But some of the people who left comments suggested that I could get a ball headpin with heavier (lower gauge) wire if I held the wire parallel with the flame. At first, I thought that would mean my hand would be closer to the flame, but it really isn't very different. As you can see, I use some long needle nosed pliers to do this. And the amazing thing is how well it works!!
The force of the flame seems to push the end of the wire to melt, and not only was I able to ball 16g wire, but I was able to get some pretty good sized balls on the end. The picture above shows a variety of headpins that were made using 16g to 20g wire.
Thank you for your comments and help!!