Some people say “[I] have no dog in this hunt” (Google = 10.4K), and others “[I] have no dog in this fight” (Google = 116K). Now, competent readers recognize both expressions as colorful ways of disclaiming interest, so it would be mistaken to ask “which is right?” I wonder, though, whether one arose out of the other, or each developed more or less simultaneously/autonomously.
Given the tradition of hunting with packs of dogs in England, I’d suspect (without direct evidence) that “hunt” came first: “I can speculate disinterestedly about which dog will turn up a pheasant first, because I don’t have a dog in the contest.” Among people less accustomed to hunting with packs of dogs, the more intelligible version would transmute the trope to the practice of dog fighting. But that’s 100% speculation. If there were a convenient way to send up a flare to Language Log in general, rather than picking on one of the Loggers, I’d refer it to them — but for now, I’ll just leave my ruminations here.