"Nevertheless, in a single game, the best strategy is to snitch because it guarantees that you don't get the maximum jail term. "

Huh? That isn't correct is it? Snitching is best because it gives the highest payout regardless of what the other person does (snitching strictly dominates not snitching).

If the other does not snitch and you do, you go free. If the other person does snitch, you should as well, because it gives you 3 months instead of 6.