Then (I think) for your inequality to hold, it must be that
U_B = f(3x+y), where f’ >= 0
If U_B care about x and y in any other proportion, then B can make trade-offs between x and y which makes things better for B, but worse for A.
This will be true (in theory) even if both A and B are satisfisers. You can see this by assuming replacing y and x with sigmoids of some other variables.
Lets say that
U_A = 3x + y
Then (I think) for your inequality to hold, it must be that
U_B = f(3x+y), where f’ >= 0
If U_B care about x and y in any other proportion, then B can make trade-offs between x and y which makes things better for B, but worse for A.
This will be true (in theory) even if both A and B are satisfisers. You can see this by assuming replacing y and x with sigmoids of some other variables.