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 \(T=2\):

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

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:

    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)