Prediction markets are a type of trading platform that allows participants to trade contracts based on expected outcomes of real-world events, such as political elections, sports competitions, or economic trends. Prices formed through free trading among participants reflect the likelihood of events occurring. In simple terms, prediction markets transform collective forecasting abilities into a tool for probability measurement.

Unlike traditional financial markets, prediction markets revolve directly around the events themselves, rather than indirectly investing in related assets. This mechanism meets diverse speculative needs and aggregates market participants’ collective beliefs about event probabilities through pricing.

However, prediction markets are also financial markets that require appropriate pricing mechanisms to encourage trading, attract more participants’ judgments, and distill this information to form the latest probability predictions. This article will introduce the pricing mechanisms behind prediction markets.

Why Can Prices in Prediction Markets Reflect the Probabilities of Events?

Unlike traditional gambling industries where players bet against bookmakers, prediction markets are open and decentralized mechanisms. In gambling, odds are set and dynamically adjusted by bookmakers. In prediction markets, prices are naturally formed through trading among participants, reflecting the collective wisdom’s assessment of event probabilities.

Imagine a binary prediction scenario: next month, there will be a football match between Germany and Spain. People can create a trading market on the prediction platform and issue two result tokens representing “Germany Wins” and “Spain Wins.” If the initial prices of both tokens are equal, it indicates the market believes the two teams have a 50-50 chance of winning.

As the match approaches, if a key German player is injured, more traders might predict a higher probability of Spain winning and buy “Spain Wins” tokens. Changes in supply and demand for the tokens will adjust their prices in real time, reflecting the increased likelihood of Spain’s victory. Similarly, during the match, if Germany scores consecutively, the demand for “Germany Wins” tokens rises, and their price will increase until the match ends when probabilities converge to actual results—100%.

When the match result is determined (e.g., Germany wins), the value of “Spain Wins” tokens will drop to zero, and holders of “Germany Wins” tokens will share profits from the total liquidity pool based on their shares. This dynamic price adjustment mechanism based on trading enables prediction markets to flexibly and efficiently reflect the probabilities of future events.

Common Pricing Mechanisms in Prediction Markets

The operation of prediction markets typically relies on two major pricing mechanisms: Continuous Double Auction (CDA) and Automated Market Maker (AMM).

Most decentralized prediction markets on the blockchain still use order books to provide liquidity, unlike the AMM widely applied in decentralized exchanges (DEXs). This may be due to the unique characteristics of result tokens: their value can fluctuate significantly with real-world events and drop to zero if the prediction is incorrect after the event concludes. Since the value of result tokens is closely tied to event outcomes, the potential losses for AMMs are significantly impacted, posing substantial risks.

To address this challenge, prediction markets introduced a specially designed automated market maker mechanism, the Logarithmic Market Scoring Rule (LMSR), to balance market liquidity and risk, supporting the stable operation of prediction markets.

Continuous Double Auction (CDA)

The Continuous Double Auction (CDA) is the most common pricing mechanism in financial markets and has been widely adopted in prediction markets. Its basic principle involves recording all unmatched orders in an order book, with buy and sell orders arranged on opposite sides. Traders can submit limit orders to the order book, and when the highest bid price matches the lowest ask price, a transaction is triggered and executed.

This mechanism is popular for its simple and clear design. However, in prediction markets with a limited number of participants, CDA may face liquidity shortages. Low liquidity often leads to wide bid-ask spreads, making pricing difficult and reducing market efficiency. In such cases, prediction markets’ price discovery and probability forecasting functions may struggle to perform effectively.

Logarithmic Market Scoring Rule (LMSR)

Unlike CDA, LMSR introduces a central automated market maker as the counterparty to all traders. The Logarithmic Market Scoring Rule (LMSR) is an automated market maker (AMM) mechanism specifically designed for prediction markets. One of its key features is that it does not rely on a liquidity pool, making it suitable for low-liquidity markets, including prediction markets. LMSR uses a logarithmic scoring rule to generate quotes, effectively preventing excessive price fluctuations. This approach provides sufficient liquidity while keeping the market maker’s potential losses within a controllable range.

The table below shows the main differences between LMSR and traditional AMMs.

To understand the uniqueness of LMSR, it is helpful to first review common AMM mechanisms. Most AMMs use a constant product formula:

x⋅y=k

In the formula, x and y represent the quantities of two tokens in the liquidity pool, and k is a constant. For example, in an ETH/DAI liquidity pool with an initial state of 100 ETH and 10,000 DAI, k=1,000,000. To keep k constant, when traders deposit ETH into the pool, the corresponding amount of DAI must decrease. Ultimately, the quote for any given trade is a function of the constant product formula and the token ratio in the pool. The graph below approximates the exchange relationship between two asset types under this model.

Source: news.marsbit.co

In contrast, LMSR’s pricing mechanism is more complex. Its formula is as follows:

qA: the quantity of outcome A (the number of shares already purchased for that outcome).

b: the “liquidity parameter” set by the market maker, which affects the sensitivity of prices to changes in trading volume.

n: the total number of possible outcomes.

In addition, LMSR defines a cost function to calculate the total cost of a trade:

This function helps market makers understand the potential losses they may face when providing liquidity. The logarithmic function included here means that as the number of contracts favoring a particular outcome increases, the price of that outcome rises at a diminishing rate. This mechanism provides more precise price adjustments and limits the market maker’s potential losses, ensuring the market’s long-term stability.

Further Improvements to Prediction Market AMMs

Prediction market AMMs have seen various improvements, with Paradigm, a well-known crypto investment firm, recently proposing its latest pricing model, pm-AMM. Paradigm aims to develop this model into a unified framework for prediction markets. The company has compared pm-AMM to other AMMs, suggesting it could also be applied to other asset types, such as bonds, options, and derivatives.

Source: paradigm.xyz

1. Optimization for Outcome Tokens
   The pm-AMM is specifically designed to handle outcome tokens, whose value is 1 if the event occurs and 0 if it does not. Traditional AMMs often face inconsistent liquidity issues with such tokens. By introducing a Gaussian distribution model, pm-AMM captures the relationship between token prices and event probabilities, providing more stable and consistent liquidity.

The Gaussian distribution, also known as the normal distribution, is assumed to govern the price fluctuations of outcome tokens in each prediction market (e.g., “event happens” and “event does not happen”). This assumption helps concentrate liquidity around more probable outcomes as the event approaches resolution (i.e., prices near 0 or 1), avoiding problems like liquidity shortages or excessive slippage during extreme scenarios while minimizing losses for market makers.

Returning to the earlier example of a football match between Germany and Spain, most market participants might initially predict Spain’s victory, leading to higher token prices for Spain’s win. However, if Germany begins to perform strongly during the match, market expectations may quickly shift in favor of Germany. Traditional AMMs might react sluggishly, leaving market makers holding a significant number of Spain victory tokens that eventually lose all value. By contrast, pm-AMM uses the Gaussian model to rapidly adjust liquidity, concentrating it around the more likely outcome, thereby reducing market maker losses and enhancing market efficiency and reliability.

1. Dynamic Liquidity Adjustment
   The pm-AMM employs a dynamic liquidity adjustment mechanism, modifying liquidity levels as the event nears its resolution. This means liquidity decreases as the prediction market approaches expiration, reducing the risk of losses for liquidity providers due to arbitrage. This mechanism ensures liquidity adapts to market volatility, maintaining stability during turbulent periods.

2. A Unified AMM Framework
   Paradigm’s pm-AMM aims to establish a unified AMM framework that extends beyond prediction markets to other asset classes, such as bonds, options, and derivatives. This versatility enhances pm-AMM’s applicability across various financial products, increasing its flexibility and utility.

3. Loss vs. Rebalancing Trade-off (LVR)
   The pm-AMM introduces the Loss vs. Rebalancing (LVR) concept, which evaluates potential losses AMMs may face due to arbitrage activities. By optimizing the AMM structure to minimize LVR, pm-AMM ensures robust liquidity while reducing potential losses, thereby improving returns for liquidity providers.

4. Enhanced User Experience
   By streamlining the trading process and improving price discovery, pm-AMM offers a more user-friendly experience. Users can intuitively understand market dynamics, with transactions automatically executed via smart contracts, eliminating delays and uncertainties caused by manual intervention.

Paradigm’s pm-AMM significantly improves traditional AMM mechanisms in prediction markets. Through innovations such as optimization for outcome tokens, dynamic liquidity adjustment, unified framework design, and the introduction of LVR, pm-AMM enhances the efficiency and stability of prediction markets while opening new use cases for other financial products. These advancements will drive the development of decentralized finance (DeFi), enabling prediction markets to reflect public sentiment and support decision-making processes better.

To learn more about the pm-AMM design principles and modeling, refer to the articles linked below.

References:

1. Paradigm’s Latest Research: A Unified Automated Market Maker for Prediction Markets – pm-AMM
2. pm-AMM: A Uniform AMM for Prediction Markets