Automated Market Makers in prediction markets function differently than their spot-market counterparts. While a Constant Product Market Maker (CPMM) is the simplest model, its properties are often suboptimal for probabilistic assets. The CPMM maintains a constant product of two asset reserves, $x$ and $y$, following the equation $x \cdot y = k$. The price is simply the ratio $y/x$. This design provides liquidity across an infinite price range, from zero to infinity. For a prediction market where outcomes are bounded between 0% and 100%, this implies that a vast portion of the capital is deployed in extremely unlikely, near-zero or near-one probability scenarios, leading to poor capital efficiency.

The Logarithmic Market Scoring Rule (LMSR) offers a more tailored approach. Developed by Robin Hanson, LMSR is an automated market maker specifically designed for prediction markets. Instead of a constant product, it uses a cost function based on the quantity of shares outstanding for each outcome. For a binary market with outcomes A and B, the cost of purchasing a set of shares $(q_A, q_B)$ is given by the change in the cost function $C$. The price of an outcome is the partial derivative of the cost function with respect to the quantity of that outcome's shares.

The instantaneous price for outcome $i$ is derived from this cost function:

The primary drawback of both basic CPMM and full-range LMSR models is their inefficient use of capital. Most of the liquidity sits waiting for extreme, low-probability price movements. To solve this, modern prediction market AMMs implement concentrated liquidity a concept popularized by Uniswap v3 but adapted for probabilistic outcomes. Instead of providing liquidity across the entire [0, 1] probability range, liquidity providers (LPs) can allocate their capital to specific, narrower probability bands. For example, if a market implies an event has a 40-60% chance of occurring, an LP can focus their entire capital within that range.

This concentration creates significantly more market depth for the same amount of capital within the active range, resulting in lower price impact for traders. However, this efficiency comes with a significant risk. If the event's perceived probability moves outside the LP's chosen range, their position becomes inactive, stops earning fees, and is left holding 100% of the less valuable asset—the one that is trending towards a value of zero. This exposes the LP to maximal settlement loss.

In this context, the risk for liquidity providers is more accurately described than 'impermanent loss'. The term Loss-Versus-Rebalancing (LVR), pronounced "lever," quantifies the adverse selection cost LPs incur. This cost arises because AMM prices are static between trades. Informed arbitrageurs can exploit the difference between the stale AMM price and the true market price (updated by new information) to profit at the LP's expense. LVR is the cumulative profit of these arbitrageurs, which is a direct loss to the LPs compared to a strategy of simply holding the assets and rebalancing them to match the new market price.

Essentially, LVR is the cost of providing continuous liquidity. For a constant product market maker, LVR can be modeled based on the volatility of the asset and the time duration. In a prediction market, this volatility is driven by new information changing the probability of the outcome. The trade-off is stark: higher fees from concentrated liquidity are a compensation for taking on higher LVR. A narrow liquidity range amplifies fee revenue but also amplifies potential losses from arbitrage when the price moves.

Probability-weighted Automated Market Makers (pm-AMMs) are a proposed solution to manage this risk. These systems dynamically adjust the liquidity concentration based on the market's current price. For example, as the probability of a 'YES' outcome increases, the AMM can automatically shift the concentrated liquidity band higher. This proactive inventory management aims to keep the LP's capital in the most active trading range while minimizing the risk of the position becoming worthless at settlement. The goal is to balance the inventory of 'YES' and 'NO' shares to track the market's consensus, reducing the LP's exposure to one-sided settlement risk.

Finding the optimal balance between earning fees and mitigating LVR is the core challenge for liquidity providers in prediction markets. It requires sophisticated modeling of event probability and dynamic risk management.