Why is choosing each option one-third of the time a Nash equilibrium in rock-paper-scissors?
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, a Nash equilibrium is a state where no player can improve their expected outcome by unilaterally changing their strategy. In the game of rock-paper-scissors, there is no "pure" strategy—such as consistently choosing rock—that serves as an equilibrium, because an opponent could easily exploit that predictability to win every match.
Instead, the game relies on a "mixed" equilibrium. By choosing each option (rock, paper, or scissors) exactly one-third of the time, a player ensures their moves are perfectly unpredictable. If a player were to deviate from this distribution by choosing one option more frequently than the others, the opponent could identify that tendency and adjust their own strategy to win a higher percentage of matches. Therefore, the one-third distribution is the only strategy that prevents an opponent from gaining a statistical advantage, making it the Nash equilibrium for this game.1