# Nash Equilibrium

Nash Equilibrium is a term used in game theory to describe an equilibrium where each player's strategy is optimal given the strategies of all other players. A Nash Equilibrium exists when there is no unilateral profitable deviation from any of the players involved. In other words, no player in the game would take a different action as long as every other player remains the same. Nash Equilibria are self-enforcing; when players are at a Nash Equilibrium they have no desire to move because they will be worse off.

Necessary Conditions

The following game doesn't have payoffs defined:

L

R

T a,b c,d

B

e,f g,h In order for (T,L) to be an equilibrium in dominant strategies (which is also a Nash Equilibrium), the following must be true: a > e c > g b > d f > h

In order for (T,L) to be a Nash Equilibrium, only the following must be true: a > or = e b > or = d

Prisoners' Dilemma (Again)

If every player in a game plays his dominant pure strategy (assuming every player has a dominant pure strategy), then the outcome will be a Nash equilibrium. The Prisoners' Dilemma is an excellent example of this. It was reviewed in the introduction, but is worth reviewing again. Here's the game (remember that in the Prisoners' Dilemma, the numbers represent years in prison):

Jack

C

NC

Tom

C

-10,-10

0,-20

NC

-20,0

-5,-5

In this game, both players know that 10 years is better than 20 and 0 years is better than 5; therefore, C is their dominant strategy and they will both choose C (cheat). Since both players chose C, (10,10) is the outcome and also the Nash Equilibrium. To check whether this is a Nash Equilibrium, check whether either player would like to deviate from this position. Jack wouldn't want to deviate, because if he chose NC and Tom stayed at C, Jack would increase his prison time by 10 years.

Iterated Deletion of Dominated Strategies

Here's another game that doesn't have dominant pure