Understanding the Nash Equilibrium Concept | Generated by AI
Question: What is the Nash Equilibrium concept?
Answer:
The Nash Equilibrium is a key concept in game theory, named after mathematician John Nash. It describes a situation in a non-cooperative game where no player can improve their payoff by unilaterally changing their strategy, assuming all other players keep their strategies unchanged.
Formal Definition
In a game with multiple players, a set of strategies (one for each player) forms a Nash Equilibrium if, for every player \(i\):
- The strategy chosen by player \(i\) is a best response to the strategies chosen by all other players.
- Deviating alone to any other strategy would yield a payoff that is equal or worse for player \(i\).
In other words, every player is playing optimally given what everyone else is doing.
Example: Prisoner’s Dilemma
Using the classic Prisoner’s Dilemma payoff matrix (payoffs are negative years in prison; lower is better):
| B Silent (Cooperate) | B Confess (Defect) | |
|---|---|---|
| A Silent (Cooperate) | (–1, –1) | (–3, 0) |
| A Confess (Defect) | (0, –3) | (–2, –2) |
- Both confessing (Defect, Defect) is the Nash Equilibrium.
- If A is confessing, B’s best response is to confess (–2 > –3).
- If B is confessing, A’s best response is to confess (–2 > –3).
- Neither can improve by switching alone.
- Both staying silent (Cooperate, Cooperate) is better for both (–1 > –2), but it is not a Nash Equilibrium because each has an incentive to defect unilaterally (0 > –1 if the other cooperates).
Key Properties
- A game can have zero, one, or multiple Nash Equilibria.
- Nash Equilibria can be Pareto inefficient (as in Prisoner’s Dilemma, where mutual cooperation is better but unstable).
- They can be in pure strategies (deterministic choice) or mixed strategies (randomizing over choices with certain probabilities).
- The concept applies to non-zero-sum games (most real-world interactions) and assumes players are rational and have complete information.
Real-World Applications
- Oligopoly pricing (e.g., firms choosing high or low prices).
- Arms races or international negotiations.
- Traffic flow (e.g., everyone choosing the same congested route).
- Evolutionary biology (stable strategies in populations).
Nash proved that every finite game with a finite number of players and strategies has at least one Nash Equilibrium (possibly in mixed strategies)—a result that earned him the Nobel Prize in Economics in 1994.
Connection to Other Concepts
- Dominant Strategy Equilibrium (stronger): A Nash Equilibrium where each player’s strategy dominates all others regardless of opponents’ actions.
- The Prisoner’s Dilemma has a dominant strategy equilibrium, which is also its unique Nash Equilibrium.