Goal. Motivate fair division and the question the Shapley value answers: a team produces one thing together, the parts reinforce each other, and we have to split the proceeds fairly when the parts did not contribute equally.
Run "The revision aid". The whole room plays at once in teams of three (about twenty teams in a class of sixty). In roughly twelve minutes each team makes a one-page revision aid for a game theory topic, on a single poster with three clearly separated parts, one per student:
Set the same topic for a fair contest, or give each team a different topic so the gallery ends up covering the syllabus.
Put every poster on the wall. The class is the market: each student walks the gallery and places one vote on the aid they would most want to revise from, but not their own. The aid with the most votes wins, and its team takes the bribentive. Photograph the winners and share them: they are real revision material for the class.
Now the part that makes it a cooperative game. Take the winning aid and split its prize fairly among the three students by the Shapley value, with the characteristic function supplied live by the class. Cover the poster and reveal the parts in combination, asking the class to score what is on show out of ten:
These seven numbers are the worth of each coalition of the three students, and they are not additive: a picture on its own is a cryptic diagram, but beside the maths and a plain-English story it makes the topic click. Write them on the board, compute the Shapley value, and hand out the bribentive in those shares.
Debrief. The three parts are worth only a few points on their own, yet the whole aid scores ten, so most of the value comes from the parts reinforcing one another. The Shapley value is what shares that synergy fairly: it pays each student the average value their part adds over every order in which the aid could come together, so the part that combines best with the others is rewarded most, above what it is worth alone. This is exactly Question 1, now with numbers the class produced.
Work through the Cooperative Games chapter.
Discussion Point: After the definition of a characteristic function game, point out that the seven scores the class gave are exactly the characteristic function.
Discussion Point: After the definition of marginal contribution, ask how much the maths is worth on its own and how much it adds once the picture and story are already there; the same part contributes more in company.
Discussion Point: After the definition of the Shapley value, ask for the steps, then work through the six orderings for the winning aid.
The activity above is written up as a marked exam question: Question 1 (the in-class activity) on the Cooperative 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 Cooperative 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 Cooperative Games.
A recording is available here [RECORDING LINK] and on Learning Central.
All class resources are available at https://vknight.org/gt/.
Thanks,
Vince