Exercises

Exercises#

Below, we have several practice problems and solutions. All of them can be solved by starting with Reduction by Dominance. They will reduce to, typically, a \(3\times 3\) matrix game that can be solved by Equalized Expectations.

Exercise 1

Find the mixed strategy solution by reducing the matrix game by dominance and then solving by equalized expectations.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrrr}&A&B&C\\ \hline A&10&11&12&3\\ B&7&-3&6&-5\\ C&10&2&11&6\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \vec r = \left[\begin{array}{c}\frac{1}{3} \\0 \\\frac{2}{3}\end{array}\right] &, \hspace{5mm} \vec c = \left[\begin{array}{c}0 \\\frac{1}{4} \\0 \\\frac{3}{4}\end{array}\right] \\[3mm]v&=5 \end{align*}\end{split}\]

Exercise 2

Find the mixed strategy solution by reducing the matrix game by dominance and then solving by equalized expectations.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrr}&A&B&C\\ \hline A&0&12&0\\ B&6&8&-1\\ C&3&5&10\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \vec r = \left[\begin{array}{c}0 \\\frac{1}{2} \\\frac{1}{2}\end{array}\right] &, \hspace{5mm} \vec c = \left[\begin{array}{c}\frac{11}{14} \\0 \\\frac{3}{14}\end{array}\right] \\[3mm]v&=\frac{9}{2} \end{align*}\end{split}\]

Exercise 3

Find the mixed strategy solution by reducing the matrix game by dominance and then solving by equalized expectations.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrr}&A&B&C\\ \hline A&5&12&3\\ B&12&6&11\\ C&9&1&-5\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \vec r = \left[\begin{array}{c}\frac{5}{14} \\\frac{9}{14} \\0\end{array}\right] &, \hspace{5mm} \vec c = \left[\begin{array}{c}0 \\\frac{4}{7} \\\frac{3}{7}\end{array}\right] \\[3mm]v&=\frac{57}{7} \end{align*}\end{split}\]

Exercise 4

Find the mixed strategy solution by reducing the matrix game by dominance and then solving by equalized expectations.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrrr}&A&B&C&D\\ \hline A&-1&7&2&-2\\ B&-5&0&1&3\\ C&-4&12&8&3\\ D&9&-5&9&6\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \vec r = \left[\begin{array}{c}0 \\0 \\\frac{7}{15} \\\frac{8}{15}\end{array}\right] &, \hspace{5mm} \vec c = \left[\begin{array}{c}\frac{17}{30} \\\frac{13}{30} \\0 \\0\end{array}\right] \\[3mm]v&=\frac{44}{15} \end{align*}\end{split}\]

Exercise 5

Find the mixed strategy solution by reducing the matrix game by dominance and then solving by equalized expectations.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrrr}&A&B&C&D\\ \hline A&1&11&1&4\\ B&9&0&5&9\\ C&5&8&-4&11\\ D&4&8&-5&-3\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \vec r = \left[\begin{array}{c}\frac{1}{3} \\\frac{2}{3} \\0 \\0\end{array}\right] &, \hspace{5mm} \vec c = \left[\begin{array}{c}0 \\\frac{4}{15} \\\frac{11}{15} \\0\end{array}\right] \\[3mm]v&=\frac{11}{3} \end{align*}\end{split}\]

Exercise 6

Find the mixed strategy solution by reducing the matrix game by dominance and then solving by equalized expectations.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrrr}&A&B&C&D\\ \hline A&5&-3&-3&7\\ B&3&9&5&8\\ C&-5&0&-1&3\\ D&1&4&6&-4\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \vec r = \left[\begin{array}{c}\frac{1}{5} \\\frac{4}{5} \\0 \\0\end{array}\right] &, \hspace{5mm} \vec c = \left[\begin{array}{c}\frac{4}{5} \\0 \\\frac{1}{5} \\0\end{array}\right] \\[3mm]v&=\frac{17}{5} \end{align*}\end{split}\]

Exercise 7

Find the mixed strategy solution by reducing the matrix game by dominance and then solving by equalized expectations.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrrrr}&A&B&C&D&E\\ \hline A&5&3&1&5&0\\ B&10&12&-2&-1&6\\ C&7&-3&2&1&-1\\ D&-2&10&3&5&9\\ E&1&6&-3&-1&12\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \vec r = \left[\begin{array}{c}0 \\0 \\\frac{1}{2} \\\frac{1}{2} \\0\end{array}\right] &, \hspace{5mm} \vec c = \left[\begin{array}{c}\frac{1}{10} \\0 \\\frac{9}{10} \\0 \\0\end{array}\right] \\[3mm]v&=\frac{5}{2} \end{align*}\end{split}\]