I make these notes available with the intent of making it easier to plan and/or take notes from class.
Student facing resources for each topic are all available at vknight.org/gt/.
Duration: 50 minutes
Use matching pennies have students play in pairs. Following each game:
Look at definition for strategies.
Consider:
\[\begin{aligned} A = \begin{pmatrix} 2 & -2\\ -1 & 1 \end{pmatrix}\qquad B = \begin{pmatrix} -2 & 2\\ 1 & -1 \end{pmatrix} \end{aligned}\]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...)
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])
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