Short answers, pulled from the story.
Game theory is the study of mathematical models of strategic interactions. It has applications across the social sciences and is used extensively in economics, logic, systems science, and computer science. It is now an umbrella term for the science of rational decision making in humans, animals, and computers.
John von Neumann founded modern game theory, publishing On the Theory of Games of Strategy in 1928 and using Brouwer's fixed-point theorem in his proof. His work culminated in the 1944 book Theory of Games and Economic Behavior, co-authored with Oskar Morgenstern.
The Nash equilibrium is a set of strategies, one for each player, such that no player can improve their payoff by unilaterally changing their strategy. John Nash developed it in 1950 and proved that every finite n-player, non-zero-sum, non-cooperative game has a Nash equilibrium in mixed strategies.
Fifteen game theorists had won the Nobel Prize in economics as of 2020. They include John Nash, Reinhard Selten, and John Harsanyi in 1994, and most recently Paul Milgrom and Robert B. Wilson.
The prisoner's dilemma is a non-zero-sum game in which two arrested gang members each face the choice to stay silent or betray the other. The dominant strategy is to betray, yet both staying silent would yield a greater reward for both than mutual betrayal.
In biology, game theory interprets payoffs as fitness and centers on the evolutionarily stable strategy, where every ESS is a Nash equilibrium. It has been used to explain roughly 1:1 sex ratios, animal communication, and biological altruism through kin selection and Hamilton's rule. John Maynard Smith applied it in his 1982 book Evolution and the Theory of Games.