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

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.