# 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 $q=\frac{1}{3}$.

Title | example of Nash equilibrium |
---|---|

Canonical name | ExampleOfNashEquilibrium |

Date of creation | 2013-03-22 12:52:48 |

Last modified on | 2013-03-22 12:52:48 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 6 |

Author | Henry (455) |

Entry type | Example |

Classification | msc 91A99 |