> For the complete documentation index, see [llms.txt](https://docs.choice.markets/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.choice.markets/lmsr-explained.md).

# LMSR Explained

#### What LMSR Is

LMSR stands for **Logarithmic Market Scoring Rule**.

It is a cost-function-based automated market maker designed for prediction markets. Instead of requiring a live counterparty for every trade, LMSR uses a mathematical rule to price outcomes continuously.

Key properties:

* always-on liquidity
* prices behave like probabilities
* smooth repricing after trades
* bounded loss for the market maker

#### Intuition

Each outcome has an outstanding quantity of shares, denoted by `q_i`.

The LMSR cost function is:

```
C(q) = b * ln(sum_i exp(q_i / b))
```

Where:

* `q_i` is the quantity of shares for outcome `i`
* `b` is the liquidity parameter

The instantaneous price for outcome `i` is:

```
p_i(q) = exp(q_i / b) / sum_j exp(q_j / b)
```

This formula ensures that:

* each outcome price stays between `0` and `1`
* all outcome prices sum to `1`
* more demand for one outcome raises its price

#### Cost of a Trade

If the market state changes from `q` to `q'`, the trader pays:

```
Trade Cost = C(q') - C(q)
```

This is the amount required to move the market from the old state to the new state.

#### The Role of `b`

The liquidity parameter `b` controls price sensitivity.

* larger `b` means deeper liquidity and smaller price movement per trade
* smaller `b` means thinner liquidity and sharper price movement per trade

For a market with `n` outcomes, LMSR gives a bounded worst-case loss of:

```
b * ln(n)
```

This bounded-loss property is one reason LMSR is so useful for long-tail prediction markets.

#### Why LMSR Fits Choice Markets

Choice Markets is designed to support a large number of socially sourced markets, including markets that may not immediately attract dense liquidity.

LMSR is a strong fit for that environment because it:

* keeps new markets tradable from the start
* supports long-tail market supply
* turns social attention into executable pricing quickly


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