A pure Nash equilibrium is an action pair in which neither player can improve their payoff by changing only their own action. The simplest way to check a small game is to mark best responses one player at a time.
Do not start by looking for the largest sum of payoffs. Equilibrium concerns unilateral incentives, while efficiency compares outcomes differently.
Specify the payoff table
Player 1 chooses Top or Bottom. Player 2 chooses Left or Right. Each payoff pair lists player 1 first and player 2 second.
Top–Left gives (3, 3). Top–Right gives (0, 2). Bottom–Left gives (2, 0). Bottom–Right gives (1, 1).
Write down the ordering of the payoff pair before calculating. Reversing the players’ payoffs is a common source of errors.
Mark player 1’s best responses
If player 2 chooses Left, player 1 compares 3 from Top with 2 from Bottom. Top is the best response.
If player 2 chooses Right, player 1 compares 0 from Top with 1 from Bottom. Bottom is the best response. These comparisons stay within a column because player 2’s action is held fixed.
Mark player 2’s best responses
If player 1 chooses Top, player 2 compares 3 from Left with 2 from Right. Left is the best response.
If player 1 chooses Bottom, player 2 compares 0 from Left with 1 from Right. Right is the best response. These comparisons stay within a row because player 1’s action is held fixed.
Find the intersections
Top–Left and Bottom–Right are both mutual best responses. At either outcome, a unilateral deviation reduces the deviator’s payoff. There are therefore two pure Nash equilibria.
The two off-diagonal outcomes fail the test. At Top–Right, either player can improve by switching. The same is true at Bottom–Left.
Separate stability from welfare
Top–Left gives both players 3, whereas Bottom–Right gives both 1. The latter is Pareto dominated by the former, yet it remains a Nash equilibrium because one player cannot move both players together.
Multiple equilibria also leave a coordination question unresolved. Finding equilibrium action pairs does not explain which one real players will choose or how they will learn it.
This game also has an interior mixed equilibrium: each player assigns probability one-half to their first action. That follows from making the other player indifferent. Pure and mixed calculations answer related but separate parts of the game.
See Stanford’s game theory reference and ask Game Theory Chat about best-response tables.