Give the definition of a dominated strategy.
Bookwork: https://vknight.org/gt/chapters/03/#Definition-of-a-strictly-dominated-strategy
Give the definition of a weakly dominated strategy.
Bookwork: https://vknight.org/gt/chapters/03/#Definition-of-a-weakly-dominated-strategy
Give the defininition of common knowledge of rationality.
Bookwork: https://vknight.org/gt/chapters/03/#Definition-of-a-weakly-dominated-strategy
For the following games predict rational behaviour or explain why this cannot be done:
We see that $r_1$ weakly dominates $r_2$ so we have:
$$A=(2,1)\qquad B =(1,1)$$
There are no further strategies that can be eliminated.
We see however that $c_2$ weakly dominates $c_1$ which would give:
$$\begin{pmatrix}
(1,1)\\
(1,3)\\
\end{pmatrix}$$
Again, there are no further strategies that can be eliminated.
We see that $c_2$ is weakly dominated by $c_3$ so we have:
$$ A = \begin{pmatrix} 2 & 3 & 17\\ 27 & 1 & 1\\ 4 & 7 & 18 \end{pmatrix} \qquad B = \begin{pmatrix} 11 & 10 & 22\\ 0 & 1 & 0\\ 2 & 12 & 0 \end{pmatrix} $$
Now $r_3$ strictly dominates $r_1$ so we have:
$$ A = \begin{pmatrix} 27 & 1 & 1\\ 4 & 7 & 18 \end{pmatrix} \qquad B = \begin{pmatrix} 0 & 1 & 0\\ 2 & 12 & 0 \end{pmatrix} $$
Now $c_3$ stricly dominates $c_1$ and $c_4$ so we have:
$$ A = \begin{pmatrix} 1\\ 7 \end{pmatrix} \qquad B = \begin{pmatrix} 1 \\ 12 \end{pmatrix} $$
Thus the predicted rational behaviour is $(r_3, c_3)$.
$ A = \begin{pmatrix} 3 & 3 & 2 \\ 2 & 1 & 3 \end{pmatrix} \qquad B = \begin{pmatrix} 2 & 1 & 3 \\ 2 & 3 & 2 \end{pmatrix} $
$c_1$ is weakly dominated by $c_3$:
$$A = \begin{pmatrix} 3 & 2 \\ 1 & 3 \end{pmatrix} \qquad B = \begin{pmatrix} 1 & 3 \\ 3 & 2 \end{pmatrix}$$
There are no further dominated strategies.
$ A = \begin{pmatrix} 3 & -1\\ 2 & 7\end{pmatrix} \qquad B = \begin{pmatrix} -3 & 1\\ 1 & -6\end{pmatrix} $
There are no dominated strategies.