The Nash Equilibrium Symbol Analysis

In A Beautiful Mind, mathematical ideas—especially the “Nash equilibrium,” Nash’s most important contribution to game theory—come to represent Nash’s own independent identity, his competitive and often vindictive spirit, and his self-interested approach to life. Nash becomes famous for his advances in game theory, a field dedicated to decision-making, interactions, and strategies among rational actors in competitive scenarios. Game theory applies the idea of “games” broadly, studying negotiations in various situations in which individuals are attempting to win, make a profit, or otherwise gain something. In his own life, Nash is intensely competitive, often challenging other mathematicians to “games.” He also develops rivalries with his colleagues and strategizes about how best to one-up his opponents or exploit others for his own benefit. Additionally, Nash often uses the language of mathematics to explain himself and his own relationships. In a letter, he describes himself as a “singularity,” a special point around which “variables,” other people, are arranged—a comparison that speaks powerfully to his own narcissism. Indeed, as a younger man, Nash’s relationships were unbalanced. He views himself as superior to his romantic interests, considering his own needs, or “payoffs,” before theirs. Alicia, Nash’s wife, sacrifices her own life for him, sparing no cost to help him recover from the devastating effects of schizophrenia; he rarely acknowledges her efforts.

Yet for all of his arrogance, Nash also longs for human connection. He wishes to love and be loved, in spite of his flaws, and the Nash equilibrium might be seen as a symbol for the kind of life he eventually learns to lead. In a Nash equilibrium between two players in a game, neither player can improve their strategies; they have both made the best decisions possible to them, and they share the payoff. As Nash slowly begins to return to society after years of suffering from schizophrenic delusions that severed his connection to reality, he realizes the error of his ways, and he attempts to make amends with those he has wronged, including Alicia. Nash acknowledges the importance of caring for those who have cared for him, and he comes to value love over rivalry and competition. By the end of A Beautiful Mind, Alicia and Nash are living together in a kind of harmony, an “equilibrium”: they take care of each other and their son, Johnny, sharing the “payoff” of this life—happiness.