Based on the NFL rules, and the NYT article on how Eli coats his balls with "the precious" and what-not, it leads me to think these things:
1. If the balls were really deflated, it would have been done by ball attendant after the referee checked the ball pressure (assuming the refs properly did their job). Is the ball attendant a home team employee?
2. If they were deflated intentionally, hard to believe that it was done without knowledge of the QB, higher-ups, etc.
3. Road teams have their own balls, and they use them in games (for instance, the mention of the Giants intercepting their own monikered ball because KC didn't change it to the KC ball when the latter was on offense.
Presumably the comparative advantage might be a slightly better grip on the ball in bad weather conditions, particularly for passing.