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Nash Equilibrium Condition

ui(si∗,s−i∗)≥ui(si,s−i∗)  ∀siu_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \;\forall s_i

No player gains by unilaterally changing strategy when others hold theirs fixed.

By John Nash

Social Sciences
Nash Equilibrium Condition
1950 · John Nash
Why it matters: Explains oligopoly pricing, arms races, auctions, and evolutionary stability.

Discoverers: John Nash (1950)

What does it mean?

No player gains by unilaterally changing strategy when others hold theirs fixed.

Why should I care?

Explains oligopoly pricing, arms races, auctions, and evolutionary stability.

Variables & Units

SymbolNameUnitMeaning
uiu_iPayoff—Player i utility
si∗s_i*Strategy—Equilibrium action
s−i∗s_{-i}*Others' strategies—Fixed opponents

Worked Example

Prisoner's dilemma: mutual defection is Nash though cooperation Pareto-dominates.

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Sources & further reading

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Nash Equilibrium Condition

ui(si∗,s−i∗)≥ui(si,s−i∗)  ∀siu_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \;\forall s_i

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No player gains by unilaterally changing strategy when others hold theirs fixed.

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