# examples of normal form games

A few example normal form games:

Prisoner’s dilemma^{}

Probably the most famous game theory^{} example, the prisoner’s dilemma is a two player game where ${S}_{1}={S}_{2}=C,D$ and:

$${u}_{1}({s}_{1},{s}_{2})=\{\begin{array}{ccccc}\hfill 5\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=C\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=C\hfill \\ \hfill 10\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=D\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=C\hfill \\ \hfill -5\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=C\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=D\hfill \\ \hfill 0\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=D\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=D\hfill \end{array}$$ |

$${u}_{2}({s}_{1},{s}_{2})=\{\begin{array}{ccccc}\hfill 5\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=C\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=C\hfill \\ \hfill 10\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=C\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=D\hfill \\ \hfill -5\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=D\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=C\hfill \\ \hfill 0\hfill & \hfill \text{if}\hfill & \hfill {s}_{1}=D\hfill & \hfill \text{and}\hfill & \hfill {s}_{2}=D\hfill \end{array}$$ |

Traditionally this is interpreted as the case of two criminal partners separately being interrogated and asked to give up the other partner. $C$ stands for cooperating (with their partners) by refusing to give up information, and $D$ stands for defecting and agreeing to testify against the partner. The different payoffs reflect different jail sentences^{}, ranging from nothing (+10) to a long jail sentence (-5), with amounts in between depending on the evidence against them.

The (much more convenient) normal form is: C D C 5,5 -5,10 D 10,-5 0,0

Notice that $(C,C)$ Pareto dominates $(D,D)$, however $(D,D)$ is the only Nash equilibrium^{}.

Battle of the Sexes

Another traditional two player game. The normal form is: O F O 2,1 0,0 F 0,0 1,2

A Degenerate Example

One more, rather pointless, example which illustrates a game where one player has no choice:

X | Y | Z | |

A | 2,100 | 1,7 | 14,-5 |

Undercut

A game which illustrates an infinite^{} (indeed, uncountable) strategy space. There are two players and ${S}_{1}={S}_{2}={\mathbb{R}}^{+}$.

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Title | examples of normal form games |
---|---|

Canonical name | ExamplesOfNormalFormGames |

Date of creation | 2013-03-22 12:51:27 |

Last modified on | 2013-03-22 12:51:27 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 9 |

Author | Henry (455) |

Entry type | Example |

Classification | msc 91A10 |

Classification | msc 91A06 |

Classification | msc 91A05 |

Defines | prisoners dilemma |

Defines | battle of the sexes |