Which AMM Pool is More profitable?

Note

Suppose that Alice and Bob both open weighted constant sum market maker pools containing Token X and Token Y. Alice decides to use the equation 2x + 5y = k, and Bob decides to use the equation 5x + 2y = k. If someone wants to trade Token X into the pool to receive Token Y in exchange, which of the pools would be more attractive? (You may assume that each pool has a sufficient amount of Token Y for the desire trade.)

Alice: $2 x + 5 y = k$
$2 (x + Δ x) + 5 (y - Δ y) = k$
$2 x + 2 Δ x + 5 y - 5 Δ y = k$
$- 5 Δ y = k - 5 y - 2 x - 2 Δ x$
$Δ y = (- \frac{1}{5}) \cdot (k - 5 y - 2 x - 2 Δ x)$

The above tells me how much Y I get back given a trade $Δ x = 1$ .

The amount depends upon the (x,y) pair being traded. Let's see if I can get them to cancel out if I am comparing.

Bob: $5 x + 2 y = k$
$5 (x + Δ x) + 2 (y - Δ y) = k$
$2 (y - Δ y) = k - 5 (x + Δ x)$
$2 y - 2 Δ y = k - 5 (x + Δ x)$
$- 2 Δ y = k - 5 x + 5 Δ x - 2 y$
$Δ y = - \frac{1}{2} \cdot (k - 5 x + 5 Δ x - 2 y)$

If we can compare the price as $Δ y$ where $Δ x = 1$ can we compare Alice and Bob's equations without knowing (x,y)?

In both cases, we probably could substitute their respective constant product equation.

This would allow us to remove y and subtitute x.

However, it would still mean that in both cases $Δ y$ is in terms of x.