# Definition:Nash Equilibrium

## 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.