In my head canon (read: something I made up at some point to explain stuff) the short end of the stick originated from that game where you have a stick, and the first person grabs the end in their fist, then the second person grabs just above the first person's fist, allowing the first person to let go and grab above the second person's fist, and so on up the stick. And the person who ends up with the short end of the stick loses.
(In fact, before reading this conversation, I didn't even realize this must be just head canon. I thought it was real . . . )