Admittedly I am not from America and I know absolutely nothing about America football so please excuse me if this is the stupidest question ever, and I am quoting you simply because it seems like you know what you're talking about.
You say deflated balls are easier to grip so it is an unfair advantage. But then is it not easier to grip for both teams? So does the advantage one team gains not equal the advantage that the other team gains?
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Each team is responsible for providing their own footballs for offense (kicking balls are separate). So the Colts footballs for offense are different than the Patriots. QBs have preferences for how they like their footballs (Aaron Rodgers likes his more inflated, Eli Manning likes his scuffed up) so they provide their own and the refs inspect them before the game to make sure they'll valid. This talks about it a bit from a former ball boy:
Deflate-Gate: An NFL Ball Boy's Perspective on Preparing Footballs - NBC News
"We would then work with the quarterbacks to customize the balls to their liking. This involved scrubbing them with stiff horsehair brushes to rub off the leather's slippery silicone sheen, and occasionally inflating or deflating the balls a very small amount, which I believe is legal to a degree. Quarterbacks are very particular about the way a ball feels in their hand, and we worked meticulously to match their particular preferences," Kester said.
Two hours before kickoff, he would bring the balls to the referees' locker room for inspection.
"I recall them having a pressure gauge in the locker room, but most often they just squeezed the balls, turned them over in their hands a few times each, and inspected the laces. I don't recall them ever rejecting one of our balls," he said.
"My thought process was, 'Let's get the balls exactly the way our quarterback wants them, and if the refs reject one or two before the game, no big deal. But there's no harm giving them our ideal balls and hoping they make it through inspection.'"