Logarithmic Market Scoring Rule (LMSR)
Hanson's logarithmic market scoring rule turns the log score into an automated market maker with a closed-form cost function over outstanding shares $q$:
$C(q)=b\,\log\textstyle\sum_i e^{q_i/b}, \qquad
p_i(q)=\dfrac{e^{q_i/b}}{\sum_j e^{q_j/b}}$
Prices are non-negative and sum to one — the implied probabilities — and the market maker's worst-case loss is bounded by $b\log n$. The liquidity parameter $b$ trades depth against that subsidy. LMSR is the entropic special case of the generic cost-function market maker.
Code: mechanisms/lmsr.py ·
Demo: examples/sim_lmsr.py ·
Research: market-scoring-rules-and-amms.md