Understanding Nash Equilibrium
Most games have no dominant strategies, so the best choice depends on what others do, and their best choices depend on yours. Nash equilibrium resolves this circularity by looking for mutual consistency: a set of strategies where each is a best response to the others.
Russell and Norvig characterize it as the condition that no player can benefit by switching strategies while every other player keeps theirs unchanged, and describe it geometrically as a local optimum in the space of policies, a peak that slopes downward along every dimension, where each dimension is one player’s strategy choices. The unilateral clause is essential: the definition says nothing about what a coordinated group could achieve by deviating together.
The existence result is what makes the concept broadly usable. Nash proved that every finite game has at least one equilibrium, provided mixed strategies, randomizations over pure strategies, are permitted. Without that allowance, games such as matching pennies have no equilibrium at all, since whatever one player picks, the other prefers to switch.
Two cautions matter in application. Equilibrium is a stability property and carries no promise of a good outcome, as the prisoner’s dilemma demonstrates with an equilibrium both players would gladly trade away. And many games have several equilibria, so the concept alone does not predict which will occur, leaving open the question of how players coordinate.
Example of Nash Equilibrium
In the prisoner’s dilemma, mutual testimony is a Nash equilibrium: given that Bob testifies, Alice gets −5 by testifying and −10 by switching to refusal, so she does not switch, and symmetrically for Bob. Mutual refusal is not an equilibrium, because either could improve from −1 to 0 by deviating.
Matching pennies illustrates why mixed strategies are needed. Two players simultaneously show heads or tails; one wins on a match, the other on a mismatch. No pair of fixed choices is stable, since the loser always prefers to switch.
Its equilibrium is in mixed strategies: each plays heads with probability one half. Given a genuinely random opponent, every choice yields the same expected payoff, so neither has any incentive to deviate. This is the sense in which Nash’s theorem guarantees an equilibrium exists.
Frequently Asked Questions
Does a Nash equilibrium have to be good for the players?
No. It guarantees only that no individual can improve by acting alone. The prisoner’s dilemma has an equilibrium that both players would prefer to escape, and they still cannot, because escaping requires simultaneous coordinated deviation.
Can a game have more than one Nash equilibrium?
Yes, and many do. Coordination games typically have several, which is why the concept alone does not always predict play; additional considerations such as risk dominance, focal points, or communication are needed to say which is reached.
What is the relationship to dominant strategy equilibrium?
A dominant strategy equilibrium is always a Nash equilibrium, since a strategy that is best against everything is in particular best against what the others actually chose. The converse fails: most Nash equilibria involve strategies that are best responses only to the specific opposing strategies in that equilibrium.
The Bottom Line
A Nash equilibrium is a profile from which no player can profitably deviate alone. Nash’s theorem guarantees one exists in every finite game once mixed strategies are allowed, and the concept describes stability rather than efficiency or fairness.