Exercises

Exercises#

Below, we have several practice problems and solutions. All of them can be solved down to a single Equilbrium Pair by using Reduction by Dominance.

Exercise 1

Find the equilibrium pair by reducing the matrix game by dominance.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrr}&A&B&C\\ \hline A&-2&-4&10\\ B&-1&2&7\\ C&5&5&9\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \text{Equilibrium Pair} &: (\text{Rose }C,\text{Colin }B) \\[3mm]v&=5 \end{align*} \end{split}\]

Exercise 2

Find the equilibrium pair by reducing the matrix game by dominance.

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

Solution
\[\begin{split}\begin{align*} \text{Equilibrium Pair} &: (\text{Rose }C,\text{Colin }B) \\[3mm]v&=5 \end{align*} \end{split}\]

Exercise 3

Find the equilibrium pair by reducing the matrix game by dominance.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrr}&A&B&C\\ \hline A&-3&6&-2\\ B&9&8&1\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \text{Equilibrium Pair} &: (\text{Rose }B,\text{Colin }C) \\[3mm]v&=1 \end{align*} \end{split}\]

Exercise 4

Find the equilibrium pair by reducing the matrix game by dominance.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrr}&A&B&C\\ \hline A&3&4&6\\ B&2&4&1\\ C&3&9&9\\ D&0&9&4\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \text{Equilibrium Pair} &: (\text{Rose }C,\text{Colin }A) \\[3mm]v&=3 \end{align*} \end{split}\]

Exercise 5

Find the equilibrium pair by reducing the matrix game by dominance.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rr}&A&B\\ \hline A&12&-4\\ B&7&7\\ C&4&-5\\ D&3&-4\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \text{Equilibrium Pair} &: (\text{Rose }B,\text{Colin }B) \\[3mm]v&=7 \end{align*} \end{split}\]

Exercise 6

Find the equilibrium pair by reducing the matrix game by dominance.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrr}&A&B&C\\ \hline A&5&0&2\\ B&-5&-3&-4\\ C&2&-2&5\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \text{Equilibrium Pair} &: (\text{Rose }A,\text{Colin }B) \\[3mm]v&=0 \end{align*} \end{split}\]

Exercise 7

Find the equilibrium pair by reducing the matrix game by dominance.

\(\begin{array}{cc}&\text{Colin}\\\text{Rose}&\begin{array}{r|rrr}&A&B\\ \hline A&-4&-1&10\\ B&-1&7&1\\\end{array}\end{array}\)

Solution
\[\begin{split}\begin{align*} \text{Equilibrium Pair} &: (\text{Rose }B,\text{Colin }A) \\[3mm]v&=-1 \end{align*} \end{split}\]