# AB - Revisiting Repeated games¶

## 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 \(T=2\):

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

**Ask students in pairs to:**

- 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:

For the row player:

\[(\emptyset, \emptyset) \to r_2\]\[(r_2, c_1) \to r_1\]\[(r_2, c_2) \to r_1\]\[(r_2, c_3) \to r_1\]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?

- 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.

- 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)