Use when expressing neutrality or lack of bias in a dispute or competition. In the debate between vegetarians and meat-eaters, I don't have a dog in the fight.
Pro Tip 2/3
Conflict Avoidance
Implies the speaker wants to remain uninvolved to avoid conflict or stress. They were arguing over the project's direction, but I didn't have a dog in the fight.
Pro Tip 3/3
Emotional Distance
Suggests the speaker has no emotional connection to the outcome of a situation. As a foreigner, I don't have a dog in the fight regarding local elections.