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Lots of specific comments below. My experience was that game theory changed the way I think about almost any aspect of life, by giving me a set of simple tools. It's not the only toolkit you should have, but it's very powerful. "This is a coordination game; this is signalling; that is a prisoner's dilemma..." turns out to be a very useful vocabulary for thinking about situations, from everyday life up to superpower conflict.

For a great introduction from this point of view, check out Thomas Schelling's Strategy of Conflict. No maths, just a great read from a Nobel prize winner.