what is the math formula behind getAmountIn ?

[nzhl]

I am struggling to figure out how can we deduce the formula used in getAmountIn. Basically here if we want to maximize our profit those two pools must have almost the same price at final stage. Hence we have:

(x1 + delta_x1) / (y1 - delta_y1) = (x2 - delta_x2) / (y2 + delta_y1)

Also we know how the pair pool is updating accordingly after swap:

(x1 + 0.997 * delta_x1) (y1 - delta_y1) = x1y1
(x2 - delta_x2)(y2 + 0.997 * delta_y1) = x2y2

Based on equations mentioned above, i just have no clue how can i simplify it to the formula in the code. Do i miss anything here ?

simple-blind-arbitrage/src/blindBackrun.sol

Line 83 in 6258205

function getAmountIn(PairReserves memory firstPairData, PairReserves memory secondPairData) internal pure returns (uint256) {

[bertmiller]

See section 2.1 of this paper. Then it's a matter of calculus!

https://arxiv.org/pdf/1911.03380.pdf

[githuber1001]

When listening to the Flashbots MEV-Share Matchmaker on the Goerli network this transaction was returned, 0xe678b8a63274d0a14c5b9438997157d5212cfaddb79151effd36afa291e8e9e6. This arbitrage appears to be losing WETH. Is the arbitrage formula reversed or am I just reading this wrong (more likely scenario)?

[WXF-WangXuFeng]

See section 2.1 of this paper. Then it's a matter of calculus!

https://arxiv.org/pdf/1911.03380.pdf

Hey handsome, I understand calculus, but I've looked through the link you gave me three times and still can't find the derivation process of the calculation formula in the BlindBackrun contract. This article is about the arbitrage process from the Uniswap market to a related market with infinite liquidity, while the calculation formula in your BlindBackrun contract refers to two identical Uniswap markets, such as the arbitrage between the currency pairs of Uniswap and Sushiswap. Would you be able to update an article showing how to derive the calculation formula inside the GatAmountIn method? This would be very helpful for us to research new arbitrage methods. Thank you very much!

[hthuz]

Suppose pair1 has reserve $x_1,y_1$ and pair 2 has reserve $x_2,y_2$ and $x_1, x_2$ corresponds to WETH. Give $\Delta x_1$ to pair1 as WETH, we can get $\Delta y_2$ (or $\Delta y_1$, whatever). Then swap $\Delta y_2$ for $\Delta x_2$ WETH in pair2. Our profit is $\Delta x_2 - \Delta x_1$. If we take derivate of profit with respect to $\Delta x_1$ and set it to 0, we can get the formula behind getAmountIn.

To derive the exact formula for profit in terms of $\Delta x_1$, we can refer to Section 1 of https://raw.githubusercontent.com/runtimeverification/verified-smart-contracts/uniswap/uniswap/x-y-k.pdf, which gives the formula concerning how much $\Delta y$ if we give $\Delta x$ to a pair:
$\Delta y = \frac{\alpha\gamma}{1 + \alpha\gamma}y, where \ \alpha = \frac{\Delta x}{x}, \gamma=1-\rho$
and $\rho$ is the fee.
So we have $\Delta y_2 = \frac{\alpha\gamma}{1 + \alpha\gamma}y1, where\ \alpha=\frac{\Delta x_1}{x_1}$ and profit is $\frac{\alpha'\gamma}{1 + \alpha'\gamma}x2-\Delta x_1, where\ \alpha'=\frac{\Delta y_2}{y_2}$. Here the profit can be simplified as a function of $\Delta x_1$. To find the maximal profit, take the derivate and set it 0, solving the $\Delta x_1$.

Python code to solve the mathematical equation is below. It gives two solution, one solution is exactly what getAmountIn does (the other is a negative value).

import sympy as sy
deltaX1 = sy.Symbol('x')
x1,y1,x2,y2 = sy.symbols("x1,y1,x2,y2")
rho = 0.003
gamma = 1 - rho

alpha = deltaX1 / x1
deltaY2 = alpha * gamma / (1 + alpha * gamma) * y1

pair2_alpha = deltaY2 / y2
profit = pair2_alpha * gamma / (1 + pair2_alpha * gamma) * x2 - deltaX1
derivate = sy.diff(profit, deltaX1)
solution = sy.solve(derivate, deltaX1)
sy.pprint(solution[0])
sy.pprint(solution[1])

Here I only consider the case when token0 in both pairs is WETH. It should be similar if token1 is WETH.