Calculating utilities of strategies - Solutions

1. Give the definition of a mixed strategy.

   Bookwork: https://vknight.org/gt/chapters/02/#Definition-of-mixed-strategies

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. $\sigma=(1, 0, 0, 0, 0)$: Valid.
   2. $\sigma=(1/4, 1/4, 0, 0, 1/4)$: Not valid, do not sum to 1.
   3. $\sigma=(1/4, 1/4, 1/2, -1/2, 1/2)$: Not valid, non positive values.
   4. $\sigma=(1/4, 1/4, 0, 0, 11/20)$: Not valid, do not sum to 1.
   5. $\sigma=(1/5, 1/5, 1/5, 1/5, 1/5)$: Valid.

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. $\sigma_r = (.2, .8)\qquad\sigma_c = (.6, .4)$

      $$u_r(\sigma_r, \sigma_c) = \sigma_r A \sigma_c ^T = .2\times.6\times 1 + .2\times .4\times (-1) + .8\times.6\times (-3) + .8\times .4\times 1=-1.08$$

      $$u_c(\sigma_r, \sigma_c) = \sigma_r B \sigma_c ^T = .2\times.6\times (-1) + .2\times .4\times 2 + .8\times.6\times 1 + .8\times .4\times (-1)=0.2 $$

   2. $\sigma_r = (.3, .7)\qquad\sigma_c = (.2, .8)$

      $$u_r(\sigma_r, \sigma_c) = -0.04$$

      $$u_c(\sigma_r, \sigma_c) = 0$$

   3. $\sigma_r = (.9, .1)\qquad\sigma_c = (.5, .5)$

      $$u_r(\sigma_r, \sigma_c) = -0.1$$

      $$u_c(\sigma_r, \sigma_c) = 0.45$$

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 $\sigma_c=\frac{z^{(1)} + z^{(2)}}{2}$:

   $$E(u_r(\sigma_r)) = 1/2u_r(\sigma_r, z^{(1)}) + 1/2u_r(\sigma_r, z^{(2)}) = 1/2\sigma_rA{z^{(2)}}^T + 1/2\sigma_rA{z^{(1)}}^T = \sigma_rA\left(1/2{z^{(2)}}^T + 1/2{z^{(1)}}^T\right)=\sigma_rA\sigma_c^T$$