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\centerline{Math 36 --- Weekly Homework}

\centerline{Assigned: 10/29} 

\centerline{Due: 11/5}
  

\begin{questions}

\question 
Prove that strict domination is a transitive relation. First, suppose that $R$, $S$, and $T$ are strategies for a player in an asymmetric game where $R$ is strictly dominated by $S$ and $S$ is strictly dominated by $T$. Now, prove that $R$ is strictly dominated by $T$.

\vspace{3 in}

\question
Give an example of a game (not an example from class) with a Nash equilibrium that is removed by iterated elimination of \emph{weakly} dominated strategies.

\newpage


\question

Use iterated elimination of strictly dominated strategies on the following game. Write the order in which you eliminated strategies, and which strategies remain after there are no more strictly dominated strategies.

  \begin{table}[h]
    \setlength{\extrarowheight}{2pt}
    \begin{tabular}{cc|c|c|c|c|}
      & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Player $2$} & \multicolumn{1}{c}{}\\
      & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$E$}  & \multicolumn{1}{c}{$F$} & \multicolumn{1}{c}{$G$} & \multicolumn{1}{c}{$H$}\\\cline{3-6}
      & $A$ & $(5,2)$ & $(3,1)$ & $(4,0)$ & $(6,5)$\\\cline{3-6}
      \multirow{2}*{Player $1$} & $B$ & $(3,4)$ & $(1,6)$ & $(8,3)$ & $(2,5)$\\\cline{3-6}
      & $C$ & $(7,6)$ & $(4,3)$ & $(2,2)$ & $(4,4)$\\\cline{3-6}
      & $D$ & $(4,7)$ & $(5,6)$ & $(6,5)$ & $(3,8)$\\\cline{3-6}
    \end{tabular}
  \end{table}

\vspace{3 in}

\question

Identify the strictly dominated strategies in this game, and the mixed strategies that dominate them.

  \begin{table}[h]
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    \begin{tabular}{cc|c|c|c|}
      & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Player $2$} & \multicolumn{1}{c}{}\\
      & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$D$}  & \multicolumn{1}{c}{$E$} & \multicolumn{1}{c}{$F$}\\\cline{3-5}
      & $A$ & $(4,4)$ & $(2,4)$ & $(4,3)$\\\cline{3-5}
      \multirow{2}*{Player $1$} & $B$ & $(4,5)$ & $(6,3)$ & $(5,3)$\\\cline{3-5}
      & $C$ & $(5,4)$ & $(5,6)$ & $(4,4)$\\\cline{3-5}
    \end{tabular}
  \end{table}



\end{questions}


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