Comparing AMM Liquidity Models: LMSR vs LSAMM vs CPMM vs Hybrid

Why the choice of AMM matters

Automated market maker design is the single biggest engineering decision a prediction market platform makes. Pick the wrong AMM and you ship a product that is either too stiff to update on news, too expensive to subsidise, or too rigid for the long-tail markets where prediction platforms create unique value. Pick the right one and your platform feels alive — prices update in real time, spreads stay reasonable across the lifecycle, and your subsidy budget lasts.

This post compares the four AMM families that dominate prediction markets today: fixed LMSR, dynamic LSAMM, constant-product CPMM, and hybrid AMM-order book designs. For each we look at how the algorithm behaves at scale, how operators tune it, and where it fits in a real product. If you want the foundational math, our LSAMM algorithm deep dive is the canonical reference. For the broader category context, see introduction to prediction markets.

The four families at a glance

Most AMMs in production today are variations of these four mechanisms.

AMM family	Liquidity source	Curve type	Best for
LMSR (Hanson)	Operator subsidy, fixed b	Log-sum-exp	Cold-start markets, bounded subsidies
LSAMM	Operator subsidy, dynamic α·Q_total	Log-sum-exp with scaling	Production prediction platforms
CPMM (Uniswap)	LP-funded, x·y = k	Constant product	Token swaps, hybrid prediction markets
Hybrid AMM + order book	Operator + market makers	AMM seed → order book takeover	High-volume binary markets

The math behind LMSR and LSAMM is in the LSAMM explainer; the original LMSR specification is in Robin Hanson's 2003 paper; the CPMM math comes from the Uniswap v2 whitepaper. The hybrid pattern is best documented in the dYdX v3 design notes and the early Polymarket migration to a hybrid CTF + book on Polygon.

Fixed LMSR: simple, bounded, predictable

LMSR remains the cleanest mental model for prediction market AMMs. The cost function is a single log-sum-exp over share counts, scaled by a fixed liquidity parameter b:

C(q) = b * ln(Σ exp(qi / b))

The operator's maximum loss is b · ln(n) for n outcomes. That is the single biggest reason LMSR was the dominant design from 2003 to roughly 2020 — operators could plan their subsidy budget on day one and never worry about runaway losses.

Where fixed LMSR wins:

• Academic and research deployments. The Iowa Electronic Markets and most university course markets still use LMSR because the bounded-loss property maps cleanly to a fixed academic budget.
• Internal corporate prediction markets. A company running an internal market on a single question (e.g., "Will we ship Q4 OKRs?") wants a fixed subsidy and a small set of traders. LMSR is the right call.
• Cold-start markets where you do not yet know if there will be any volume. You commit a small b and walk away.

Where fixed LMSR loses:

• Production retail markets. The fixed b means the 10,000th trader pays the same slippage as the 1st trader. That is fine for a 50-person research panel and catastrophic for a market with $1M+ in volume.
• News-driven markets. A fixed b cannot widen its quote when news arrives; either prices move violently or they do not move at all, depending on whether b was set high or low.
• Multi-outcome markets at scale. Subsidy scales as b · ln(n), so the operator has to commit increasing subsidy for every outcome added. That gets expensive fast.

The LMSR trade-off is essentially: one knob, one budget, lifetime. If your market only needs to serve a few dozen traders for a fixed window, LMSR is hard to beat.

Dynamic LSAMM: liquidity that scales with volume

LSAMM keeps the LMSR cost function but replaces the fixed b with L_dynamic = α · Σ qi, so liquidity grows as the market accumulates volume. The result is a self-regulating market: very responsive when small, robust when large.

Where LSAMM wins:

• High-volume retail prediction markets. The reason platforms like the modernised Polymarket variants and several next-gen prediction venues have migrated to LSAMM is exactly this — late traders see tight spreads, while early traders still get price discovery.
• Markets with uncertain demand. If you do not know whether a market will pull $1k or $1M in volume, LSAMM gives you a single configuration that handles both.
• Multi-outcome markets. Subsidy still scales with ln(n), but the effective b adapts to the actual volume, so you do not pay for capacity you never use.

Where LSAMM struggles:

• Bootstrapping with truly zero volume. When Q_total = 0, the cost function evaluates 0/0. Production code has to short-circuit to uniform prices 1/n, which is fine but adds an edge case.
• Operator subsidy is not bounded in the same neat way. The subsidy depends on α and the actual trajectory of Q_total. You can still bound it analytically, but it is no longer a single number on day one.
• Implementation complexity. Lazy updates of q_max, caching of L_dynamic, fixed-point arithmetic — there are more sharp edges than in LMSR.

If you are building a serious prediction market product today, LSAMM is the default to beat. The full math, calibration loop, and production patterns are in our LSAMM algorithm deep dive.

Constant-product CPMM: LP-funded, not probability-aware

Uniswap-style constant-product market makers price two assets with the invariant x · y = k. The price of asset A in units of asset B is y / x, which shifts as traders swap between the two. There is no operator subsidy — liquidity providers (LPs) deposit pairs and earn fees on trades.

In a prediction market context, CPMM is sometimes used as the underlying primitive for binary outcome contracts. Polymarket's early architecture sat a CPMM on top of Gnosis Conditional Tokens to provide liquidity for Yes/No pairs.

Where CPMM wins:

• LP-funded liquidity. You do not need an operator subsidy budget. Users provide liquidity and get paid by fees.
• Composable with DeFi. CPMM positions are ERC-20 LP tokens, easy to integrate into yield farms, lending markets, and aggregators.
• Two-outcome binary markets can sit on a single CPMM pair, which is the simplest possible implementation.

Where CPMM struggles for prediction markets:

• Prices do not naturally sum to 1. A CPMM on Yes/No can produce a price pair like 0.55/0.40, which sums to 0.95 — there is missing probability that comes from the impermanent-loss curve and the LP fee structure. Operators have to add normalisation logic or accept the leak.
• Impermanent loss is a feature, not a bug, that you do not want in prediction markets. In token swaps, IL is a price for LPs offering arbitrage liquidity. In a prediction market, IL becomes a hidden tax on the LP that materialises exactly when the market is most informative.
• No bounded loss for the platform. LPs absorb the risk, which is fine for tokens but can be socially unattractive for prediction markets where users associate the platform with the outcome.

CPMM remains a useful primitive for hybrid designs, but a pure CPMM is rarely the right choice for a prediction market on its own.

Hybrid AMM + order book: best of both worlds

The most sophisticated production designs combine an AMM seed with an order book layer. The AMM provides initial liquidity and ensures the market is always tradable; the order book takes over as traders post tighter quotes. Kalshi's hybrid model, the modernised Polymarket stack, and several institutional venues all converge on this pattern.

How it works in practice:

1. The market launches with an AMM (LMSR or LSAMM) providing the initial quote.
2. Market makers begin posting limit orders inside the AMM spread.
3. As the order book deepens, the AMM becomes a backstop — only filling orders when the book has nothing better.
4. At maturity, almost all flow goes through the order book; the AMM stays as a tail-risk insurance against book emptying.

Where hybrid designs win:

• High-volume markets get the tight spreads of an order book without losing the bootstrap-from-zero property of an AMM.
• Long-tail markets never go dry, because the AMM always quotes something.
• Institutional traders can post resting orders, which is impossible in pure AMM designs.

Where hybrid designs are hard:

• Implementation complexity. You are running two pricing engines side by side and need consistent state.
• MEV exposure. Hybrid systems on-chain need careful protection against sandwich attacks that arbitrage the gap between AMM and book quotes.
• Operational overhead. Market makers must be onboarded, monitored, and incentivised, which is more work than just running an AMM.

If your platform is sophisticated enough to handle the complexity, hybrid designs are the long-term destination. Most platforms start with LSAMM and move to hybrid once volume justifies it.

How to choose for your platform

The right AMM depends on your product, your audience, and your operational maturity.

Choose fixed LMSR if:

• You are running a research or academic market with a known, finite participant pool.
• Your subsidy budget is fixed and small.
• You do not expect volume to grow beyond ~10x your seed.

Choose LSAMM if:

• You are building a production retail prediction market.
• You expect volume to vary by orders of magnitude across markets.
• You want a single configuration that handles both cold-start and mature phases.

Choose CPMM if:

• You are building inside the DeFi stack and need LP-funded liquidity.
• Your outcome contracts are already tokenised (e.g., on Conditional Tokens).
• You are willing to accept some probability leakage and add normalisation.

Choose hybrid if:

• You are an established platform with market-making partnerships.
• Your top markets do >$100k daily notional and you can support an order book.
• You have engineering capacity to run two pricing engines safely.

For most teams reading this, the realistic path is LSAMM today, hybrid tomorrow. That is the pattern several mature prediction platforms have followed, and the math behind both is reasonably well understood.

Subsidy math by family

A practical comparison: how much subsidy does each design need to deliver, say, 1% round-trip slippage on a $10,000 trade?

AMM	Subsidy to deliver 1% slippage on $10k binary trade
LMSR	b ≈ $10,000, fixed for lifetime
LSAMM	α ≈ 0.05, scales with Q_total
CPMM	LP-funded; subsidy is the deposit
Hybrid	AMM seed ≈ $5k, rest covered by makers

The headline differences are not in the cost of a single trade — they are in how the cost evolves. An LMSR subsidy is paid once and sits there. An LSAMM subsidy starts small and grows as volume justifies it. A CPMM relies on LPs to materialise, and a hybrid pays only enough AMM seed to bootstrap the order book.

Operators frequently underestimate the variance in their subsidy math. A model that says "$10k of LMSR subsidy will cover us" can be off by 5x once a single news event triples the market's volume in a day. LSAMM and hybrid designs are more robust to that variance because the effective liquidity scales with what actually happens.

Mistakes operators repeatedly make

Three patterns we see across prediction market launches.

Treating b like a magic constant. New operators set b based on a friend's recommendation, ship the market, and discover at the first volume spike that the slippage was wrong by an order of magnitude. The fix is to think about b (or α) as a budget, not a style. Estimate the daily volume, the acceptable slippage, and back out the parameter.

Underestimating the variance. A market that averages $5k notional per day may see $50k in a single hour during a news event. If your subsidy is sized for the average, you ship a market that becomes extremely responsive (good!) but also extremely costly (bad!) on news days. LSAMM mitigates this naturally; LMSR does not.

Ignoring fee leakage. Every AMM has fees, and fees compound. A 1% AMM fee plus a 30 bps LP cut plus a 50 bps gas overhead can quickly eat 2–3% of a trader's edge. The most successful platforms keep total fees under 1% across the full round trip.

For trader-side counterpart to these design lessons, see market microstructure for forecasting and our risk management playbook.

Practical implications

For builders, the lesson is to treat market design, content, and liquidity as one system. A category page, a resolution rule, and an AMM parameter all influence whether users trust the probability they see. When these pieces are aligned, the market becomes easier to discover, easier to trade, and easier to explain. See our category design and binary vs multiple outcome markets posts for the upstream design decisions.

For traders, the practical implication is discipline. Do not read a price without asking how it was formed, how deep it is, how it resolves, and what cost you will pay to enter or exit. A useful forecast is not just a number; it is a number connected to process, liquidity, and incentives. Our bankroll framework and forecast calibration playbook walk through how to translate AMM properties into trade sizing decisions.

What to watch next

The next generation of prediction markets will reward teams that combine strong infrastructure with clear editorial context. The winners will not only launch more markets — they will help users understand why each market exists, what it measures, and how to act on the signal without taking unnecessary risk. We unpack the macro outlook in the future of prediction markets in 2026.

Two specific trends to watch:

1. AMM-aware fee structures. Platforms are starting to vary fees based on the cost the AMM is bearing — lower fees on deep, mature markets, higher on long-tail. Expect this to become the default.
2. Cross-market liquidity routing. Aggregators that route trades to the AMM with the best effective price across multiple venues are emerging. This will pressure single-venue AMMs to keep their math current.

Frequently Asked Questions

Which AMM is used by Polymarket?

Polymarket's current production stack uses a hybrid design: AMM-based liquidity for cold-start markets, with order book overlays on the deepest contracts. The underlying primitive is Gnosis Conditional Tokens, with custom liquidity logic on top. The exact AMM curve has evolved over time; the public technical docs are the best reference.

Is LSAMM strictly better than LMSR?

For production prediction markets, yes. For research and academic deployments with bounded subsidies and small participant pools, LMSR is still simpler and easier to reason about. The right question is "which constraints am I optimising for?" rather than "which is better in the abstract".

Can I use Uniswap directly for a prediction market?

Technically yes, but you will fight the math. A Uniswap pair on Yes/No tokens does not produce a probability simplex, prices do not sum to 1, and LPs eat impermanent loss exactly when the market is most informative. If you want CPMM-style liquidity for a prediction market, you typically wrap a custom contract around the underlying tokens to add normalisation.

What is the cheapest AMM to subsidise?

Hybrid designs, because you only need to seed the AMM until the order book takes over. LSAMM is the cheapest pure AMM, because the subsidy scales with actual volume rather than your guess at launch. LMSR is the most predictable but often the most expensive in expectation because operators tend to oversize b to avoid embarrassment.

Do AMMs work for multi-outcome markets?

Yes. LMSR and LSAMM both extend naturally to n outcomes — the cost function is just a log-sum-exp over n terms instead of 2. CPMM is harder to extend naturally; you typically need n pairs sharing a common collateral, which complicates the math. Hybrid designs handle multi-outcome cleanly by overlaying an order book on top of a multi-outcome AMM.

How do I tune α for LSAMM?

Start at α = 0.05. Increase if traders complain about slippage on small markets. Decrease if you want sharper price discovery in cold-start markets. The most rigorous tuning is an A/B test on two parallel markets at α = 0.03 and α = 0.07, measured against trader satisfaction and operator subsidy burn. The full calibration loop is in our LSAMM deep dive.

What is impermanent loss and why does it matter for prediction markets?

Impermanent loss is the unrealised loss an LP takes when the relative price of the two assets in a CPMM diverges from the deposit ratio. In a token-swap context it is the price LPs pay for offering arbitrage liquidity. In a prediction market, IL materialises exactly when the market is doing its job — pricing new information — which makes LPs reluctant to provide liquidity in informative markets. This is why pure CPMMs underperform LMSR/LSAMM for prediction.

Can I switch AMM after launch?

Not without disruption. The AMM is baked into the cost function that every trade has used, so changing it strands existing positions or requires a complex migration. Most platforms instead launch new markets on the new AMM and let old markets run to resolution on the old one. The lesson: pick your AMM carefully because reversibility is expensive.

Where to go next

You now have a working framework for picking the right AMM. The natural follow-ups:

• For the mathematical foundation, read the LSAMM algorithm deep dive.
• For market structure decisions upstream of the AMM, see binary vs multiple outcome markets and category design for market discovery.
• For how AMM choices affect actual trading, see market microstructure for forecasting.
• For the platform context, introduction to prediction markets ties everything together.

The right AMM is the one that matches your platform's audience, volume profile, and operational maturity. Start with LSAMM, add an order book when justified, and never forget that the AMM is just one part of a product that has to be discoverable, tradable, and resolvable.