Nash Equilibrium

Nash Equilibrium

Key takeaways
* A Nash equilibrium is a set of strategies in which no player can improve their payoff by unilaterally changing their own strategy, given the other players’ strategies.
* Games can have one, multiple, or no Nash equilibria.
* Nash equilibrium assumes rational players who know or anticipate others’ strategies.
* It is distinct from a dominant strategy: a dominant strategy is best regardless of opponents’ choices; a Nash equilibrium is best given the others’ choices.

Definition
Nash equilibrium (named for mathematician John Nash) is a solution concept in game theory: when all players choose strategies such that no single player can benefit by changing only their own strategy, the game is at a Nash equilibrium.

How it works
* Model the players, their possible strategies, and payoffs for each strategy profile.
* For each player, given the other players’ choices, check whether any unilateral change yields a higher payoff.
* If no player can increase their payoff by changing strategy alone, the profile is a Nash equilibrium.

Example (simple two-player game)
Suppose Tom and Sam each choose A (gain $1) or B (lose $1). If both choose A, each gets $1. Revealing each other’s choices doesn’t create an incentive for either to switch—moving to B would make them worse off. The strategy profile (A, A) is therefore a Nash equilibrium.

Prisoner’s Dilemma — classic illustration
Two prisoners cannot communicate. Payoffs (in years in prison):
* Both betray: 5 years each.
* One betrays, other remains silent: betrayer goes free (0 years), silent prisoner gets 10 years.
* Both remain silent: 1 year each.

Each prisoner’s dominant action is to betray (it yields a better outcome regardless of the other’s choice), so both betray. That mutual-betray outcome is a Nash equilibrium, even though mutual silence (1 year each) is collectively better (Pareto superior). The dilemma highlights how individual incentives can produce suboptimal group outcomes.

Nash equilibrium vs. dominant strategy
* Dominant strategy: a strategy that is best for a player no matter what opponents do.
* Nash equilibrium: a profile of strategies where each is best given the others’ strategies.
A dominant-strategy profile (if it exists) is necessarily a Nash equilibrium, but Nash equilibria can exist without any dominant strategies.

Identifying Nash equilibria — practical steps
1. List all players and possible strategies.
2. Build payoff outcomes for each combination of strategies.
3. For each player and each strategy profile, ask: “Can this player improve payoff by changing strategy while others hold theirs fixed?”
4. Profiles where no player can improve are Nash equilibria.

Properties and limitations
* Multiple or no equilibria: some games have several equilibria, others none.
* Rationality assumption: analysis relies on players being rational and correctly anticipating others’ choices.
* Static vs. dynamic games: in repeated or sequential games, equilibrium concepts extend (e.g., subgame perfect equilibrium), and behavior can differ from one-shot outcomes.

Applications
Nash equilibrium is used across economics, political science, biology (evolutionary game theory), and any setting where strategic interactions matter—market competition, auctions, bargaining, and social behavior.

Conclusion
Nash equilibrium describes a stable strategic situation where no single player benefits from changing strategy unilaterally. It provides a powerful lens for predicting behavior in strategic settings, while its limitations (multiple equilibria, rationality assumptions, and potential inefficiency) are important when applying it to real-world problems.