# Rationalisation - Solutions¶

1. Give the definition of a dominated strategy.

2. Give the definition of a weakly dominated strategy.

3. Give the defininition of common knowledge of rationality.

4. For the following games predict rational behaviour or explain why this cannot be done:

1. $A = \begin{pmatrix} 2 & 1\\ 1 & 1\end{pmatrix} \qquad B = \begin{pmatrix} 1 & 1\\ 1 & 3\end{pmatrix}$

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.

1. $A = \begin{pmatrix} 2 & 1 & 3 & 17\\ 27 & 3 & 1 & 1\\ 4 & 6 & 7 & 18 \end{pmatrix} \qquad B = \begin{pmatrix} 11 & 9 & 10 & 22\\ 0 & 1 & 1 & 0\\ 2 & 10 & 12 & 0 \end{pmatrix}$

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 $(r_3, c_3)$.

1. $A = \begin{pmatrix} 3 & 3 & 2 \\ 2 & 1 & 3 \end{pmatrix} \qquad B = \begin{pmatrix} 2 & 1 & 3 \\ 2 & 3 & 2 \end{pmatrix}$

$c_1$ 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.

2. $A = \begin{pmatrix} 3 & -1\\ 2 & 7\end{pmatrix} \qquad B = \begin{pmatrix} -3 & 1\\ 1 & -6\end{pmatrix}$

There are no dominated strategies.

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