Nash equilibrium
A Nash equilibrium is the best strategy to play given the strategy the other player is actually using.
This is weaker than a dominant strategy — a dominant strategy is the best strategy regardless of what the other player does, whereas a NE strategy is only optimal in response to a specific opponent strategy. A dominant strategy outcome is always a Nash equilibrium, but a Nash equilibrium is not necessarily a dominant strategy.
Nash proved that all finite competitive games have at least one Nash equilibrium.
How to find Nash equilibria
Given a payoff matrix where rows = Player 1’s strategies and columns = Player 2’s strategies:
- Go down the first column. Find the highest payoff to Player 2 (the column player) for that strategy.
- Check whether Player 1’s (the row player’s) payoff in that box is also the highest payoff to Player 1 in that row. If so, that box is a NE, and the two strategies that produce it are NE strategies; otherwise there is no NE in that column.
- Repeat for each column.
Limitations
A Nash equilibrium describes a stable outcome, not necessarily a good one:
- In the prisoner’s dilemma, the unique NE leaves both players worse off than mutual cooperation — both players prefer box II but end up in box IV.
- When multiple NE exist (as in Stag Hunt or Battle of the Sexes), theory alone does not determine which equilibrium players will coordinate on.
In multi-NE games, a focal point (aka Schelling point) — a NE with some conspicuous, prominent, or salient feature that each player can reasonably expect the other to notice — can increase both players’ propensity to choose it, even without communication.