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.

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