Definition:Nash Equilibrium

This page is about Nash Equilibrium. For other uses, see 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.

Historical Note

The concept of Nash equilibrium was defined by John Forbes Nash in $1950$.

Sources

• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): $\text I$ Strategic Games: Chapter $2$ Nash Equilibrium: $2.2$: Nash Equilibrium: Definition $14.1$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Nash equilibrium (J.F. Nash, 1950)
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Nash equilibrium