Nash Equilibrium

Micro GameTheory

See Pure-Strategy Nash Equilibrium, Mixed-Strategy Nash Equilibrium, Subgame Perfect Nash Equilibrium and so on.

Definition

A strategy profile $s^{\ast }=(s_1^{\ast },s_2^{\ast })$ is a Nash Equilibrium if

1. $u_1(s^{\ast })\ge u_1(s_1,s_2^{\ast })$ for all $s_1$ and
2. $u_2(s^{\ast })\ge u_2(s_1^{\ast },s_2)$ for all $s_2$

⇒ No player can be strictly better off by solely changing her strategy.

本质上，这是划线法的一种数学体现。

We need to make sure that if the other player plays this strategy, we would like to play this certain strategy. So preferably, in a $2\times 2$ game, each student would like to think about it twice.

In a Nash equilibrium, each player maximizes her payoff given a certain prediction.

If no one wants to deviate from the strategy profile $s^{\ast }$ unilaterally, then $s^{\ast }$ is the Nash Equilibrium.

Example:

	L	R
U	1,0	4,1
M	2,2	3,1
D	0,4	5,1

We can figure out there is no dominated strategy here, (see Iterated Dominance)

However, there exists a Nash Equilibrium.

Assume that we fix player 2 to play $L$, for player 1, her best response is $M$. If we fix player 2 to play $R$, for player 1, her best response is $D$.

1. If player 1 plays $U$, player 2 plays $R$
2. …
3. …

So we could easily get a Nash Equilibrium: $(M,L)$