# AB - Revisiting Repeated games¶

## Corresponding chapters¶

Duration: 50 minutes

## Objectives¶

• Re visit repeated games and ensure able to compute Nash equilibria for games that are not stage nash.

## Notes¶

Consider the $$(A,B)\in{\mathbb{R}^{2\times 2}}^2$$ game with $$T=2$$:

$\begin{split}A = \begin{pmatrix} 200 & 4 & 3 \\ 1 & 0 & 2 \end{pmatrix} \qquad B = \begin{pmatrix} 0 & -3 & 0 \\ -10 & 0 & -10 \end{pmatrix}\end{split}$

• Identify pure Nash equilibria of stage game:
$\begin{split}A = \begin{pmatrix} \underline{200} & \underline{4} & \underline{3} \\ 1 & 0 & 2 \end{pmatrix} \qquad B = \begin{pmatrix} \underline{0} & -3 & \underline{0} \\ -10 & \underline{0} & -10 \end{pmatrix}\end{split}$
• Identify repeated game Nash equilibria:

• $$(r_1r_1, c_1c_1)$$ with utility: (400, 0).
• $$(r_1r_1, c_1c_3)$$ with utility: (203, 0).
• $$(r_1r_1, c_3c_1)$$ with utility: (203, 0).
• $$(r_1r_1, c_3c_3)$$ with utility: (6, 0).
• Identify an equilibria that is not stage Nash:

1. For the row player:

$(\emptyset, \emptyset) \to r_2$
$(r_2, c_1) \to r_1$
$(r_2, c_2) \to r_1$
$(r_2, c_3) \to r_1$
2. For the column player:

$(\emptyset, \emptyset) \to c_2$
$(r_2, c_2) \to c_1$
$(r_1, c_2) \to c_3$

This corresponds to the following scenario:

> Play $$(r_2, c_2)$$ in first stage and $$(r_1,c_1)$$ in second stage unless the row player does not cooperate in which case play $$(r_1, c_3)$$.

This gives a utility of $$(200, 0)$$. Is this an equilibrium?

1. If the row player deviares, they would only be rational to do so in the
first state, if they did they would gain 4 but lose 197.
2. If the column player deviates they would do so in the first round and gain
no utility.

Potentially discuss how this repeated game framework can/does correspond to a normal normal form game.

• Writing all strategies down;
• Obtaining large matrix (very large)