Noun
(idiomatic) Something that may lead to gain, depending on the outcome; a position for which to campaign or cheer.
Synonyms: dog in the fight, dog in the race, horse in the race
A national political party is unlikely to feel it has a particular dog in the hunt for a typical small town mayoral race.
Used other than figuratively or idiomatically: see dog, hunt.
Source: en.wiktionary.org