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What is Nash Equilibrium?

Nash Equilibrium is a strategic outcome where each player's choice is optimal given the choices of others. It's the stable point where no one can improve by unilaterally switching strategies.

Short answer

Nash Equilibrium is a strategic configuration where no player benefits from changing strategy alone, given the others' choices—the cornerstone of game theory.

2×2 Game Matrix Example

Player A cooperates

•Both gain equally
•Vulnerable to betrayal

Player A defects

•Higher personal gain
•Mutual loss risk

Step-by-step worked examples

In the Prisoners' Dilemma, each player can confess or stay silent. If both stay silent, each gets 1 year. If one confesses and the other doesn't, the confessor gets 0 years and the other gets 5. If both confess, each gets 3 years. Where is the Nash Equilibrium?

Check each player's incentive:
If A stays silent → B confesses (0 years vs 5)
If A confesses → B confesses (3 years vs 5)
Both confess is NE (each prefers it given the other's choice)
But both silent is socially better!

Two firms choose to enter or stay out of a market. Entering costs 10, and profit depends on competition. If both enter: profit 15 each. If one enters, one out: profit 30 vs 0. If both stay out: profit 0. Find the Nash Equilibrium.

If rival stays out → I enter (30 > 0)
If rival enters → I enter (15 > 0)
Both firms enter is NE (dominant strategy)

Rock-Paper-Scissors: each option beats one and loses to one. Is there a pure strategy Nash Equilibrium?

If opponent plays Rock → I play Paper
But then opponent switches to Scissors
No pure NE exists
Mixed strategy NE: play each 1/3 of the time

Flashcards

Quick quiz

Q1.Nash Equilibrium means…

Correct answer: B. NE is individually stable—not everyone's best off, just that no one can improve alone.

Q2.In the Prisoners' Dilemma, is mutual silence a Nash Equilibrium?

Correct answer: B. If one stays silent, the other benefits from confessing. So mutual silence is not NE.

Q3.Can a Nash Equilibrium be worse for everyone than another outcome?

Correct answer: C. Prisoners' Dilemma: both confess (NE) yields 3 years each; both silent yields 1 year each.

Q4.What does 'unilaterally' mean in Nash Equilibrium?

Correct answer: B. Unilateral = one-sided. At NE, one player can't improve by changing alone.

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Common mistakes

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Confusing Nash Equilibrium with Pareto optimality. — Correct: NE is individually stable; Pareto is collectively efficient. They often differ.

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Assuming Nash Equilibrium is always unique. — Correct: Many games have multiple NE; some have none in pure strategies.

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Thinking players can communicate at NE. — Correct: NE assumes non-binding play—each player acts independently.

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Assuming NE is always the 'best' outcome. — Correct: NE is stable, not optimal. Prisoners' Dilemma shows worse-off outcomes at NE.

FAQ

What is Nash Equilibrium?

A strategic outcome where no player can improve by unilaterally changing strategy. It's the stable point in game theory.

How do you find Nash Equilibrium?

For each player, identify their best response to each rival strategy. NE is where all best responses intersect.

Is Nash Equilibrium always optimal?

No. It's individually rational but may be collectively suboptimal—Prisoners' Dilemma is the classic example.

What's the difference between Nash Equilibrium and Pareto efficiency?

NE is about stability (no one wants to change alone); Pareto is about efficiency (can't help one without hurting another).