By Cyberhang
Here we assume there is a liquidity pool on Ethereum. Two assets GCR-USDC, with a relation xy=k, where x,y represent the quantity of asset GCR, USDC in the pool, and k is the product of their quantity, which is a constant given the property of the AMM mechanism. (We just ignore pool transaction fee since it is quite small and negligible in this question)
Sandwich Attacks is basically a front-running strategy to transact just before a large trade in a pool. For example, Alice trades GCR-USDC pair in Uniswap V2. She wants to buy Δx amount of GCR tokens with Δy USDC. After she approves her transaction with an input of Δy amount of USDC, her transaction is pending and waiting for a miner to package her transaction. Just right before her transaction is packaged, a front-running bot discovered her transaction and traded before her transaction is finalized. We assume the front-running bot puts Δ y{a} USDC into the pool at this time(an “a” here means attack, to distinguish with Alice’s money), with a higher network transaction fee to front-run Alice. What happened then is the bot’s transaction is packaged before Alice’s, which means Alice trades at a higher price than she expected because here is a bot traded before her. We just use the following formulas to describe it in a formalized manner:
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