From AMM to Order Book: Exploring the Transformation of Polymarket’s Pricing Mechanism and the Possibility of Integration with DEX

Author: @BlazingKevin_, Researcher at Movemaker

In Polymarket, each prediction market is essentially a "probability exchange for future events." Users can express their bets on a particular event by buying an option (e.g., "Trump will win the 2024 US election").

Because buying probabilistic events differs from typical trading, Polymarket's initial pricing and liquidity mechanisms differed from common AMM algorithms. Polymarket's pricing mechanism has undergone significant changes since its initial release. Initially, it employed an AMM mechanism for real-time liquidity and pricing, called the Logarithmic Market Scoring Rule, or LMSR. This algorithm is currently used by other crypto protocols, such as Ethos.

Understanding the characteristics of LMSR will help you understand Polymarket's pricing mechanism for most of its history, why other protocols choose LMSR, and why Polymarket upgraded from LMSR to an off-chain order book.

LMSR Features and Advantages and Disadvantages

What is LMSR?

LMSR is a pricing mechanism designed specifically for prediction markets. It allows users to buy "shares" of an option based on their own judgment, while the market automatically adjusts the price based on aggregate demand. The most significant feature of LMSR is that it allows transactions to be completed without relying on a counterparty. Even if you're the first trader, the system can price and execute the trade for you. This gives prediction markets "perpetual liquidity" similar to Uniswap. In simple terms, LMSR is a cost function model that calculates prices based on the "shares" of each option held by the user. This mechanism ensures that prices always reflect the current market's expected probabilities for different outcomes.

LMSR Core Formula

LMSR's cost function, C, is calculated based on the number of shares sold for all possible outcomes in a market. The formula is:

The symbols here represent:

• C(…): Cost function, representing the total cost incurred by the market maker to maintain the current share distribution of all outcomes.
• n: The total number of possible outcomes in the market (e.g., for a "yes/no" market, n=2).
• qi: Represents the current share of the i-th option purchased (which can be understood as "voting rights" or "number of bets")
• b: A liquidity parameter; a larger value indicates a more stable market and less sensitive prices to new transactions
• C(q): Represents the cost of moving the market from its current state to q

The most important property of this formula is that the sum of all resulting prices is always equal to 1 (∑Pi=1). When a user purchases a "yes" share, $q(YES) increases, causing P(YES) to rise while P(NO) decreases, thus maintaining the sum of prices at 1.

How is pricing generated?

Another key point of LMSR is that price is the marginal derivative of the cost function. That is, the price pi of option i is the marginal cost you pay to buy one more unit of that option:

This means:

• If the purchase volume of an option increases (that is, more people bet on it happening), its price will gradually rise;
• Ultimately, the price will approach the market's subjective probability of each option happening.

For example, in a "yes/no" prediction market, if most people buy "yes," the price of "yes" might rise to 0.80, while "no" might fall to 0.20. This is like saying, "The market believes the probability of the event occurring is 80%."

Furthermore, regardless of liquidity, the cost function curve is always upward. This means that the more shares purchased, the higher the total cost.

The role of the liquidity parameter b: The value of b directly determines the "flatness" of the curve, or in other words, the liquidity or "thickness" of the market.

• High liquidity (left image, b=100): The curve is relatively flat. This means that even if you purchase a large number of shares, the price will rise slowly. Such a market can absorb large transactions without causing drastic price fluctuations.
• Low Liquidity (right chart, b=20): The curve is very steep. This means that even a small purchase can cause a sharp price increase. This indicates a very sensitive market with low liquidity.

High liquidity (a large b value) acts as a "cushion," allowing the market to absorb greater buying power without drastic price fluctuations (a flat curve); low liquidity is very sensitive (a steep curve).

LMSR's Mechanism Tradeoffs and Polymarket's Paradigm Shift

Before discussing Polymarket's evolution toward an order book model, it's necessary to first analyze the LMSR mechanism it adopted in its early days. LMSR isn't a simple technical option, but rather a set of underlying protocols with a clear design philosophy and inherent tradeoffs. Its characteristics determine its historical position in different stages of prediction market development.

LMSR's Core Mechanisms and Design Tradeoffs

LMSR's fundamental design goal is information aggregation, not market maker profitability. Through an automated mathematical model, it solves the most vexing "cold start" problem for prediction markets: liquidity provision in the early stages when there are no counterparties.

1. Advantage Analysis: Unconditional Liquidity Provision and Controllable Market Making Risk LMSR's core contribution is that it ensures the presence of counterparties at all times. No matter how unpopular or extreme a market view, market makers can always provide a buy or sell quote. This fundamentally addresses the dilemma of traditional order books in early markets, which were unable to execute trades due to thin liquidity.

Conversely, the maximum potential loss for market makers guaranteeing this "unlimited" liquidity is predictable and bounded. This maximum loss is determined by the liquidity parameter "b" and the number of market outcomes "n," using the formula "maximum loss = b⋅ln(n)." This certainty of risk makes the cost of sponsoring a prediction market manageable and eliminates the risk of unlimited losses, which is crucial for parties or organizations launching new markets.

2. Inherent Flaws: Static Liquidity and Non-Profit Orientation However, the advantages of LMSR also bring with them structural flaws that cannot be overcome.

• The b-parameter dilemma and static liquidity: This is the core limitation of LMSR. The liquidity parameter "b" is set at market creation and typically remains constant throughout its lifecycle. A large "b" value indicates deep liquidity and stable prices, but a slow response to new information; a small "b" value indicates price sensitivity and rapid convergence of opinions, but also market fragility and volatility. This static setting prevents the market from adaptively adjusting its depth and sensitivity based on actual increases and decreases in liquidity and changes in information flows.
• The Subsidy Role of Market Makers: The LMSR model has a theoretical mathematical expectation of loss. The market maker's losses are considered the "information fee" they pay to access the collective market wisdom (i.e., the final, accurate price determined by all trades). This positioning makes it essentially a system where the initiator subsidizes transactions. It is not suitable for a profit-seeking market maker model and it is difficult to build a profitable ecosystem with the participation of a large number of decentralized limited partners.

Furthermore, when implementing LMSR on-chain, the logarithmic and exponential operations involved consume more gas than the four arithmetic operations commonly used in DEXs, further increasing transaction friction in a decentralized environment.

Paradigm Shift: The Logical Inevitability of Polymarket's Abandonment of LMSR

In summary, LMSR was an effective and practical tool in the early stages of a platform's liquidity shortage. However, once Polymarket's user and capital base exceeded a critical mass, its design, which sacrificed efficiency for liquidity, transformed from an advantage into a hindrance to its development. Its migration to an order book model was based on the following strategic considerations:

• Fundamental Purpose of Capital Efficiency: LMSR requires market makers to provide liquidity across the entire price range, from 0% to 100%. This results in a large amount of capital being accumulated at price points with extremely low transaction probabilities, resulting in low capital efficiency. The order book allows market makers and users to precisely concentrate liquidity in the most active price ranges, which is highly consistent with professional market making strategies.
• Optimizing the Trading Experience: The algorithmic characteristics of LMSR mean that trades of any size will inevitably incur slippage. In a market with increasing liquidity, this inherent trading friction can hinder the entry of large funds. A mature order book market, however, can absorb large orders through a dense counterparty depth, providing lower slippage and a better execution experience.
• Strategic Need to Attract Professional Liquidity: The order book is the most common and familiar market model for professional traders and market makers. By shifting to the order book, Polymarket is sending a clear invitation to professional liquidity providers from both the crypto world and traditional finance. This is a critical step for the platform from attracting retail investors to building professional-grade market depth.

Current Polymarket Pricing and Liquidity Mechanism

Polymarket's upgrade is a necessary step after reaching a critical mass of user scale and platform maturity. This transformation reflects a systematic consideration of the three objectives of trading experience, gas costs, and market depth. Its current architecture can be analyzed from two perspectives: liquidity mechanism and price anchoring logic.

Hybrid Model of On-Chain Settlement and Off-Chain Order Book

Polymarket's liquidity mechanism utilizes a hybrid on-chain and off-chain architecture, designed to balance the security of decentralized settlement with the smooth experience of centralized trading.

• Off-Chain Order Book: User limit orders are submitted and matched instantly on off-chain servers, with no gas costs. This makes the Polymarket trading experience similar to that of centralized exchanges, allowing users to intuitively view the market depth (buy and sell orders) comprised of all limit orders. Liquidity therefore comes directly from all trading participants, rather than from a passive liquidity pool.
• On-Chain Settlement: When buy and sell orders in the off-chain order book are successfully matched, the final asset delivery step is executed on the Polygon chain through smart contracts. This "off-chain matching, on-chain settlement" model ensures the finality of trade results and the immutability of asset ownership while preserving the flexibility of the order book. The displayed "price" is the midpoint between the bid and ask prices in the off-chain order book.

The Underlying Logic of Price Anchoring: Minting of Share Pairs and the Arbitrage Cycle

For prediction markets, the core mechanism is how to ensure that the sum of the probabilities of "yes" and "no" outcomes is always equal to 100% (i.e., "$1"). The order book model itself does not impose a mandatory limit on the order price through code. Instead, through a sophisticated set of underlying asset design and arbitrage mechanisms, leveraging the inherent market correction force, the price sum is ensured to always converge towards "$1."

1. Core Foundation: Minting and Redemption of Complete Share Pairs The cornerstone of this mechanism is the unshakable value equation established by the Polymarket contract layer.

• Minting: Any participant can deposit $1 USDC into the contract and receive one YES share and one NO share simultaneously. This operation establishes the underlying value anchor of "1 YES share + 1 NO share = $1."
• Redemption: Similarly, any participant holding one YES share and one NO share can combine them and return them to the contract at any time to redeem $1 USDC.

This two-way channel ensures that the total value of the complete set of outcomes is firmly anchored at $1.

2. Price Discovery: Independent Order Book Trading Based on the above foundation, YES shares and NO shares are two independent assets, trading against USDC on their own order books. Participants are free to place limit orders at any price, with no protocol restrictions. This free pricing mechanism inevitably leads to price deviations, creating opportunities for arbitrageurs.

3. Price Constraints: Market-Based Arbitrage Correction The profit-seeking behavior of arbitrageurs (typically automated bots) is key to ensuring price reversion. Once the sum of the trading prices of YES and NO shares deviates from $1, a risk-free arbitrage window opens.

• When "P(YES) + P(NO) > $1" (for example, "$0.70 + $0.40 = $1.10"), arbitrageurs will execute a "mint-sell" operation: depositing $1 into the contract, minting one YES and one NO share, then immediately selling them on the order book at $0.70 and $0.40, respectively, for a risk-free profit of $0.10. This frequent occurrence increases selling pressure in the market, driving the prices of YES and NO down simultaneously until their sum returns to $1.
• When "P(YES) + P(NO) < $1" (for example, "$0.60 + $0.30 = $0.90"), arbitrageurs will execute a buy-and-redemption operation: buying one YES share and one NO share on the order book at $0.60 and $0.30, respectively, then combining them and redeeming them for $1 through the contract, earning a risk-free profit of $0.10. This action increases market demand, driving the prices of both shares up simultaneously until the total returns to $1.

The key design principle of this mechanism lies in the fact that the protocol itself does not act as an arbitrator. Instead, by establishing a solid value anchor and open arbitrage channels, the profit-seeking behavior of market participants becomes the decisive force in maintaining system price stability.

Possible Integration of Polymarket and DEX

Polymarket chose to upgrade from an AMM to an order book model. Firstly, the platform's user base has exploded, resulting in ample liquidity and a relatively stable order book experience. Secondly, the upgraded pricing mechanism makes it more suitable for professional market makers.

With X officially announcing its partnership with Polymarket, Polymarket has become X's official prediction market. The asymmetric user base between Polymarket and X will undoubtedly attract further new users to Polymarket. In this process, Polymarket will also create an asymmetry in user base with crypto protocols, and user traffic will flow into the crypto industry through X through Polymarket.

Against this backdrop, we need to consider new possibilities between Polymarket and crypto protocols, and further, the potential integration of Polymarket and DEXs.

First, Polymarket provides DEX ecosystem participants with a native and efficient risk hedging tool. Asset holders and limited partners in DEXs generally face exposure to impermanent loss, protocol risk, or macroeconomic volatility. Traditional hedging tools feel disconnected in DeFi, but Polymarket's event contracts serve as a mirror layer for risk pricing. For example, prediction contracts predicting whether a stablecoin will depeg or whether a protocol upgrade will succeed can be directly used by DEX users to hedge potential losses on their on-chain positions. This model shifts risk management from passive acceptance to active allocation, becoming a composable financial building block in the DeFi ecosystem.

Secondly, price data from prediction markets can serve as a valuable leading indicator for centralized liquidity management on DEXs. In centralized liquidity models like Uniswap V3, LPs' capital efficiency is positively correlated with risk, and their profitability is determined by their speed of responsiveness to market changes. Polymarket's real-time odds for key events are essentially the market's collective consensus on future probabilities, often changing before on-chain asset price fluctuations. Automated strategies can capture this leading signal and dynamically adjust LPs' position ranges—widening or withdrawing positions when risk probability increases, and narrowing them when certainty increases. This transforms LPs from passive liquidity "sandbags" into active, probabilistic risk managers.

Furthermore, by tying the core DEX metrics to Polymarket's event outcomes, new structured financial products can be created. Protocol growth needs to be deeply tied to community interests, and Polymarket provides a transparent and impartial external verification mechanism for this purpose. Protocols can design a "conditional" revenue distribution model: for example, tying the distribution of a large portion of trading fees to the outcome of a Polymarket event: "Can trading volume exceed $N billion this quarter?" If the outcome is "yes," stakers share in the excess returns; if "no," the proceeds are used for buyback and burn. This design transforms the protocol's KPIs into financial products that the community can directly participate in, creating a more direct shared interest and closed-loop value capture.

In summary, the integration of Polymarket and DEXs is not a simple functional addition, but a deep integration at the infrastructure level. Polymarket is evolving into a "risk pricing layer" and "information oracle" for the entire crypto industry. As the traffic brought by X gradually penetrates, its integration with basic protocols such as DEX will no longer be an option, but a key variable that determines whether the future DeFi ecosystem can become more efficient, mature, and resilient.