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.
The deck. There is a write-on deck at /decks/repeated-games/main.pdf: the stage game printed, boxes for \(\delta\) and the expected number of rounds under both dice, and the Grudger comparison built up to the threshold.
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
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:
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. The two dice are chosen to sit either side of the threshold: the first sustains cooperation and the second does not, so the room should feel the difference rather than be told about it. Contrast this with a game of a fixed, commonly known length, where backward induction unravels cooperation from the last round.
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.
The stage game. Low dominates High (\(5 > 3\) and \(1 > 0\)), so played once you both bid Low and get \(1\) each, when mutual High would have paid \(3\).
The die. Our first die stops on a 1, so \(\delta = 5/6\) and the expected number of rounds is \(1/(1 - \delta) = 6\). Carrying on only for a 5 or a 6 gives \(\delta = 1/3\) and one and a half rounds.
Grudger against Grudger. Conforming for ever pays \(3 + 3\delta + 3\delta^{2} + \cdots = 3/(1 - \delta)\). Deviating once pays \(5\) now and then \(1\) for ever, which is \(5 + \delta/(1 - \delta)\).
The threshold. Cooperation is worth it when
At \(\delta = 5/6\) conforming is worth \(18\) against \(10\) for deviating, so cooperation holds. At \(\delta = 1/3\) it is \(9/2\) against \(11/2\), so it does not. The two dice sit either side of \(\delta = 1/2\) deliberately: this is the moment the activity is built around, and it is worth asking the room to predict which way the second run will go before playing it.
Watch for a tempting near miss. Ending on a 1 or a 2 gives \(\delta = 2/3\), which still sustains cooperation (\(9\) against \(7\)), so a die that merely feels shorter is not enough. With a six-sided die anything above \(\delta = 1/2\) holds, so the second run has to carry on for at most two of the six faces.
A known last round. In a round known to be the last the future is worth nothing, so both bid Low. Both of you know that, so the round before it is effectively last, and so on: cooperation unravels all the way back. The die was doing its work not by making the game long but by making the end uncertain.
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.
General email templates to send before and after this class. Fill in the bracketed placeholders before sending.
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
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