Repeated games

Note: These are not designed to be student facing.

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/.

Playing repeated games in pairs

  • Explain that we are about to play a game twice.
  • Explain that this has to be done SILENTLY

In groups we are going to play:

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

In pairs:

  • Decide on row/column player (recall you don’t care about your opponents reward).
  • We are going to play the game TWICE and write down both players cumulative scores.
  • Define a strategy and ask players to write down a strategy that must describe what they do in both stages by answering the following question:
    • What should the player do in the first stage?
    • What should the player do in the second stage given knowledge of what both players did in the first period?
  • SILENTLY, after having written down a strategy: show each other your strategies and SILENTLY agree on the pair of utilities. If you are unable to agree on a utility this indicates that the strategies were not descriptive enough. SILENTLY start again :)

As a challenge: repeat this (so repeatedly play a repeated game, repeatedly write down a new strategy) and make a note when you arrive at an equilibria (where no one has a reason to write a different strategy down)

If anyone arrives at an equilibria where the row player scores more than 24 and the column player more than 4 stand up as a pair.

Following this, assuming a pair has arrived at such an equilibrium discuss this.

Then work through the notes:

  • Definition of a repeated game;
  • Definition of a strategy;

  • Theorem of sequence of stage Nash (relate this back to the utilities of our game):

    • $(r_1r_1, c_1c_1)$ 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).

Now discuss the potential of a different equilibrium:

  1. For the row player:
\[(\emptyset, \emptyset) \to r_2 (r_2, c_1) \to r_3 (r_2, c_2) \to r_1\]
  1. For the column player:
\[(\emptyset, \emptyset) \to c_2 (r_1, c_2) \to c_1 (r_2, c_2) \to c_1 (r_3, c_2) \to c_1\]

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?

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

Thus this is Nash equilibrium.

  • Discuss how to identify such an equilibria: which player has an incentive to build a reputation? (The column player want to prove to be trustworthy to gain 2 in the final round).
  • Mention how this shows how game theory studies the emergence of unexpected behaviour.

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