# 07 Repeated games¶

## Corresponding chapters¶

## Objectives¶

- Define repeated games and strategies in repeated games
- Understand proof of theorem for sequence of stage Nash
- Understand that other equilibria exist

## Notes¶

### 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:

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:

For the row player:

\[(\emptyset, \emptyset) \to r_2\]\[(r_2, c_1) \to r_3\]\[(r_2, c_2) \to r_1\]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?

- If the row player deviates, they would do so in the first round and gain no utility.
- 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.