Skip to main content
The math behind Skepsis pricing β€” no PhD required.

​
What is LMSR?

LMSR stands for Logarithmic Market Scoring Rule. It’s the algorithm that:
  1. Sets prices for every range
  2. Updates prices when people bet
  3. Guarantees there’s always liquidity
  4. Ensures the market stays solvent
Think of it as the automated market maker that runs every Skepsis market.

​
The Simple Explanation

​
The Core Idea

Imagine a market with 5 possible outcome ranges: 
A: $94K-$95K
B: $95K-$96K  
C: $96K-$97K  ← Current price is here
D: $97K-$98K
E: $98K-$99K
LMSR assigns each range a probability based on how many people have bet on it: 
More bets on C β†’ C's probability goes up
C's probability up β†’ C's odds go down
Other ranges β†’ Their probabilities go down
Other ranges β†’ Their odds go up

​
Why Logarithmic?

The β€œlogarithmic” part means:
  • Small bets move prices a little
  • Large bets move prices more, but with diminishing impact
  • No bet can move prices to 0% or 100%
This prevents manipulation and keeps the market stable.

​
A Visual Example

​
Starting State

New market, no bets yet. All ranges equally likely: 
Range A: 20% (5x odds)  β–ˆβ–ˆβ–ˆβ–ˆ
Range B: 20% (5x odds)  β–ˆβ–ˆβ–ˆβ–ˆ
Range C: 20% (5x odds)  β–ˆβ–ˆβ–ˆβ–ˆ
Range D: 20% (5x odds)  β–ˆβ–ˆβ–ˆβ–ˆ
Range E: 20% (5x odds)  β–ˆβ–ˆβ–ˆβ–ˆ

​
After Alice Bets $100 on Range C

Range A: 17% (5.9x odds)  β–ˆβ–ˆβ–ˆ
Range B: 17% (5.9x odds)  β–ˆβ–ˆβ–ˆ
Range C: 32% (3.1x odds)  β–ˆβ–ˆβ–ˆβ–ˆβ–ˆβ–ˆ  ← Alice's bet
Range D: 17% (5.9x odds)  β–ˆβ–ˆβ–ˆ
Range E: 17% (5.9x odds)  β–ˆβ–ˆβ–ˆ
What happened:
  • Range C probability increased (more demand)
  • Range C odds decreased (worse deal now)
  • Other ranges’ odds improved (less demand)

​
After Bob Bets $200 on Range C

Range A: 14% (7.1x odds)  β–ˆβ–ˆ
Range B: 14% (7.1x odds)  β–ˆβ–ˆ
Range C: 44% (2.3x odds)  β–ˆβ–ˆβ–ˆβ–ˆβ–ˆβ–ˆβ–ˆβ–ˆβ–ˆ  ← Alice + Bob
Range D: 14% (7.1x odds)  β–ˆβ–ˆ
Range E: 14% (7.1x odds)  β–ˆβ–ˆ
What happened:
  • Range C moved even more
  • But not twice as much (logarithmic diminishing returns)
  • Other ranges now have great odds for contrarians

​
Key Properties of LMSR

​
1. Infinite Liquidity

You can always place a bet. There’s no order book, no waiting for counterparties. 
Traditional exchange: "No sellers at this price, order not filled"
LMSR: "Here's your quote, always available"

​
2. Bounded Loss

The market maker (protocol) has a known maximum loss. It can’t go bankrupt. 
Initial liquidity: $10,000
Max possible loss: ~$5,000 (50%)
This is calculated mathematically, guaranteed.

​
3. Path Independence

The final market state depends only on total bets, not the order they came in. 
Scenario A: Alice bets $100, then Bob bets $100
Scenario B: Bob bets $100, then Alice bets $100

Final prices: SAME in both scenarios
This means no one gets an unfair advantage by timing.

​
4. No Manipulation

You can’t β€œpump and dump” the market: 
Whale buys $10K to move price up
Whale tries to sell at new price
Selling moves price back down
Whale ends up roughly where they started (minus fees)

​
The Alpha Parameter

You might hear about β€œalpha” (Ξ±) in Skepsis discussions. Here’s what it means:

​
What Alpha Does

Alpha controls how sensitive prices are to bets:
AlphaEffectWho Benefits
High Ξ±Prices move slowlyLarge traders, stable odds
Low Ξ±Prices move quicklySmall traders, volatile odds

​
The Skepsis Approach

We use dynamic alpha that adjusts based on market liquidity: 
Formula: Ξ± = pool_balance / (2 Γ— ln(n))

Where:
- pool_balance = current liquidity in the market
- n = number of buckets/ranges
- ln = natural logarithm

​
Why This Matters

  • More liquidity = higher alpha = more stable prices
  • Less liquidity = lower alpha = prices move more per bet
This means:
  • Popular markets are more stable
  • Small markets are more volatile (but potentially more profitable)

​
How Prices Actually Calculate

Let’s peek under the hood (simplified):

​
The Cost Function

Cost to buy shares = C(after) - C(before)

Where C(q) = Ξ± Γ— ln(Ξ£ e^(qi/Ξ±))

Don't panic! Let's break it down:
- q = shares in each bucket
- Ξ± = liquidity parameter
- The formula sums up exponentials

​
In Plain English

  1. Look at current state of all buckets
  2. Calculate a β€œcost score” (C_before)
  3. Add your shares to your bucket
  4. Calculate new β€œcost score” (C_after)
  5. You pay the difference

​
Why Exponentials?

The exponential function creates:
  • Smooth price transitions
  • No sudden jumps
  • Prices always between 0and0 and 1 per share
  • Natural probability interpretation

​
LMSR vs Other Systems

​
vs Order Books (Traditional Exchanges)

FeatureOrder BookLMSR
LiquidityDepends on tradersAlways available
SpreadVariablePredictable
Fill guaranteeNot guaranteedAlways fills
ComplexitySimple conceptMath-heavy

​
vs AMMs (Uniswap-style)

FeatureUniswap AMMLMSR
Use caseToken swapsPredictions
Price range0 to ∞0% to 100%
Impermanent lossYesNo (bounded loss)
Sum to 100%?NoYes (probabilities)

​
Why LMSR is Perfect for Prediction Markets

  1. Probabilities naturally sum to 100% β€” Unlike token prices, predictions must add up
  2. Bounded loss β€” Market can always pay winners
  3. Always liquid β€” No waiting for counterparties
  4. Incentivizes truth β€” Honest predictions are rewarded

​
The Bottom Line

You don’t need to understand the math to use Skepsis. Just know:
  • βœ… Prices automatically adjust based on betting activity
  • βœ… More popular ranges = lower odds
  • βœ… Less popular ranges = higher odds
  • βœ… You can always bet (infinite liquidity)
  • βœ… Your payout is guaranteed (deterministic)
The LMSR handles all the complexity behind the scenes.

​
Want to Go Deeper?

For the mathematically curious:

​
Next Steps

Continuous vs BinaryWhy distributions beat yes/noContinuous vs Binary
EconomicsWhere does the money come from?Economics