# Definition:Nash Equilibrium

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## Definition

Let a strategic game $G$ be modelled by:

- $G = \stratgame N {A_i} {\succsim_i}$

A **Nash equilibrium** of $G$ is a profile $a^* \in A$ of moves which has the property that:

- $\forall i \in N: \forall a_i \in A_i: \tuple {a^*_{-i}, a^*_i} \succsim_i \tuple {a^*_{-i}, a_i}$

Thus, for $a^*$ to be a **Nash equilibrium**, no player $i$ has a move yielding a preferable outcome to that when $a^*_i$ is chosen, given that every other player $j$ has chosen his own equilibrium move.

That is, no player can profitably deviate, if no other player also deviates.

## Source of Name

This entry was named for John Forbes Nash.

## Sources

- 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $2.2$: Nash Equilibrium: Definition $14.1$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**natural deduction**(J.F. Nash, 1950)