Moderate -0.3 This is a standard D2 game theory question requiring routine application of taught algorithms: finding saddle points via maximin/minimax (part a), then setting up and solving linear equations for mixed strategies (part b). While multi-step, it involves direct application of memorized procedures with no novel insight or complex problem-solving required, making it slightly easier than average.
Two people, Raj and Cal, play a zero-sum game. The game is represented by the following pay-off matrix for Raj.
Cal
\cline { 2 - 5 }
Strategy
X
Y
Z
Raj
I
- 7
8
- 5
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II
6
2
- 1
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III
- 2
4
- 3
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\cline { 2 - 5 }
Show that this game has a stable solution and state the play-safe strategy for each player.
Ros and Carly play a different zero-sum game for which there is no stable solution. The game is represented by the following pay-off matrix for Ros, where \(x\) is a constant.
4
\begin{enumerate}[label=(\alph*)]
\item Two people, Raj and Cal, play a zero-sum game. The game is represented by the following pay-off matrix for Raj.
\begin{center}
\begin{tabular}{ l | c | c | c | c | }
& \multicolumn{3}{c}{Cal} & \\
\cline { 2 - 5 }
& Strategy & X & Y & Z \\
\hline
Raj & I & - 7 & 8 & - 5 \\
\cline { 2 - 5 }
& II & 6 & 2 & - 1 \\
\cline { 2 - 5 }
& III & - 2 & 4 & - 3 \\
\cline { 2 - 5 }
& & & & \\
\cline { 2 - 5 }
\end{tabular}
\end{center}
Show that this game has a stable solution and state the play-safe strategy for each player.
\item Ros and Carly play a different zero-sum game for which there is no stable solution. The game is represented by the following pay-off matrix for Ros, where $x$ is a constant.
\begin{center}
\begin{tabular}{ l | c | c | c | }
& \multicolumn{2}{c}{Carly} & \\
\cline { 2 - 4 }
& \multicolumn{2}{c}{Strategy} & $\mathbf { C } _ { \mathbf { 1 } }$ \\
\cline { 2 - 4 }\cline { 2 - 3 }
$\operatorname { Ros }$ & $\mathbf { R } _ { \mathbf { 1 } }$ & 5 & $\mathbf { C } _ { \mathbf { 2 } }$ \\
\cline { 2 - 4 }
& $\mathbf { R } _ { \mathbf { 2 } }$ & - 2 & $x$ \\
\cline { 2 - 4 }
& & & 4 \\
\hline
\end{tabular}
\end{center}
Ros chooses strategy $\mathrm { R } _ { 1 }$ with probability $p$.
\begin{enumerate}[label=(\roman*)]
\item Find expressions for the expected gains for Ros when Carly chooses each of the strategies $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$.
\item Given that the value of the game is $\frac { 8 } { 3 }$, find the value of $p$ and the value of $x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2009 Q4 [10]}}