example of Nash equilibrium

Consider the first two games given as examples of normal form games.

In Prisoner’s Dilemma the only Nash equilibrium is for both players to play $D$: it’s apparent that, no matter what player $1$ plays, player $2$ does better playing $D$, and vice-versa for $1$.

Battle of the Sexes has three Nash equilibria. Both $(O,O)$ and $(F,F)$ are Nash equilibria, since it should be clear that if player $2$ expects player $1$ to play $O$, player $2$ does best by playing $O$, and vice-versa, while the same situation holds if player $2$ expects player $1$ to play $F$. The third is a mixed equilibrium; player $1$ plays $O$ with $\frac{2}{3}$ probability and player $2$ plays $O$ with $\frac{1}{3}$ probability. We confirm that these are equilibria by testing the first derivatives (if $0$ then the strategy is either maximal or minimal). Technically we also need to check the second derivative to make sure that it is a maximum, but with simple games this is not really necessary.

Let player $1$ play $O$ with probability $p$ and player $2$ plays $O$ with probability $q$.

$$u_1(p,q)=2pq+(1-p)(1-q)=2pq-p-q+pq=3pq-p-q$$

$$u_2(p,q)=pq+2(1-p)(1-q)=3pq-2p-2q$$

$$\frac{\partial u_1(p,q)}{\partial p}=3q-1$$

$$\frac{\partial u_2(p,q)}{\partial q}=3p-2$$

And indeed the derivatives are $0$ at $p=\frac{2}{3}$ and $q=\frac{1}{3}$.