared stochastic model for how probabilities evolve across time, shocks, or related events.· Dalen et al., *ibid.*

Uses LMSR's mathematical identity with the softmax function to bridge quant finance and prediction market pricing.· gemchanger, *Your Hedge Fund's Sharpe Ratio Is a Lie*

Many automated market maker mechanisms have been designed to provide (effectively infinite) liquidity for prediction markets.· Chen & Pennock, *Designing Markets for Prediction*

Where it matters

LMSR is the mechanism every prediction-market designer benchmarks against. The orthodoxy is "LMSR for low-liquidity / long-tail, CLOB for liquid markets" · Polymarket validated this by migrating to CLOB once liquidity was sufficient. But the failure of LMSR on binaries does not generalize to continuous markets, where the partition-of-unity structure (and Dekant's L2-norm CFAMM specifically) keeps the AMM solvent. LMSR remains the conceptual ancestor of every cost-function AMM in the space, and the softmax identity is what lets traditional quant tools transfer directly. Dalen's logit jump-diffusion framework is a strong candidate for the missing institutional pricing kernel sitting on top of an LMSR-compatible state.

Connections

Proper scoring rules · LMSR is the cost-function dual of the log scoring rule
Market scoring rules · LMSR is the canonical instance
Market making · LMSR is an automated market maker (no inventory risk in the usual sense)
Order book · Polymarket abandoned LMSR for CLOB
Combinatorial prediction markets · LMSR decomposes naturally for combinatorial pricing
Liquidity provision · LMSR's `b` parameter sets subsidized liquidity
Incentive compatibility · LMSR inherits IC from the log scoring rule

Platforms linked to this concept

Dekant · addresses · Dekant's L2-norm CFAMM extends the LMSR family to continuous outcomes where the structural-loss issue does not apply
Melee Markets · studies · Melee's 'Why AMMs Failed Prediction Markets' is the primary diagnostic of LMSR's binary-market failure
Polymarket · affected-by · Polymarket abandoned LMSR for a CLOB in late 2022 · the canonical inflection point cited for LMSR's binary failure
Augur · implements · Augur's REP-resolved markets sit downstream of LMSR-style scoring-rule designs
Manifold Markets · implements · Manifold uses a Maniswap variant derived from LMSR cost-function logic
Zeitgeist · implements · Zeitgeist uses LS-LMSR for scalar markets

Related concepts

Sources

Why AMMs Failed Prediction Markets · Melee · X · Apr 13, 2026
Your Hedge Fund's Sharpe Ratio Is a Lie. Prediction Markets Are the Only Place It Can't Hide. · gemchanger · X · Feb 25, 2026
Toward Black–Scholes for Prediction Markets · Shaw Dalen, Daedalus Research Team · arXiv · Oct 17, 2025
The Game Theory Behind Prediction Markets · Baheet · X · Sep 10, 2025
Designing Markets for Prediction · Yiling Chen, David M. Pennock · Harvard · Jan 14, 2025
Combinatorial Prediction Markets: An Experimental Study · Powell, Hanson, Laskey & Twardy · Springer · Sep 16, 2013

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