Best response polytopes - exercises

  1. Define a polytope as a convex hull
  2. Define a polytope as an intersection of halfspaces
  3. For the following set of vertices draw the corresponding polytope and obtain their definition as an intersection of halfspaces:
    1. V = $\{(0, 0), (0, 1), (1, 0), (1, 1)\}$
    2. V = $\{(0, 0), (0, 1/4), (1, 2/3), (2, 1/5)\}$
    3. V = $\{(0, 0), (0, 1/4), (1, 2/3), (2, 1/5), (1, 0)\}$
  4. Define the best response polytopes.
  5. For the following games, obtain the best response polytopes, label the vertices and identify all Nash equilibria:
    1. $ A = \begin{pmatrix} 3 & -1\\ 2 & 7\end{pmatrix} \qquad B = \begin{pmatrix} -3 & 1\\ 1 & -6\end{pmatrix} $
    2. $ A = \begin{pmatrix} 2 & -1\\ 1 & 3\end{pmatrix} \qquad B = \begin{pmatrix} -2 & 2\\ 1 & -2\end{pmatrix} $
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