Calculating utilities of strategies - Exercises

1. Give the definition of a mixed strategy.
2. For the following vectors explain which ones are valid mixed strategy vectors for a strategy set of size 5. If there are not: explain why.
   1. $\sigma=(1, 0, 0, 0, 0)$
   2. $\sigma=(1/4, 1/4, 0, 0, 1/4)$
   3. $\sigma=(1/4, 1/4, 1/2, -1/2, 1/2)$
   4. $\sigma=(1/4, 1/4, 0, 0, 11/20)$
   5. $\sigma=(1/5, 1/5, 1/5, 1/5, 1/5)$

3. Calculate the utilities (for both the row and column player) for the following game for the following strategy pairs:

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

   1. $\sigma_r = (.2, .8)\qquad\sigma_c = (.6, .4)$
   2. $\sigma_r = (.3, .7)\qquad\sigma_c = (.2, .8)$
   3. $\sigma_r = (.9, .1)\qquad\sigma_c = (.5, .5)$

4. Consider two column player strategies $z^{(1)}$ and $z^{(2)}$, obtain a linear algebraic expression for the expected utility to the row player playing against both these strategies, in terms of a row strategy $\sigma_r$, a payoff matrix $A$ and $\sigma_c=\frac{z^{(1)} + z^{(2)}}{2}$