Why? It's just a waste of time really! Especially since it wastes a good few seconds for the in game battle for when you're EV training. And, if you accidentally touch the touch screen after battle, it accidentally pets your Pokemon's head.
Bonus question: Why are some people nicknaming their Aromatisse, Aunt Jemima? I don't get :/
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There's more to the game than just the competitive. I only have one Garchomp which also happens to be my competitive one, so if I want to see Garchomp in Amie then obviously I'll use my competitive one.