Logic aims at truth, or more accurately, at deriving some truths from other truths. But why are we interested in truth in the first place? Surely one reason is that relying on truth makes it easier to make better choices.
One could think of Game theory as a tool which bridges the gap between logic and rationality.
Decision theory - or single agent game theory tells us when to make the best choice in a game of us against nature. But nature has no desire to further or frustrate our efforts. Nature is mysterious but not malign. Things change when there are other agents involved. Then the best thing for us to do will depend on what they do. And they will think the same. Issues like common knowledge and rationalizability will then arise.