Prediction market AMMs — LMSR, pm-AMM, and volume accounting

1. TL;DR

An outcome token — worth $1 if an event happens and $0 if it doesn’t — behaves nothing like an ordinary asset: its volatility is lowest when the outcome is a foregone conclusion and spikes exactly when the market matters most, right before expiry with the odds near 50/50. Ordinary constant-product pools and even the classic prediction-market mechanism, Hanson’s logarithmic market scoring rule (LMSR), give liquidity providers wildly uneven losses as a result; Paradigm’s newer pm-AMM invariant is designed instead to lose value to arbitrage at a constant, predictable rate regardless of the current probability. Despite this research, most real prediction-market volume — Polymarket, Kalshi — still clears on order books rather than AMMs, and even measuring that order-book volume correctly turns out to be its own non-trivial problem.

2. Explain it simply

Analogy

Imagine a fairground booth taking side bets on a coin that is about to be flipped live on stage. In the minutes before the flip, tiny scraps of new information — someone glimpses which side is up on the referee’s practice flip — can swing the true odds wildly, so the booth needs a completely different pricing approach right before the flip than a shopkeeper selling apples, whose price drifts slowly and predictably all year regardless of the time of day.

ELI5

A prediction market sells two tokens tied to one yes-or-no question: one pays out if the answer turns out to be “yes,” the other if it’s “no.” Right when almost everyone already knows the answer, the price barely moves no matter who trades. But right before the answer becomes public — and especially if the odds are close to a coin flip — a trade based on slightly better information can swing the price a lot, and whoever supplied the tokens for people to trade against loses money exactly in that moment. Building a market maker that doesn’t get badly hurt in that one narrow, dangerous window turns out to be a different design problem than building one for ordinary buying and selling.

Step-by-step walkthrough

Scenario: a market on “will it rain in this city tomorrow,” with an automated market maker holding reserves of RAIN and NO-RAIN tokens, both worth $1 at resolution for the correct side.

1. Far from expiry (state: 24 hours to go, market at 50% probability). The pool holds roughly equal value in RAIN and NO-RAIN reserves; a trade of ordinary size barely moves the reported probability, because there’s a full day left for the outcome to swing back.
2. A weather model updates (state: new forecast data becomes public). A trader who has seen the update buys RAIN tokens from the pool, which raises the implied probability of rain and shrinks the pool’s RAIN reserves relative to NO-RAIN.
3. Close to expiry (state: 1 hour to go, market at 50%). The same-sized trade now swings the probability much further, because there is far less remaining time for new information to arrive and be priced in — the pool’s tokens are much more sensitive to trading near the deadline.
4. At resolution (state: it rains; RAIN pays $1, NO-RAIN pays $0). Whatever value the pool had left in NO-RAIN tokens goes to zero; the liquidity provider’s total exposure to this event is now fully realized.

Common misconceptions

• Myth: Prediction markets are just AMMs like Uniswap. Reality: “Most crypto prediction market liquidity is available on orderbooks, rather than AMMs, despite AMMs being dominant for most other decentralized exchange volume” (Moallemi & Robinson, Paradigm, 2024-11).
• Myth: LMSR and pm-AMM solve the same problem the same way. Reality: LMSR concentrates liquidity around 50/50 regardless of time to expiry; pm-AMM’s entire innovation is making liquidity concentration time-aware, shrinking as expiry approaches (Moallemi & Robinson, 2024-11).
• Myth: Polymarket’s headline volume figure is real, one-sided trading volume. Reality: Summing Polymarket’s on-chain OrderFilled events double-counts both the number of contracts and the USD flow, because the same trade emits a maker-side and a taker-side event (Slivkoff, Paradigm, 2025-12).
• Myth: Providing liquidity to a prediction market is like providing it to any stable, low-volatility pair. Reality: Outcome-token volatility spikes exactly at the moment the market matters most — near 50/50, near expiry — which is also when an LP’s expected losses are worst (Moallemi & Robinson, 2024-11).

If you only remember one thing

An outcome token’s danger to liquidity providers isn’t constant like a normal asset’s — it’s concentrated right at the moment the coin is about to land, which is exactly the moment an ordinary AMM handles worst.

3. How it works

From LMSR to a general AMM

Hanson’s logarithmic market scoring rule prices $n$ outcomes via a cost function $C(q)=b\ln\sum_i e^{q_i/b}$, where $q_i$ is the quantity of outcome $i$ sold and $b$ is a liquidity parameter; the implied price of outcome $i$ is $p_i = \partial C/\partial q_i = e^{q_i/b}/\sum_j e^{q_j/b}$. Angeris and Chitra note LMSR can be written as an invariant-based AMM satisfying $x\ln x + y\ln y = L$ for a two-outcome market, with pool value function $V(P) = L + P\ln P + (1-P)\ln(1-P)$ — proportional to the binary entropy of the implied probability $P$. A plain constant-product market maker (CPMM) with invariant $xy=L$ has pool value $V(P)=L+P\sqrt{L/P}-\sqrt{LP}$.

Loss-versus-rebalancing and “uniformity”

Moallemi and Robinson define a uniform AMM for an asset as one whose expected loss-versus-rebalancing (LVR) is a constant fraction of the pool’s current value, regardless of the current price — Milionis et al. showed geometric-mean market makers are the essentially unique uniform AMMs for assets following ordinary geometric Brownian motion. Outcome tokens don’t follow that process: under Gaussian score dynamics, an underlying score $Z_t$ follows Brownian motion $dZ_t=\sigma\,dB_t$, and the outcome-token price is the probability the score finishes positive, $P_t = \Phi(Z_t/(\sigma\sqrt{T-t}))$ where $\Phi$ is the standard normal CDF. Applying Itô’s lemma, the price process itself simplifies to $dP_t = \dfrac{\phi(\Phi^{-1}(P_t))}{\sqrt{T-t}}\,dB_t$ (with $\phi$ the standard normal density) — volatility that explodes as $T-t\to 0$ exactly at $P=0.5$, which is why the closer to expiry and the closer to a coin flip, the more dangerous the market becomes for an LP. The instantaneous LVR of any AMM with pool value function $V(P)$ under this process is $\mathsf{LVR}_t = -\tfrac12\dfrac{\phi(\Phi^{-1}(P_t))^2}{T-t}V''(P_t)$, and neither the CPMM nor LMSR keeps this ratio constant across prices — both suffer disproportionate losses near $P=0.5$ close to expiry.

The static and dynamic pm-AMM

Solving for the pool value function that makes LVR uniform at every price, for a fixed time-to-expiry, gives the static pm-AMM: $V(P) = L\,\phi(\Phi^{-1}(P))$, with optimal reserves $x^*(P) = L\{\Phi^{-1}(P)P + \phi(\Phi^{-1}(P)) - \Phi^{-1}(P)\}$ and $y^*(P) = L\{\Phi^{-1}(P)P + \phi(\Phi^{-1}(P))\}$, which together satisfy the invariant $(y-x)\,\Phi\!\left(\dfrac{y-x}{L}\right) + L\,\phi\!\left(\dfrac{y-x}{L}\right) - y = 0$. Because the static design’s loss rate relative to its own value still rises as expiry approaches ($\mathsf{LVR}_t = V_t/(2(T-t))$), the dynamic pm-AMM shrinks its liquidity parameter over time, $L_t = L_0\sqrt{T-t}$, which makes the expected LVR constant in absolute terms right up until expiry: $\mathbb{E}[V_t]$ decays linearly to zero and $\mathbb{E}[\mathsf{LVR}_t]=V_0/(2T)$ stays flat — with exactly half the pool’s starting wealth lost to arbitrage in expectation by the time the market resolves.

4. Worked numeric example

Take a liquidity parameter $L=1{,}000$ and compare pool value $V(P)$ across the CPMM, LMSR, and static pm-AMM at $P=0.5$ (a coin-flip market) and $P=0.9$ (a near-resolved one), using $\Phi^{-1}(0.5)=0,\ \phi(0)=0.3989$ and $\Phi^{-1}(0.9)=1.2816,\ \phi(1.2816)=0.1754$:

$P$	CPMM $V=2L\sqrt{P(1-P)}$	LMSR $V=L\{-P\log_2P-(1-P)\log_2(1-P)\}$	pm-AMM $V=L\,\phi(\Phi^{-1}(P))$
0.5	1,000.0	1,000.0	398.9
0.9	600.0	469.0	175.4

Moving from a coin-flip to a near-resolved market, the CPMM retains 60% of its value (600/1,000) and LMSR retains 46.9%, while the pm-AMM retains only 44% (175.4/398.9) — the pm-AMM gives up relatively more value at the extremes and holds relatively more at $P=0.5$, exactly the “concentrate liquidity around 50/50, thin it out at the extremes” behavior the design targets.

Checking uniformity directly. With $T-t=1$, the static pm-AMM’s loss rate is $\mathsf{LVR}_t = V_t/(2(T-t)) = V_t/2$ at any price. At $P=0.5$: $\mathsf{LVR}=398.9/2=199.5$, so $\mathsf{LVR}/V = 199.5/398.9 = 0.500$. At $P=0.9$: $\mathsf{LVR}=175.4/2=87.7$, so $\mathsf{LVR}/V=87.7/175.4=0.500$ — identically 50% of the pool’s value lost per unit time at both prices, even though the dollar amounts differ by more than 2x. That equal ratio, not equal dollar loss, is what “uniform” means.

Reserves at the two prices. At $P=0.5$, symmetry gives $x^*=y^*=L\phi(0)=398.9$. At $P=0.9$: $x^*(0.9)=1{,}000\{1.2816\times0.9+0.1754-1.2816\}=47.2$ and $y^*(0.9)=1{,}000\{1.2816\times0.9+0.1754\}=1{,}328.8$ — checking $y^*-x^*=1{,}281.6=L\Phi^{-1}(0.9)$, matching the invariant exactly.

5. Where it’s used

Ethereum

• No confirmed production deployment of the pm-AMM invariant. As of publication, Moallemi and Robinson describe pm-AMM as a design “that may be useful for bootstrapping passive liquidity on onchain prediction markets” — a research proposal, not a stated live integration (Paradigm, 2024-11).
• Polymarket runs on Polygon (an Ethereum-anchored PoS chain) using a central limit order book (CLOB) via CTF/NegRisk exchange contracts, not an AMM; it saw “over $2 billion in volume in just October 2024” (Paradigm, citing Polymarket, 2024-11). docs.polymarket.com
• LMSR-style AMMs (Augur v1, Gnosis) were the historical precedent that motivated pm-AMM’s comparison baseline, discussed in /exchange/cfmm-math/ and /derivatives/prediction-markets/.

Solana

• Jupiter Forecast — launched 4 June 2026 as “Solana’s first native prediction market,” using competing “Prop AMMs” (proprietary automated market makers) that independently post YES/NO prices; Jupiter’s system “surveys the live quotes from each participating market maker and routes the order to the most competitive one,” the same proprietary-AMM order-flow pattern used elsewhere in Solana spot trading (see /exchange/lp-toxicity-and-jit/). Jup Predict, which wraps Forecast alongside integrated Kalshi (since October 2025) and Polymarket (since February 2026) markets, had accumulated about $17 million in total volume before the Forecast launch, including $5.3 million in April 2026 alone (Solana Compass, secondary, as of 2026-06).
• Prop AMMs quote outcome tokens competitively rather than implementing the static or dynamic pm-AMM invariant directly — no source confirms Jupiter Forecast uses the pm-AMM formula specifically.

6. Risks, attacks, and incidents

• Polymarket/UMA governance attack, 24–25 March 2025. A single actor controlling roughly 5 million UMA tokens (about 25% of the vote) across three wallets manipulated the dispute-resolution vote on a $7 million market, “Will Ukraine agree to Trump’s mineral deal before April?” — the reported odds moved from 9% to 100% and the market resolved “Yes” despite no such agreement having actually occurred at the time (Orochi Network, secondary, accessed 2026-08-29).
• Volume-measurement risk. Polymarket’s trade-settlement contracts emit two representations of the same fill — a “maker-focused” OrderFilled event and a redundant “taker-focused” one — so naively summing all OrderFilled events roughly doubles both the reported token volume and USD cash flow; Slivkoff gives a concrete example where a real $4.13 sale of YES tokens was recorded as $8.26 of volume by this common (flawed) method (Paradigm, 2025-12).
• Adverse selection is inherent, not incidental. pm-AMM’s own motivation is that outcome-token volatility — and therefore LP losses to better-informed traders — peaks exactly when new information is about to become public (e.g. election results, a game’s final minutes), which is structurally the worst possible moment for a passive liquidity provider (Moallemi & Robinson, 2024-11).
• Bounded but real subsidizer risk in LMSR. Hanson’s cost-function design guarantees the market maker’s worst-case loss is bounded (by $b\ln n$ for $n$ outcomes), but that bound is still real capital a subsidizing party must be willing to lose to guarantee the market has liquidity from the outset. mason.gmu.edu/~rhanson/mktscore.pdf

7. Open problems

• Does the “price equals probability” assumption hold? Moallemi and Robinson flag that their model “assumes away important features of the market, including risk and time preferences,” and explicitly leave examining that gap for future study (Paradigm, 2024-11).
• Which events actually fit Gaussian score dynamics? The authors note basketball (frequent scoring) is a better fit than soccer (infrequent scoring), and that one-off surprise events like earthquakes don’t fit the model at all — leaving open how to model AMM liquidity for that broader class of prediction markets (Paradigm, 2024-11).
• Extending “uniform AMM” methodology beyond outcome tokens. The paper’s stated conclusion is that the same methodology “may also be more widely applicable” to bonds, options, and other derivatives whose price processes aren’t geometric Brownian motion — left as future work rather than solved (Paradigm, 2024-11).
• Standardizing volume measurement across platforms. Slivkoff frames the correct fix (one-sided, e.g. taker-side or maker-side volume) as necessary for “apples-to-apples comparisons to other prediction market platforms,” implying today’s cross-platform volume comparisons are not yet standardized (Paradigm, 2025-12).
• Continuous-outcome markets. White’s Distribution Markets paper proposes AMMs over full probability distributions (not just binary or discrete outcomes) using a constant-$L^2$-norm invariant over functions, but notes the continuous case remains the harder, less-explored generalization of the discrete mechanism (Paradigm, 2024-12).

8. Ethereum vs Solana

Aspect	Ethereum	Solana
Dominant mechanism	Central limit order book (Polymarket, on Polygon)	Prop AMMs (Jupiter Forecast, launched 2026-06) plus aggregated CLOB liquidity (Kalshi, Polymarket via Jup Predict)
AMM-native invariant research	LMSR, CPMM, static/dynamic pm-AMM, Distribution Markets (all Paradigm/academic proposals)	No source confirms a live deployment of these specific invariants
Headline volume, dated	Polymarket: “$2 billion in volume in just October 2024” (Paradigm, 2024-11)	Jup Predict: ~$17 million total, $5.3 million in April 2026 alone (Solana Compass, secondary, as of 2026-06)
Known integrity incident	Polymarket/UMA governance attack, March 2025, $7 million market	None found in this research
Volume accounting	Documented double-counting bug in common on-chain analysis methods (Polymarket, 2025-12)	Not analyzed in sources found here

Ethereum-anchored Polymarket is the dominant venue by volume and the one with the most scrutinized market microstructure — both its double-counted OrderFilled events and its UMA oracle’s governance-attack surface are consequences of running a large, order-book-based, oracle-resolved system at scale. Solana’s approach, still emerging as of this research (Jupiter Forecast launched June 2026), routes prediction-market orders through the same competing-market-maker “Prop AMM” pattern that has come to dominate Solana spot trading generally, rather than adopting either LMSR or pm-AMM directly.

9. Reference doc

The reference

pm-AMM: A Uniform AMM for Prediction Markets — Ciamac Moallemi, Dan Robinson (Paradigm), 5 November 2024. paradigm.xyz/writing/pm-amm

Summary of the reference

The post opens by observing an apparent contradiction: prediction-market volume in crypto has taken off, yet most of it clears on order books rather than AMMs, “despite AMMs being dominant for most other decentralized exchange volume.” It argues existing AMMs are a poor fit for outcome tokens because their volatility depends heavily on both the current implied probability and the time remaining, making liquidity providers’ losses inconsistent across the life of a market.

Model. The paper introduces Gaussian score dynamics: an unobserved score process $Z_t$ follows Brownian motion, the event resolves based on its sign at expiry $T$, and the outcome-token price is $P_t=\Phi(Z_t/(\sigma\sqrt{T-t}))$. This is proposed as a reasonable model for markets like basketball games, elections, or asset-price threshold bets, while explicitly not fitting events like earthquakes.

Uniform AMMs. Building on Milionis et al.’s loss-versus-rebalancing (LVR) framework, the paper defines a uniform AMM as one whose expected LVR is a constant fraction of pool value regardless of price, and notes this property is asset-specific: geometric-mean market makers are uniform for assets following geometric Brownian motion, but neither the CPMM nor LMSR is uniform for outcome tokens under Gaussian score dynamics — both lose disproportionately more value at extreme prices, especially near expiry.

Static and dynamic pm-AMM. Solving the ODE for a price-uniform pool value function yields the static pm-AMM, $V(P)=L\phi(\Phi^{-1}(P))$; because its per-dollar loss rate still rises as $T-t\to0$, the paper derives a dynamic version that withdraws liquidity over time, $L_t=L_0\sqrt{T-t}$, achieving constant expected LVR right up to expiry, with half of initial wealth lost to arbitrage in expectation by resolution.

Prior work and conclusion. The paper situates pm-AMM against LMSR (Hanson), StableSwap (correlated/mean-reverting assets) and YieldSpace (bonds) as prior attempts to customize AMMs for non-GBM assets, and against Goyal et al.’s liquidity-maximizing framework, which sometimes reaches opposite conclusions about where to concentrate liquidity. It closes by suggesting the underlying “uniform AMM” methodology generalizes beyond prediction markets to other derivatives.

Key quotes

“But ironically, despite prediction market volumes taking off in crypto, most of it uses orderbooks, not AMMs.” (Introduction)

“We define a uniform AMM for a particular asset as an AMM whose expected LVR is a constant fraction of the value of the current value of the pool, regardless of the current price of the asset.” (Loss-vs-rebalancing and uniformity)

“Polymarket saw over $2 billion in volume in just October 2024.” (Background, Prediction markets)

“The dynamic pm-AMM isn’t magic—it prevents LVR from increasing as expiration approaches by providing a decreasing amount of liquidity.” (Contributions)

How to read the original

Background needed: loss-versus-rebalancing (see /exchange/impermanent-loss-vs-lvr/), the normal distribution’s CDF/PDF notation, and a passing sense of what an “invariant” AMM is. Skip the full derivation of the dynamic-liquidity ODE on a first pass and focus on the two boxed invariant formulas and Figures 1–2 (liquidity concentration comparison). The hardest paragraph is the Itô’s-lemma step deriving $dP_t$‘s volatility term — it’s easiest to accept the punchline (volatility is $\phi(\Phi^{-1}(P_t))/\sqrt{T-t}$, independent of $\sigma$) rather than re-derive it, since the paper itself only sketches the steps.

What changed since

• White’s “Distribution Markets” (Paradigm, 2024-12) extended the same market-scoring-rule lineage to continuous-outcome events, generalizing LMSR-style mechanics to a constant-$L^2$-norm AMM over probability distributions.
• Slivkoff’s “Polymarket Volume Is Being Double-Counted” (Paradigm, 2025-12) shifted attention from AMM design to the more basic problem of correctly measuring the order-book volume that still dominates the space.
• Jupiter Forecast (June 2026) became, per available sources, the first named Solana-native prediction-market product, using a competing-market-maker “Prop AMM” model rather than the pm-AMM invariant specifically.

Secondary references

1. Slivkoff, “Polymarket Volume Is Being Double-Counted” (Paradigm, 2025-12) — read if you work with prediction-market on-chain data and need to avoid the OrderFilled double-count.
2. White, “Distribution Markets” (Paradigm, 2024-12) — read for the continuous-outcome generalization of the same market-scoring-rule lineage.
3. Hanson, “Combinatorial Information Market Design” (mktscore.pdf) — read for the original LMSR cost-function derivation.
4. Solana Compass, “Jupiter Forecast” coverage (2026-06) — read for the most current Solana-native implementation details (secondary source, volume figures dated).

10. Sources

• pm-AMM: A Uniform AMM for Prediction Markets — Ciamac Moallemi, Dan Robinson (Paradigm) — 2024-11-05 — https://www.paradigm.xyz/writing/pm-amm
• Distribution Markets — Dave White (Paradigm) — 2024-12-10 — https://www.paradigm.xyz/writing/distribution-markets
• Polymarket Volume Is Being Double-Counted — Storm Slivkoff (Paradigm) — 2025-12-08 — https://www.paradigm.xyz/writing/polymarket-volume-is-being-double-counted
• Improved Price Oracles: Constant Function Market Makers — Guillermo Angeris, Tarun Chitra — 2020-03-22 — https://arxiv.org/abs/2003.10001
• Automated Market Making and Loss-Versus-Rebalancing — Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden, Anthony Lee Zhang — 2022-08-11 — https://arxiv.org/abs/2208.06046
• Polymarket Documentation — docs.polymarket.com — fetched 2026-08-29 — https://docs.polymarket.com/
• Governance Attack or Smart Trade? UMA’s Oracle Just Got Outplayed on Polymarket (secondary) — Orochi Network — accessed 2026-08-29 — https://orochi.network/blog/oracle-manipulation-in-polymarket-2025
• Jupiter Launches Forecast: Solana’s First Native Prediction Market (secondary) — Solana Compass — accessed 2026-08-29 — https://solanacompass.com/news/jupiter-launches-forecast-solanas-first-native-prediction-market-with-multi-market-maker-quoting
• Solana’s Proprietary AMM Revolution — Helius — 2025-08 — https://www.helius.dev/blog/solanas-proprietary-amm-revolution