If you meant that B always loses every time, and A always wins, that isn't fair. If both are trying to send a lot of traffic, A will take over the entire capacity of the network, and B will get nothing -- not very satisfactory.
If you mean that each PC remembers who won the last time and you alternate, that is also problematic, for a different reason. It doesn't scale well to more than two PCs on the same network.
The standard protocol is memoryless; it doesn't require keeping track of past history. This makes it scalable, efficient, and simple.
Your understanding of the standard protocol seems faulty. In the standard protocol, if there is a collision, both parties back off; so both A and B would be picking from {0,1,2,3} (rather than A picking from {0,1} and B from {0,1,2,3}).