I have 4 different profit functions and I would like to compare them analytically and find under which conditions what is that ranks of profit functions. For example, I expect an outcome as follows: if xxx is met then profit1>profit3>profit2>profit4
Below you can find the 4 profit functions. Thanks in advance!
Profit 1: $π_T^*$=$\frac{(2-δ)^2}{(12-δ ) } $
Profit2:
$π_T^*=\frac{-(F(δ- 2)^2 (5(δ^2- 4b^2 δ + 4b^2 )+ Fδ - 8F))}{(b^2 δ^2- 4b^2 δ + 4b^2+ 8Fδ - 12F+7)^2} $
Profit3
$π_T^*=\frac{-(α^2- 2F + 2Fδ)^2}{5(- α^2+ 4F)(α^2+6F + 2Fδ) }$
Profit 4
$π_T^*=\frac{-(- b^2+ Fα^2- 5Fδ + 4F)}{4(b^2- 3F + 5Fδ) } $