Repeated games

In class we spoke about repeated games.

You can see a recording of this here: https://cardiff.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=f8ffef2b-2a36-4bf4-ae9f-b11600a56276.

I asked you all to write strategies for the game defined by playing the following stage game twice:

\[A = \begin{pmatrix} \underline{12} & 6\\ 0 & \underline{24}\\ \underline{12} & 23\\ \end{pmatrix} \qquad B = \begin{pmatrix} \underline{2} & 1\\ \underline{5} & 4\\ \underline{0} & 0\\ \end{pmatrix}\]

We discussed things at length and found 4 Nash equilibria which corresponds to both players playing a strategy that did not depend on actions of either player:

• \((r_1r_1, c_1c_1)\) with utility: (24, 4).
• \((r_1r_3, c_1c_1)\) with utility: (24, 2).
• \((r_3r_1, c_1c_1)\) with utility: (24, 2).
• \((r_3r_3, c_1c_1)\) with utility: (24, 0).

We then discussed the following strategy:

1. For the row player:

\[\begin{array} (\emptyset, \emptyset) \to r_2\\ (r_2, c_1) \to r_3\\ (r_2, c_2) \to r_1\\ \end{array}\]

1. For the column player:

\[\begin{array} (\emptyset, \emptyset) \to c_2\\ (r_1, c_2) \to c_1\\ (r_2, c_2) \to c_1\\ (r_3, c_2) \to c_1 \end{array}\]

This corresponds to the following scenario:

Play \((r_2, c_2)\) in first stage and \((r_1,c_1)\) in second stage unless the column player does not cooperate in which case play \((r_3, c_1)\)

This gives a utility of \((36, 6)\). Is this an equilibrium?

We discussed the incentive either play might have to deviate from this: there is none. No player has a reason to deviate so this is a Nash equilibrium.