Nash equilibria - Exercises

1. Give the definition of a best response.

2. For the following games identify the best reponses:

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

   2. $ 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} $

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

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

3. Represent the following game in normal form:

   Assume two neighbouring countries have at their disposal very destructive armies. If both countries attack each other the countries' civilian population will suffer 10 thousand casualties. If one country attacks whilst the other remains peaceful, the peaceful country will lose 15 thousand casualties but would also retaliate causing the offensive country 13 thousand casualties. If both countries remain peaceful then there are no casualties.

   1. Clearly state the players and strategy sets.
   2. Plot the utilities to both countries assuming that they play a mixed strategy while the other country remains peaceful.
   3. Plot the utilities to both countries assuming that they play a mixed strategy while the other country attacks.
   4. Obtain the best responses of each player.

4. State and prove the best response condition.

5. Define Nash equilibria.