What did Nash prove about equilibria in finite noncooperative games?
The normal-form table illustrates the concept of a saddlepoint, or entry, in a payoff matrix at which the expected gain of each participant (row or column) has the highest guaranteed payoff.
In game theory, John Nash proved a fundamental existence theorem regarding Nash equilibria. He demonstrated that for every noncooperative game involving a finite number of players, where each player chooses from a finite number of strategies, there exists at least one Nash equilibrium, provided that mixed strategies are allowed.
A Nash equilibrium represents a stable outcome in which no player can improve their expected payoff by unilaterally changing their own strategy. While some games possess a "pure" equilibrium—where players consistently employ a single strategy—others require a "mixed" equilibrium, in which players must employ a specific probability distribution over their available strategies to remain in a stable state.1