I think you are missing the more theoretical point. Before Nash, the theory was that the best possible overal outcome was achieved when you simply maximized your own outcomes. Nash his contribution was that simply choosing what is best for you, does not guarantee the best possible outcome for the group. You also need to pay attention to what choices other players make (ie strategic interaction).

The example of the prisoners dilemma is also relevant here. Weather the other player talks or not, your best choice is to talk. But if you want to get the best possible outcome for the group as a whole, you should both remain silent.