The Logarithmic Market Scoring Rule (LMSR) is the most taught AMM for prediction markets. Invented for research and corporate forecasting tools, it gives a closed-form price for each outcome with a tunable liquidity parameter covered deeply in the next chapter.
This chapter builds intuition and shows the core math without drowning in proofs. LMSR is the intellectual ancestor of many pool-priced crypto markets; a regulated CLOB does not run LMSR on its touch—but arbitrage from LMSR pools to books still enforces one probability when rules match.
What LMSR optimizes
LMSR markets aim to always offer a price to buy any outcome share, bound market maker loss (subsidy cap), and aggregate trades into incentives aligned with proper scoring rules—experts earn more for honest probabilities. The “house” is an algorithm with a known worst-case loss, not an open-ended bookmaker taking unlimited risk.
A CLOB pays makers to quote; LMSR pays a bounded subsidy through b. Maximum loss scales on the order of b · ln(n) for n mutually exclusive outcomes, while prices are constrained to sum to one across outcomes.
State variables
For mutually exclusive outcomes (YES/NO or three candidates):
- q_i — quantity of shares outstanding for outcome i (net sold by the market maker)
- b — liquidity parameter (bigger b → prices move slower, deeper apparent liquidity)
Prices depend on all outcomes’ quantities—buying YES affects YES price and indirectly NO.
Cost function (intuition)
LMSR defines a cost function C(q) that rises as traders accumulate shares. Traders pay the difference in C when they buy. Marginal price of outcome i behaves like a softmax over quantities: weights grow with exp(q_i / b), normalized so probabilities sum to 1.
Intuition: If more YES shares have been issued (high q_YES), YES becomes more expensive—scarcer from the pool’s view. Raising b dampens the same trade’s impact.
Binary example (numbers)
Let b = 100, start at q_YES = 0, q_NO = 0 → prices near 50% / 50%. A trader buys YES until q_YES increases by 10; YES price might climb from 50% to roughly 52–55% (exact value from the full formula). A larger purchase moves it further—slippage. Heavy YES flow can approach 80–90% before settlement without a hard cap below $1.
At YES price $0.55, you read roughly 55% implied at that pool state; your next buy pays more than 55% if you size up. Exact cents depend on the platform’s implementation—the table below is for intuition, not live quoting.
| q_YES (net sold, illustrative) | Approx YES price | Comment |
|---|---|---|
| 0 | ~50% | Opening |
| 10 | ~52–55% | Small informational trade |
| 50 | ~60–65% | Moderate conviction flow |
| 150 | ~75–85% | Large move—check manipulation budget |
Worked trade — paying cost difference
Suppose baseline cost at (0,0) is zero. A trader buys Δq_YES = 10: they pay C(10,0) − C(0,0) and might lift YES from 50% toward ~53%. A second buy of 10 more when q_YES is already 10 costs C(20,0) − C(10,0) at a higher marginal price—path dependence. Two clips of 10 YES can cost more in total than one buy of 20 YES at once because price rises between clips.
Why logarithmic / exponential
The log-exp structure keeps prices between 0 and 1, forces probabilities to sum to 1 across outcomes, and bounds market maker loss by b · ln(n). That bound is why operators choose b—trade-off between subsidy risk and trader experience.
Raising b from 50 to 200 doubles subsidy headroom order-of-magnitude and dampens price impact per trade. For five candidates, the loss scale grows with ln(5) versus ln(2) for binary markets.
You do not need to implement the cost function to trade LMSR markets, but you should understand that every buy pays for the integral under the curve, not the sticker price at the start of the click. That is why platforms show simulators: the marginal price after your trade can differ materially from the price before it.
LMSR vs constant-product (DeFi AMM)
LMSR is built for probability outcomes with explicit b subsidy math. Uniswap-style x·y = k pools are built for token pairs and need extra logic for multi-outcome events. Live platforms mix ideas; read each venue’s docs. Compare pool simulators to regulated bid/ask, not only the hero percent.
Trader-facing implications
Prices are path-dependent—sequence of trades matters. There is no empty book, only a formula price you may not trust when b is tiny. Fees stack: platform fee plus implicit slippage on the curve. After your own $5,000 buy, “the market is 70%” may mean you moved it to 70%. Pool percent and book mid can lag when rules or bridges differ.
Market maker loss (sketch)
If the market starts at 50/50 and ends paying $1 on the true outcome, LMSR’s structure caps how much worse the pool can be versus the best case. Operators fund b knowing worst-case subsidy—for binary markets with b = 100, loss bound scale is on the order of tens to low hundreds of dollars before fees, a budgeting line item not an afterthought.
Arbitrage and manipulation
LMSR YES at 70% versus book YES ask at 62% invites bots to sell rich YES or buy cheap elsewhere until gas, fees, and capital stop them. LMSR does not “know” the book exists—arbitrageurs enforce consistency. With b = 20 on play money, a few hundred dollars of YES buys might swing displayed probability by fifteen points; with b = 500, the same budget moves less. Manipulation budget scales with b and opposing flow.
Multi-outcome elections
With three or more mutually exclusive outcomes, LMSR keeps a price on each branch and forces the vector to sum to one. Buying shares on Candidate A raises A’s price and slightly reallocates probability mass away from B and C. That coherence is a feature for forecasting dashboards; it also means no separate YES/NO pair per candidate unless the venue lists binaries deliberately.
When LMSR is used today
Academic and play-money forecasting, internal corporate markets, and some crypto implementations (often modified). Live billion-dollar books often add hybrids or professional makers on top. LMSR teaches why a single percent can exist without a CLOB—and why b matters more than the hero number.
Teaching versus live trading
Classroom LMSR markets often use play money with transparent b so students feel slippage. Live platforms modify fees, rebates, and UI rounding. The formulas in this chapter are the spine; your venue’s simulator is the skin. When they disagree, trust the simulator.
Proper scoring intuition (light touch)
LMSR connects to proper scoring rules: forecasters who report probabilities honestly maximize expected score in expectation. That is why corporations used LMSR internally—not because employees were betting for entertainment, but because the scoring aligned incentives. Crypto volume adds speculation, but the mathematical lineage still explains why academics reach for log-exp costs.
Before trading: find published b or equivalent depth; preview slippage for your size; compare to an external book if hedging; remember YES+NO ≈ 1 is built in internally but cross-venue arb is not automatic. The formula is elegant; your simulator is mandatory.
Key ideas
LMSR prices come from exponential weights over quantities—more bought outcome → higher price. b controls sensitivity and max loss. Slippage is built into the cost function. Great for guaranteed quotes; costly for deep liquidity at scale.
Next: how AMMs set prices and supply liquidity step by step when you click buy.