From AMM to Order Book: The Evolution of Polymarket's Pricing Mechanism and Its Potential for Integration with DEX

Polymarket, as a prediction market platform, is essentially a "future event probability exchange." Users can express their judgment on specific events by purchasing a certain option.

Unlike traditional trading, where buying is based on probabilistic events, Polymarket initially adopted a unique AMM mechanism to provide real-time liquidity and pricing, known as the Logarithmic Market Scoring Rule (LMSR). This algorithm is now also used by some other crypto protocols.

Understanding the characteristics of LMSR can help us comprehend the pricing mechanism of Polymarket during most periods, as well as the reasons other protocols choose LMSR, while also clarifying why Polymarket upgraded from LMSR to an off-chain order book.

Characteristics and Advantages/Disadvantages of LMSR

Introduction to LMSR

LMSR is a pricing mechanism specifically designed for prediction markets, allowing users to buy "shares" of a certain option based on their judgment, with the market automatically adjusting prices according to total demand. The biggest feature of LMSR is that it can complete transactions without relying on counterparties; even the first trader can have pricing and execution provided by the system. This gives prediction markets a "perpetual liquidity" similar to that of certain DEXs.

In simple terms, LMSR is a cost function model that calculates prices based on the "shares" of various options currently held by users. This mechanism ensures that prices always reflect the current market's expected probabilities of different event outcomes.

The core formula of LMSR

The cost function C of LMSR is calculated based on the quantity of shares sold for all possible outcomes in the market. Its formula is:

C(q) = b * ln(Σexp(qi/b))

Among them:

• C(...): Cost function, representing the total cost incurred by the market maker to maintain the current distribution of shares across all outcomes.
• n: The total number of possible outcomes in the market (e.g., for a "Yes/No" market, n=2).
• qi: The current purchased share of the i-th option.
• b: Liquidity parameter, the larger it is, the more "stable" the market is, and the less sensitive the price is to new trades.
• C(q): The cost of moving the market from its current state to q.

The most important characteristic of this formula is that the sum of the prices of all results is always equal to 1. When a user purchases "yes" shares, q(YES) increases, causing P(YES) to rise, while P(NO) falls, thereby maintaining the total price sum at 1.

pricing mechanism

Another key point of LMSR is that the price is the marginal derivative of the cost function. That is, the price pi of the i-th option is the marginal cost that needs to be paid to buy one more unit of that option:

pi = ∂C/∂qi = exp(qi/b) / Σexp(qj/b)

This means:

• The larger the purchase volume of a certain option, the higher its price will gradually rise.
• The final price will tend to reflect the market's subjective probabilities of each option occurring.

For example, in a "yes/no" binary prediction market, if most people buy "yes", the price of "yes" may rise to 0.80, while "no" drops to 0.20, indicating that the market believes the probability of the event occurring is 80%.

In addition, regardless of the liquidity, the curve of the cost function extends upward. This means that the more shares purchased, the higher the total cost that needs to be paid.

The role of the liquidity parameter b: The size of the b value directly determines the "smoothness" of the curve, which is the liquidity or "thickness" of the market.

• High liquidity (large b value): The curve is relatively flat. Even when purchasing a large number of shares, the price increase is relatively slow. Such a market can "absorb" large transactions without causing significant price fluctuations.
• Low liquidity (small b value): The curve is very steep. Even a small purchase can lead to a sharp increase in price. Such markets are very sensitive and have poor liquidity.

The Mechanism Trade-offs of LMSR and the Paradigm Shift of Polymarket

Before discussing Polymarket's evolution to an order book model, it is necessary to first analyze the LMSR mechanism it adopted in its early stages. LMSR is not a simple technical option, but a set of underlying protocols with a clear design philosophy and inherent trade-offs, whose characteristics determine its historical positioning at different stages of development in prediction markets.

The Core Mechanism and Design Trade-offs of LMSR

The fundamental design goal of LMSR is information aggregation, rather than profit for market makers. It solves the most challenging "cold start" problem for prediction markets, which is the provision of liquidity when there is a lack of counterparties in the early stages, through an automated mathematical model.

Advantage Analysis: Unconditional liquidity supply and controllable market making risks

The most significant contribution of LMSR is that it ensures there is always a counterparty in the market at any given point in time. Regardless of how unpopular or extreme the market view may be, market makers can always provide a buy or sell quote. This fundamentally addresses the dilemma faced by traditional order books in early markets where transactions could not be executed due to thin liquidity.

Correspondingly, the market makers providing collateral for this "infinite" liquidity have a potential maximum loss that is predictable and bounded. The maximum loss is determined by the liquidity parameter "b" and the number of market outcomes "n", with the formula being "maximum loss = b⋅ln(n)". The certainty of this risk makes the cost of sponsoring a prediction market manageable, eliminating the risk of infinite losses, which is crucial for parties or organizations needing to launch new markets.

Inherent Defects: Static Liquidity and Non-Profit Orientation

However, the advantages of LMSR also bring about structural defects that it cannot overcome.

• The b parameter dilemma and static liquidity: This is the core constraint of LMSR. The liquidity parameter "b" is set at the creation of the market and usually remains unchanged throughout the market's lifecycle. A large "b" value means deep liquidity and stable prices, but slow to respond to new information; a small "b" value means price sensitivity and the ability to quickly aggregate opinions, but a fragile market with significant volatility. This static setting prevents the market from adaptively adjusting its depth and sensitivity based on the actual changes in liquidity and the flow of information.

• The role of the market maker as a subsidy: The theoretical mathematical expectation of the LMSR model is a loss. The losses incurred by the market maker are viewed as the "information cost" paid for acquiring the collective wisdom of the market (i.e., the final accurate price formed by all transactions). This positioning determines that it is essentially a system subsidized by the initiator's transactions, which is not suitable for profit-seeking market maker models, and it is also difficult to build a profitable ecosystem involving a large number of decentralized LPs.

In addition, when LMSR is implemented on-chain, the logarithmic and exponential operations involved consume more Gas compared to the common arithmetic operations in DEX, which further increases transaction friction in a decentralized environment.

Paradigm Shift: Polymarket Abandons the Logical Necessity of LMSR

In summary, LMSR is an efficient and practical tool during the early stages of the platform when liquidity is scarce. However, once the user and capital volume of Polymarket exceed the critical point, its design that sacrifices efficiency for liquidity turns from an advantage into a constraint on development. Its shift to an order book model is based on the following strategic considerations:

• Fundamental demand for capital efficiency: LMSR requires market makers to provide liquidity across the entire price range from 0% to 100%, which leads to a significant amount of capital being locked in price points with very low transaction probabilities, resulting in low capital efficiency. Order books, on the other hand, allow market makers to precisely concentrate liquidity with users in the most active price ranges of the market, which aligns closely with professional market-making strategies.

• Optimization of trading experience: The algorithmic characteristics of LMSR dictate that trades of any scale inevitably incur slippage. In markets with increasingly thick liquidity, this inherent trading friction can hinder the entry of large funds. However, a mature order book market can absorb large orders through dense counterparty depth, providing lower slippage and a better trading execution experience.

• The strategy to attract professional liquidity requires: The order book is the most common and familiar market model for professional traders and market-making institutions. Shifting to an order book signifies that Polymarket has sent a clear invitation signal to professional liquidity providers in the crypto world and even traditional finance. This is a key step for the platform to move from attracting retail participation to building professional-level market depth.

Current Pricing and Liquidity Mechanism of Polymarket

The upgrade of Polymarket is an inevitable choice after reaching a critical point in user scale and platform maturity. Behind this transformation is a systematic consideration of three goals: trading experience, Gas costs, and market depth. Its current architecture can be analyzed from two aspects: liquidity mechanisms and price anchoring logic.

Hybrid model of on-chain settlement and off-chain order book

Polymarket's liquidity mechanism adopts a hybrid architecture that combines on-chain and off-chain elements, aiming to balance the security of decentralized settlement with the smooth experience of centralized trading.

• Off-chain order book: Users' limit orders are submitted and matched on off-chain servers, making the process instantaneous and with no Gas costs. This aligns the trading experience of Polymarket with that of centralized exchanges, allowing users to intuitively see the market depth (buy and sell orders) constituted by all limit orders. Liquidity thus directly comes from all trading participants themselves, rather than 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 retains the flexibility of the order book while ensuring the finality of trade results and the immutability of asset ownership. The displayed "price" is the midpoint between the best buy and best sell prices in the off-chain order book.

The underlying logic of price anchoring ------ Share for minting and arbitrage cycle

For prediction markets, the core mechanism is how to ensure that the probabilities of the two outcomes "yes" (YES) and "no" (NO) always sum to 100% (i.e., "$1"). The order book model itself does not enforce the limit order price through code but rather utilizes a sophisticated underlying asset design and arbitrage mechanisms, harnessing the market's own corrective power to ensure that the total price consistently converges towards "$1".

Core Basics: Minting and Redeeming Complete Shares

The cornerstone of this mechanism is an unshakeable value equation established by the Polymarket contract layer.

• Minting: Any participant can deposit "$1" USDC into the contract and simultaneously receive 1 YES share and 1 NO share. This operation establishes an underlying value anchor of "1 YES share + 1 NO share = $1".

• Redemption: Similarly, any participant holding 1 YES share and 1 NO share can combine them and return them to the contract at any time to redeem "$1" USDC.

This bidirectional channel ensures that the total value of a complete set of results is firmly anchored at "$1".

Price Discovery: Independent Order Book Trading

Based on the above foundation, YES shares and NO shares are two independent assets that trade with USDC on their respective order books. Participants can freely place limit orders at any price, with no restrictions imposed by the protocol layer. This free pricing mechanism will inevitably lead to price deviations, thus creating opportunities for arbitrageurs.

Price Constraint: Market-based Arbitrage Correction

The profit-seeking behavior of arbitrageurs (usually automated bots) is key to ensuring that prices revert. 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" ): Arbitragers will execute the "mint-sell" operation: deposit "$1" into the contract, mint 1 YES and 1 NO share, and then immediately sell them on the order book for "$0.70" and "$0.40" respectively, obtaining a risk-free profit of "$0.10". This behavior occurs frequently, increasing the selling pressure in the market, driving the prices of YES and NO down simultaneously until their total returns to "$1".

• When "P(YES) + P(NO) < $1" ( for example