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[parent] examples of normal form games (Example)

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:

\begin{displaymath} u_1(s_1,s_2)=\left\{ \begin{array}{ccccc} 5 &\text{ if } &s_... ... 0 &\text{ if } &s_1=D &\text{ and } &s_2=D \end{array}\right. \end{displaymath}
\begin{displaymath} u_2(s_1,s_2)=\left\{ \begin{array}{ccccc} 5 &\text{ if } &s_... ... 0 &\text{ if } &s_1=D &\text{ and } &s_2=D \end{array}\right. \end{displaymath}

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}^+$.

\begin{displaymath} u_1(s_1,s_2)=\left\{ \begin{array}{ccc} 1 &\text{ if } &s_1<s_2\ 0 &\text{ if } &s_1\geq s_2 \end{array}\right. \end{displaymath}
\begin{displaymath} u_2(s_1,s_2)=\left\{ \begin{array}{ccc} 1 &\text{ if } & s_2<s_1\ 0 &\text{ if } & s_2\geq s_1 \end{array}\right. \end{displaymath}



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Also defines:  prisoners dilemma, battle of the sexes

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Cross-references: strategy space, uncountable, infinite, Nash equilibrium, Pareto dominates, normal form, sentences, reflect, payoffs, information, game, player, game theory, dilemma, normal form games
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This is version 6 of examples of normal form games, born on 2002-07-23, modified 2005-11-08.
Object id is 3192, canonical name is ExamplesOfNormalFormGames.
Accessed 6382 times total.

Classification:
AMS MSC: 91A05 (Game theory, economics, social and behavioral sciences :: Game theory :: 2-person games)
 91A06 (Game theory, economics, social and behavioral sciences :: Game theory :: $n$-person games, $n>2$)
 91A10 (Game theory, economics, social and behavioral sciences :: Game theory :: Noncooperative games)
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