Is pocket determined by who connects faster to the match?

As I understand it, positions are determined by colors. If the colors are 1, 2 and 3, 2 will always be flank.

My question is: if no one picks a color, are they determined by who connects to the match faster? As in, the first one to connect will be 1, the second will be 2 and the third will be 3.

Thanks in advance!