Repeated Games

Activity (20 minutes)

Goal. Make the shadow of the future tangible: introduce repeated play and the discount factor as the probability that the game continues, before we formalise any of it.

Run "Will it end?". Students pair off. In each round both players secretly choose to bid High or Low for a contract, using the stage game

\[ A = \begin{pmatrix} 3 & 0 \\ 5 & 1 \end{pmatrix} \]

where the first action is High (cooperate) and the second is Low (undercut). After every round the pair rolls a die: on a 1 the repeated game ends, otherwise it continues to another round. So the game continues with probability \(\delta = 5/6\). Players keep a running total.

Run it a couple of times. Some pairs stop after one round, others run long. Then hand out a bribentive to the podium, the three individual students with the highest average score per round, and ask:

Did you play differently knowing the game might continue?
Would you bid High in a round you knew for certain was the last?
Now end the game on a 1 or a 2 (so \(\delta = 4/6\)) and replay. Does a shorter expected future change how willing you are to cooperate?

Debrief. Connect their experience to the chapter: \(\delta\) is the probability the game does not end, the expected number of rounds is \(1/(1 - \delta)\), and cooperation is easier to sustain when \(\delta\) is large. Contrast this with a game of a fixed, commonly known length, where backward induction unravels cooperation from the last round.

Discussion (20 minutes)

Discuss the Repeated Games chapter.

Discussion Point: After the definition of a strategy in a repeated game, ask what a strategy looked like in our dice game, given the history of play.

Discussion Point: After the definition of an infinitely repeated game with discounting, ask how our die roll corresponds to \(\delta\).

Discussion Point: After the Folk Theorem, ask what it means for cooperation in the contractor game and more generally.

Discussion Point: After the finite-horizon discussion, ask why backward induction kills cooperation when the number of rounds is known in advance.

From the activity to the exam answer

The activity above is written up as a marked exam question: Question 1 (the in-class activity) on the Repeated games page, with a full worked solution. Closing the loop here is the step that helps students who find exams hard: work through that question together, or set it as the immediate follow-up, so they see the game they just played turned into a full-mark answer.

Communications

General email templates to send before and after this class. Fill in the bracketed placeholders before sending.

Before class

Hi all,

A reminder that our next Game Theory class covers Repeated Games.

All of the course materials, including the relevant chapter, are available
at https://vknight.org/gt/. It is worth skimming the chapter beforehand.

See you in class,
Vince

After class

Dear all,

Thanks for your work in today's class on Repeated Games.

A recording is available here [RECORDING LINK] and on Learning Central.

All class resources are available at https://vknight.org/gt/.

Thanks,
Vince