LMSR (Logarithmic Market Scoring Rule): Definition & Formula

DefinitionAn automated market maker, proposed by Robin Hanson, that always quotes prices for every outcome as a softmax of outstanding shares, with a worst-case loss for the sponsor of b·ln n.

Formula#

C (q) = b ln j \sum e^{q_{j} / b}, p_{i} = \frac{e ^{q_{i} / b}}{\sum _{j} e ^{q_{j} / b}}, max loss = b ln n

A trade costs the change in $C$ . For a binary market, moving the price from $p_{0}$ to $p_{1}$ by buying Yes costs $b ln \frac{1 - p _{0}}{1 - p _{1}}$ .

Choosing b#

Larger $b$ means a deeper market and a larger subsidy. In our simulation of budget-limited traders, accuracy was best with $b$ around 10–20 times a typical trader's budget, and the realised subsidy averaged 14–20% of the $b ln 2$ bound (derivation and simulation).

Versus other designs#

Compared with a constant-product pool at matched depth, LMSR is shallower near 0 and 1 but has a hard loss cap. Compared with an order book, it never runs out of liquidity but cannot price adverse selection (comparison).

Questions#

What does LMSR stand for?

Logarithmic market scoring rule, an automated market maker for prediction markets introduced by Robin Hanson.

What is the maximum loss of an LMSR?

b·ln n for n outcomes; for a binary market with b = 100, about $69.31.