Prisoners dilemma - exercises

1. Give the general definition for a Prisoners dilemma.

2. Justify if the following games are Prisoners dilemmas or not:

   1. $ A = \begin{pmatrix} 3 & 0\\ 5 & 1 \end{pmatrix} \qquad B = \begin{pmatrix} 3 & 5\\ 0 & 1 \end{pmatrix} $
   2. $ A = \begin{pmatrix} 1 & -1\\ 2 & 0 \end{pmatrix} \qquad B = \begin{pmatrix} 1 & 2\\ -1 & 0 \end{pmatrix} $
   3. $ A = \begin{pmatrix} 1 & -1\\ 2 & 0 \end{pmatrix} \qquad B = \begin{pmatrix} 3 & 5\\ 0 & 1 \end{pmatrix} $
   4. $ A = \begin{pmatrix} 6 & 0\\ 12 & 1 \end{pmatrix} \qquad B = \begin{pmatrix} 6 & 12\\ 0 & 0 \end{pmatrix} $

3. Obtain the Markov chain representation for a match between reactive strategies with the following vectors:

   1. $p=(1/2, 1/2)\qquad q=(1/2, 1/2)$
   2. $p=(1/4, 1/2)\qquad q=(1/2, 1/4)$
   3. $p=(1/3, 1/3)\qquad q=(2/3, 1/4)$

4. Obtain the utilities for both players for the vectors of question 3.

5. Assuming $p=(x, 1/2)$, find the optimal $x$ against the following players:

   1. $q=(1, 0)$
   2. $q=(1/2, 1/2)$

   Interpret these results.