# 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:

 $u_{1}(s_{1},s_{2})=\left\{\begin{array}[]{ccccc}5&\text{ if }&s_{1}=C&\text{ % and }&s_{2}=C\\ 10&\text{ if }&s_{1}=D&\text{ and }&s_{2}=C\\ -5&\text{ if }&s_{1}=C&\text{ and }&s_{2}=D\\ 0&\text{ if }&s_{1}=D&\text{ and }&s_{2}=D\end{array}\right.$
 $u_{2}(s_{1},s_{2})=\left\{\begin{array}[]{ccccc}5&\text{ if }&s_{1}=C&\text{ % and }&s_{2}=C\\ 10&\text{ if }&s_{1}=C&\text{ and }&s_{2}=D\\ -5&\text{ if }&s_{1}=D&\text{ and }&s_{2}=C\\ 0&\text{ if }&s_{1}=D&\text{ and }&s_{2}=D\end{array}\right.$

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:

 A X Y Z 2,100 1,7 14,-5

Undercut

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

 $u_{1}(s_{1},s_{2})=\left\{\begin{array}[]{ccc}1&\text{ if }&s_{1}
 $u_{2}(s_{1},s_{2})=\left\{\begin{array}[]{ccc}1&\text{ if }&s_{2}
Title examples of normal form games ExamplesOfNormalFormGames 2013-03-22 12:51:27 2013-03-22 12:51:27 Henry (455) Henry (455) 9 Henry (455) Example msc 91A10 msc 91A06 msc 91A05 prisoners dilemma battle of the sexes