Nash Equilibrium (pure strategies)

Profiles where no player gains by switching alone.

The idea

A pure-strategy Nash equilibrium is the basic prediction for how a game will be played: a choice of strategy for each player that no player can improve on alone.

Each player $i$ chooses a strategy $s_{i}$, and a strategy profile $s = (s_{1}, \ldots, s_{n})$ names one choice per player, which determines every player's payoff $u_{i}(s)$. Write $s_{-i}$ for the choices of everyone other than $i$. Player $i s strategy is a best response to $s_{-i}$ when no other strategy of theirs pays more against it.

A profile $s$ is a pure-strategy Nash equilibrium when every player is best-responding at the same time: for each player $i$ and every alternative strategy $s_{i}'$,

$u_{i}(s_{i}, s_{-i}) \ge u_{i}(s_{i}', s_{-i}).$

The deviation is unilateral — one player switches while the rest stand still — so the condition says nothing about what a group could gain by moving together. Pure means each player names one definite strategy rather than randomizing over several.

In a two-player game the condition can be pictured on the payoff grid: with one player choosing between strategies $s$ and $s'$ and the other between $t$ and $t'$, draw an arrow from each cell to the cell the deviating player prefers. A cell that no arrow leaves is a pure-strategy Nash equilibrium.

Ways to work on it

• Walkthrough. Best responses and the equilibrium of a 2x2 game.
• Practice. Count the pure-strategy equilibria of a 2x2 game.
• Hardest. Test a candidate profile via best-response checks on both players.

Not sure where to start? Take the ten-question placement test.