Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Path Dependence in AMM-Based Markets: Mathematical Proof and Implications for Truth Discovery

Keroshan Pillay

TL;DR

The paper proves that AMM-based markets with the constant product invariant $x \times y = k$ exhibit path dependence: the final pool state hinges on the sequence of operations, not merely net activity. It combines formal non-commutativity proofs with extensive ETH/USDC empirical data to show real-world path effects and analyzes their implications for prediction markets and price interpretation. Its simulations demonstrate that identical information can yield different prices under different operation orders, challenging the notion of AMMs as pure truth machines and prompting design, governance, and transparency adjustments in DeFi protocols. The work contributes to market-efficiency discourse by highlighting memory effects in automated liquidity mechanisms and proposes practical mitigations and indicators for path sensitivity. This has broad practical impact for price discovery, risk management, and the design of robust, path-aware DeFi systems.

Abstract

This paper demonstrates that Automated Market Maker (AMM) based markets, such as those using constant product formulas (e.g., Uniswap), are inherently path-dependent. We prove mathematically that the sequence of operations in AMMs determines the final state, challenging the notion that market prices solely reflect information. This property has profound implications for decentralized prediction markets that rely on AMMs for price discovery, as it demonstrates they cannot function as pure "truth machines." Using both mathematical proofs and empirical evidence from ETH/USDC pools, we show that AMM-based markets incorporate historical path information beyond the current market beliefs. Our findings contribute to the understanding of market efficiency, mechanism design, and the interpretation of prices in decentralized finance systems.

Path Dependence in AMM-Based Markets: Mathematical Proof and Implications for Truth Discovery

TL;DR

The paper proves that AMM-based markets with the constant product invariant exhibit path dependence: the final pool state hinges on the sequence of operations, not merely net activity. It combines formal non-commutativity proofs with extensive ETH/USDC empirical data to show real-world path effects and analyzes their implications for prediction markets and price interpretation. Its simulations demonstrate that identical information can yield different prices under different operation orders, challenging the notion of AMMs as pure truth machines and prompting design, governance, and transparency adjustments in DeFi protocols. The work contributes to market-efficiency discourse by highlighting memory effects in automated liquidity mechanisms and proposes practical mitigations and indicators for path sensitivity. This has broad practical impact for price discovery, risk management, and the design of robust, path-aware DeFi systems.

Abstract

This paper demonstrates that Automated Market Maker (AMM) based markets, such as those using constant product formulas (e.g., Uniswap), are inherently path-dependent. We prove mathematically that the sequence of operations in AMMs determines the final state, challenging the notion that market prices solely reflect information. This property has profound implications for decentralized prediction markets that rely on AMMs for price discovery, as it demonstrates they cannot function as pure "truth machines." Using both mathematical proofs and empirical evidence from ETH/USDC pools, we show that AMM-based markets incorporate historical path information beyond the current market beliefs. Our findings contribute to the understanding of market efficiency, mechanism design, and the interpretation of prices in decentralized finance systems.

Paper Structure

This paper contains 34 sections, 10 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. AMM Fundamentals
  3. Constant Product Market Makers
  4. Key Operations in AMMs
  5. AMMs in Prediction Markets
  6. Mathematical Proof of Path Dependence
  7. Defining Path Dependence
  8. Non-commutativity in AMMs
  9. Theorem 1: Order of Operations Matters
  10. Numerical Example
  11. Economic Significance of Non-commutativity
  12. Implications of Non-commutativity
  13. Empirical Evidence from ETH/USDC Pools
  14. Data Collection and Methodology
  15. Data Sources:
  16. ...and 19 more sections

Figures (4)

  • Figure 1: Examples of path dependence in ETH/USDC pools. The chart shows the magnitude of price impact (as a percentage) for various combinations of swap and liquidity events that occurred within the same 60-second time window. Negative values indicate that executing a swap before adding liquidity resulted in a lower final price than the reverse sequence.
  • Figure 2: Relationship between liquidity proportion (alpha) and price impact. The scatter plot demonstrates a strong negative correlation (-0.992), confirming that larger proportional changes in liquidity lead to more significant path dependence effects.
  • Figure 3: Path dependence heatmap showing the interaction between swap amounts and liquidity proportions (alpha). Darker regions indicate stronger path dependence effects, with the most significant impacts occurring at high alpha values.
  • Figure 4: Temporal distribution of path dependence opportunities over the analysis period, showing the magnitude of price impacts for identified events. The trend suggests that path dependence opportunities occur regularly, with varying magnitudes depending on market conditions.

Theorems & Definitions (1)

  • proof