Nash Equilibrium
What is Nash Equilibrium?
Nash Equilibrium is a stable state of a game where no player can improve their payoff by unilaterally changing their strategy.
In a Nash Equilibrium, each player's strategy is optimal given the strategies of the other players. No player can benefit by changing their strategy while the other players keep theirs unchanged. Nash Equilibria can occur in non-cooperative games with two or more players.
Nash Equilibrium: a worked example
Two food trucks each pick the Beach lot or the Park lot for the day. Beach customers spend $600 in total and Park customers spend $400, and if both trucks pick the same lot they split that lot's money evenly. Say truck A takes the Beach and truck B takes the Park, so A earns $600 and B earns $400. Now test each one alone. If A switched to the Park, the two would split $400 and A would get $200, worse than $600. If B switched to the Beach, they would split $600 and B would get $300, worse than $400. Neither gains by moving alone, so this pair of choices is a Nash equilibrium.
The mistake students make with nash equilibrium
Students assume a Nash equilibrium is the outcome that is best for the group, and that each game has exactly one. Both beliefs fail. The food truck game above has two Nash equilibria, A at the Beach with B at the Park or the reverse, and a prisoner's dilemma has a single Nash equilibrium that leaves both players worse off than cooperating would. The only test is whether one player, moving alone, can do better. Total payoff never enters the check.
Nash Equilibrium questions
How do you find a Nash equilibrium in a payoff matrix?
A Nash equilibrium is found by checking one player at a time. Fix the column player's choice, then underline the row player's highest payoff in that column, and repeat for the other column. Then fix each row and underline the column player's highest payoff in it. Any cell with both numbers underlined is a Nash equilibrium, because neither player can gain by switching alone.
Is a dominant strategy the same as a Nash equilibrium?
A dominant strategy is not the same as a Nash equilibrium, though the two are related. A dominant strategy is one player's best move no matter what the other does. When every player has one, the resulting cell is automatically a Nash equilibrium. But a game can have a Nash equilibrium with no dominant strategies at all, where each player's best move depends on what the other picks.
Can a game have no Nash equilibrium?
A game can have no Nash equilibrium in pure strategies. Matching pennies is the standard case: whatever pair of choices you name, one player wants to switch, so the checking process never lands on a stable cell. Allowing mixed strategies, where players randomize between moves, restores an equilibrium, but that idea sits beyond the simple two-by-two matrices used in an introductory course.
Related terms
Common comparisons
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