# 01 Strategies and utilities¶

## Objectives¶

- Define mixed strategies
- Understand utility calculation for mixed strategies
- Understand utility calculation as a linear algebraic construct

## Notes¶

### Utility calculations¶

Use ```
matching pennies
form
```

have students play in pairs.
Following each game:

- Ask how many people won?
- Ask why they won?

### Mixed strategies¶

Look at definition for mixed strategies.

Consider:

\[\begin{split}A =
\begin{pmatrix}
2 & -2\\
-1 & 1
\end{pmatrix}\qquad
B =
\begin{pmatrix}
-2 & 2\\
1 & -1
\end{pmatrix}\end{split}\]

Let us assume we have \(\sigma_r=(.3, .7)\) and \(\sigma_c=(.1, .9)\):

\[u_r(\sigma_r, \sigma_c) = 0.3 \times 0.1 \times 2 + 0.3 \times 0.9 \times
(-2) + 0.7 \times 0.1 \times (-1) + 0.7 \times 0.9 \times 1 = 0.08\]

because the game is zero sum we immediately know:

\[u_c(\sigma_r, \sigma_c) = -0.08\]

This corresponds to the linear algebraic multiplication:

\[u_r(\sigma_r, \sigma_c) = \sigma_r A \sigma_c^T\]

\[u_c(\sigma_r, \sigma_c) = \sigma_r B \sigma_c^T\]

(Go through this on the board, make sure students are comfortable.)

This can be done straightforwardly using `numpy`

:

```
>>> import numpy as np
>>> A = np.array([[2, -2], [-1, 1]])
>>> B = np.array([[-2, 2], [1, -1]])
>>> sigma_r = np.array([.3, .7])
>>> sigma_c = np.array([.1, .9])
>>> np.dot(sigma_r, np.dot(A, sigma_c)), np.dot(sigma_r, np.dot(B, sigma_c))
(0.079..., -0.079...)
```

### Strategy profiles as coordinates on a game¶

One way to thing of any game \((A, B)\in{\mathbb{R}^{m \times n}}^2\) is as a mapping from the set of strategies \([0,1]_{\mathbb{R}}^{m}\times [0,1]_{\mathbb{R}}^{n}\) to \(\mathbb{R}^2\): the utility space.

Equivalently, if \(S_r, S_c\) are the strategy spaces of the row/column player:

\[(A, B): S_r\times S_c \to \mathbb{R} ^2\]

We can use games defined in `nashpy`

in that way:

```
>>> import nashpy as nash
>>> game = nash.Game(A, B)
>>> game[sigma_r, sigma_c]
array([ 0.08, -0.08])
```