Easy -1.3 This is a standard textbook exercise in zero-sum game theory from Decision Mathematics. Part (a) requires routine application of maximin/minimax to identify a saddle point, while part (b) involves straightforward calculation of mixed strategies using the standard 2×2 reduction formula. Both are algorithmic procedures with no novel problem-solving required.
Two people, Ann and Bill, play a zero-sum game. The game is represented by the following pay-off matrix for Ann.
\multirow{5}{*}{Ann}
Bill
Strategy
\(\mathbf { B } _ { \mathbf { 1 } }\)
\(\mathbf { B } _ { \mathbf { 2 } }\)
\(\mathbf { B } _ { \mathbf { 3 } }\)
\(\mathbf { A } _ { \mathbf { 1 } }\)
-1
0
-2
\(\mathbf { A } _ { \mathbf { 2 } }\)
4
-2
-3
\(\mathbf { A } _ { \mathbf { 3 } }\)
-4
-5
-3
Show that this game has a stable solution and state the play-safe strategies for Ann and Bill.
Russ and Carlos play a different zero-sum game, which does not have a stable solution. The game is represented by the following pay-off matrix for Russ.
3
\begin{enumerate}[label=(\alph*)]
\item Two people, Ann and Bill, play a zero-sum game. The game is represented by the following pay-off matrix for Ann.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{5}{*}{Ann} & \multicolumn{4}{|c|}{Bill} \\
\hline
& Strategy & $\mathbf { B } _ { \mathbf { 1 } }$ & $\mathbf { B } _ { \mathbf { 2 } }$ & $\mathbf { B } _ { \mathbf { 3 } }$ \\
\hline
& $\mathbf { A } _ { \mathbf { 1 } }$ & -1 & 0 & -2 \\
\hline
& $\mathbf { A } _ { \mathbf { 2 } }$ & 4 & -2 & -3 \\
\hline
& $\mathbf { A } _ { \mathbf { 3 } }$ & -4 & -5 & -3 \\
\hline
\end{tabular}
\end{center}
Show that this game has a stable solution and state the play-safe strategies for Ann and Bill.
\item Russ and Carlos play a different zero-sum game, which does not have a stable solution. The game is represented by the following pay-off matrix for Russ.
\begin{center}
\begin{tabular}{ l | c | c | c | c | }
& \multicolumn{3}{c}{Carlos} & \\
\cline { 2 - 5 }
& Strategy & $\mathbf { C } _ { \mathbf { 1 } }$ & $\mathbf { C } _ { \mathbf { 2 } }$ & $\mathbf { C } _ { \mathbf { 3 } }$ \\
\cline { 2 - 5 }
Russ & $\mathbf { R } _ { \mathbf { 1 } }$ & - 4 & 7 & - 3 \\
\cline { 2 - 5 }
& $\mathbf { R } _ { \mathbf { 2 } }$ & 2 & - 1 & 1 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Find the optimal mixed strategy for Russ.
\item Find the value of the game.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2010 Q3 [12]}}