Since the accepted answer doesn't detail any actual game theory in Starship Troopers, I'm going to nominate Larry Niven's aliens.
The Pak protectors are constantly at war because they instantly react to public knowledge in the style prescribed by game theory, just like the rational wives who kill their husbands.
The Moties are trapped in a Prisoner's Game-like matrix where every individual's incentive is to reproduce, but the payoff when everyone reproduces is recurrent societal collapse. Crazy Eddie is the figure that urges everyone to defect and try birth control or emigration via blind hyperspace jump instead.
Niven is just shot through with game theory. Earth under ARM is an unstable, non-Nash equilibrium, and they know it. Space combat involves predicting your opponent's best strategy based on when information about your last move reached them. Pierson's puppeteers have a strict code of deleting any information they use for blackmail, making for an effective reputational strategy in a repeated game. If Niven didn't use game theory first, he used it best.