Prediction Market AMMs: LMSR vs Constant Product vs Order Book

AbstractEvent markets can be run by an order book, a scoring-rule market maker (LMSR) or a constant-product pool. We derive the price-impact cost of each for a binary outcome and compare them with depth matched at 50¢. Matched so that the first cent of impact costs the same $2.02, moving the price from 50¢ to 99¢ costs $391 under LMSR (b = 100) and $895 in a fixed-product pool. The pool is far deeper in the tails, but the liquidity provider's worst-case loss is the whole $100 of funding, against the LMSR's bounded b·ln 2 = $69.31. Order books dominate once professional liquidity arrives; AMMs remain the right tool for launch and for long-tail markets.

Why an automated market maker at all#

A new event market has a cold-start problem. Nobody wants to post the first order in an empty book, and a trader with information has no counterparty. Hanson's insight (2003, 2007) was that a market scoring rule can act as a counterparty of last resort: an algorithm that always quotes, moves its price as it is traded against, and whose total loss is bounded and known in advance. The bound is the price a market sponsor pays for the information the market produces.

Three designs are in common use for binary outcomes:

Table 1. Three ways to make a binary market.
Mechanism	Who provides liquidity	Price rule	Worst-case loss	Where it fits
Order book (CLOB)	Market makers and traders	Best resting bid/ask	None for the venue; risk sits with makers	Liquid markets with professional makers
LMSR	Market sponsor (subsidy)	Softmax of outstanding shares	b ln n, fixed in advance	Launch, long-tail and combinatorial markets
Fixed-product pool	Liquidity providers (pooled)	Ratio of pool reserves	Up to the full funding	Permissionless on-chain markets

LMSR in one page#

With outstanding Yes and No shares $q_{y}, q_{n}$ and liquidity parameter $b$ , the LMSR charges traders according to the cost function

C (q_{y}, q_{n}) = b ln (e^{q_{y} / b} + e^{q_{n} / b}), p_{y} = \frac{e ^{q_{y} / b}}{e ^{q_{y} / b} + e ^{q_{n} / b}}

(1)

A trade costs $C (after) - C (before)$ . Moving the Yes price from $p_{0}$ to $p_{1} > p_{0}$ by buying Yes costs

cost_{LMSR} (p_{0} \to p_{1}) = b ln \frac{1 - p _{0}}{1 - p _{1}}

(2)

and the sponsor's worst-case loss over any sequence of trades is $b ln 2$ for a binary market ( $b ln n$ for $n$ outcomes). The LMSR note derives these and simulates how $b$ trades subsidy against accuracy.

Fixed-product pools for binary outcomes#

A fixed-product market maker (FPMM) holds reserves of Yes tokens $y$ and No tokens $n$ and keeps $y n = k$ . A Yes and a No token together are always worth $1, since exactly one of them pays. To buy Yes with $x$ dollars, the pool mints $x$ complete sets, adds them to its reserves, and releases Yes tokens until the product returns to $k$ . The marginal Yes price is

p_{y} = \frac{n}{y + n}

(3)

Solving for the reserves at a given price and differencing, the cost of moving the price from $p_{0}$ to $p_{1}$ is

cost_{FPMM} (p_{0} \to p_{1}) = k (\frac{p _{1}}{1 - p _{1}} - \frac{p _{0}}{1 - p _{0}})

(4)

Compare equation (2). LMSR cost grows like $ln \frac{1}{1 - p}$ as the price approaches 1; FPMM cost grows like $(1 - p)^{- 1/2}$ , which is much faster. Constant-product pools become very deep near the tails.

Matched depth: exact costs#

To compare like with like, we funded an FPMM so that the first cent of price impact at 50¢ costs exactly what it costs in an LMSR with $b = 100$ : $2.02. That requires $k \approx$ 100.0, a pool seeded with about $100 of complete sets.

Table 2. Dollar cost of moving the Yes price, with depth matched at 50¢ (computed in research/amm.py from equations 2 and 4).
Price move	LMSR (b = 100)	Fixed-product pool	Ratio
50¢ → 51¢	$2.02	$2.02	1.00×
50¢ → 60¢	$22.31	$22.47	1.01×
50¢ → 75¢	$69.31	$73.20	1.06×
50¢ → 90¢	$160.94	$199.99	1.24×
50¢ → 99¢	$391.20	$894.93	2.29×
90¢ → 95¢	$69.31	$135.88	1.96×
90¢ → 99¢	$230.26	$694.94	3.02×
5¢ → 1¢	$160.94	$559.06	3.47×

LMSR, b = 100Fixed-product pool, matched at 50¢

Figure 1. Marginal depth across the price range. LMSR depth is symmetric and modest at the extremes; the fixed-product pool's depth explodes toward 0 and 1.

The two are indistinguishable for small moves around the middle. They diverge in the tails: pushing a contract from 90¢ to 99¢ costs three times as much in the pool. That is good for traders who want to trade size in near-certain markets. It is bad for the liquidity provider, because the trader doing the pushing is usually the one who knows the outcome.

Inside a fixed-product trade#

A pool holds 100 Yes and 100 No tokens ( $k = 10, 000$ , price 50¢). A trader pays $10 for Yes. The pool mints 10 complete sets (10 Yes + 10 No), holding 110 Yes and 110 No, then releases Yes until the product is back to $k$ : it keeps $10, 000/110$ = 90.91 Yes and pays out 19.09 Yes tokens. The trader's average price is 52.38¢ per Yes token, and the new marginal price is 54.75¢. A second $10 at the new state costs more per token. Price impact compounds exactly as in equation (4).

Notice what the pool now holds: more No tokens than Yes. If Yes wins, the pool's No tokens are worthless and its liquidity providers have lost value relative to simply holding the original $100. That is the mechanism behind the worst-case loss below, and the reason pool fees exist.

Who pays, and how much#

The LMSR sponsor's loss is bounded by $b ln 2$ = $69.31 at $b = 100$ . A fixed-product pool funded with $100 of complete sets can lose almost all of it: if informed traders drive the Yes price to near 1 and Yes wins, the pool is left holding mostly worthless No tokens. Pool designs recover some of this through trading fees, which is why on-chain prediction-market pools typically charge a fee per trade. The underlying economics are the same as the adverse selection faced by any market maker, with the difference that the pool cannot widen its quotes or stop quoting.

In our LMSR simulation (budget-constrained traders with noisy signals, 6,000 markets per setting), the realised mean subsidy was a steady 14–20% of the worst-case bound:

Table 3. LMSR sponsor loss by liquidity parameter (research/amm.py).
b	Worst case b ln 2	Mean loss	99th percentile loss	Mean / worst case
10	$6.93	$0.95	$6.62	14%
25	$17.33	$3.12	$16.15	18%
50	$34.66	$6.24	$31.16	18%
100	$69.31	$13.26	$60.55	19%
200	$138.63	$27.29	$117.58	20%
400	$277.26	$53.88	$221.42	19%

When to use which#

Order books win wherever professional market makers are present. Makers can price adverse selection, widen into news and hedge across contracts; an AMM can do none of these, so it either subsidises informed traders or charges everyone a fee to cover them. Most high-volume event venues today run order books for this reason, and several on-chain venues that launched with pools later moved to them.

LMSR remains the best design for a sponsor who wants information and is willing to pay a known, capped amount for it: internal corporate forecasting markets, research markets, and long-tail contracts that will never attract a professional maker. It also extends cleanly to many outcomes and combinatorial questions, which order books handle badly.

Fixed-product pools make sense when liquidity must be permissionless and pooled from many small providers, and when fees can compensate for the tail exposure. They should be paired with a mechanism that pauses or re-centres the pool before resolution; otherwise the last informed trader takes most of the pool.

A hybrid is common in practice: an AMM seeds the market and an order book sits on top, with professional makers taking over as volume arrives. Our venue programme is built for exactly that handover.

Questions#

What is an automated market maker in a prediction market?

An algorithm that always quotes a price and trades against anyone, adjusting the price as it accumulates positions. In prediction markets the main designs are Hanson's logarithmic market scoring rule (LMSR) and constant-product (fixed-product) pools.

Is LMSR better than a constant-product AMM?

They make different trade-offs. With depth matched at 50¢, a fixed-product pool is far deeper near 0 and 1 but its liquidity providers can lose almost all their funding. LMSR's loss is capped at b·ln 2 for a binary market.

Why do large prediction markets use order books instead of AMMs?

Professional market makers in an order book can widen quotes into news, manage inventory and price adverse selection. A passive AMM cannot, so it either subsidises informed traders or must charge fees high enough to cover them.

References#

Hanson, R. (2003). Combinatorial information market design. Information Systems Frontiers, 5(1), 107–119.
Hanson, R. (2007). Logarithmic market scoring rules for modular combinatorial information aggregation. Journal of Prediction Markets, 1(1), 3–15.
Chen, Y. and Pennock, D. M. (2007). A utility framework for bounded-loss market makers. Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence (UAI).
Angeris, G. and Chitra, T. (2020). Improved price oracles: constant function market makers. Proceedings of the 2nd ACM Conference on Advances in Financial Technologies (AFT).

DisclosureThis note is research and education. It is not investment advice or an offer to trade, and nothing in it describes the firm's own results. Code and outputs: research/amm.py in our research repository, reproducible with a fixed seed.