Last Update: May 10, 2022
The class should be partitioned into teams with optimal size of 3, but groups of size 1,2, and 4 are also OK. Each team will have a leader. You are welcome to volunteer to be a leader, first-come first-serve. Otherwise I will pick a leader at random.
Students should pick a project no later then 10:00 pm, April 10, 2022 (it should be included in the homework due that day). But the membeship is first-come-first-serve, so you are welcome to email me as soon as you decide, in order to guarantee a place.
The team leader will be in charge of coordinating the various contributions, and writing up (with the team's help) the first draft of a paper that will be definitely posted here. The first, preliminary, "skeleton" versions should be ready by
8:00pm, May 9, 2022, emailed to ShaloshBEkhad@gmail.com
but it is hoped that they will be expanded into publishable papers (at least in arxiv.org, and possibly in a a "real" journal). Each paper should be accompanied by at least one Maple package, and a write-up in .pdf .
See the projects from Spring 2021, projects from Spring 2020, and example from 2019. Here is another example (of a different kind). See also an example from 2018 and another example from 2018.
Team leader (and only member): Yuxuan Yang
Implement (correctly!) the clever construction of Jerrold Tunnell's brilliant academic son Sasa Radomirovic.
HERE IS the Maple Code.
Team leader: George Spahn
Team members: George Tsoukalas
Systematically experiment with Shapley value and Shapley-Shubik index for both artificial and "real-life" situations. In particular write a Maple implementation of the article On Banzhaf and Shapley-Shubik Fixed Points and Divisor Voting Systems by Alex Arnell, Richard Chen, Evelyn Choi, Miroslav Marinov, Nastia Polina, Aaryan Prakash, and test their conjectures, and who knows? even prove them!
HERE ARE the Maple Code and article
Team leader: AJ Bu
Team members: Kayla Gibson, Lucy Martinez, Natasha Ter-Saakov
Systematically explore the famous Gale-Shapley algorithm, nicely described in Chapter 10 of Notes on Introductory Combinatorics by Polya, Tarjan, and Wood.
In particular, investigate the quantities: "Average happiness" of men, "Average happiness of women", for each of the n!^(2*n) possible
choices, and by simulation, and who knows, "analytically", estimate, and find formulas for the expectation, variance, and higher moments
of these quantities.
HERE ARE the Maple Code and article
Team leader: Robert Dougherty-Bliss
Team members: Rebecca Embar
Continue and expand on the preliminary study done in this gem. Possibly using the Maple package LinDiopantus.txt investigate other scenarios, e.g. the probability of there being a clear-cut first-place winner, using majority rule, generalized cut-offs. Also find bijective proofs suggested by the explicit generating functions.
[Added April 11, 2022: Rebecca Embar found the brilliant bijection! It would be the subject of a great paper (coauthored with Dr. Z.)]
Also, by simulation study more than three candidates, and the scenarios: "Clear-cut full ranking", "Clear-cut choice of first-place", "Clear-cut choice of first and second-place" etc. See the input file and the corresponding output file.
[Added April 11, 2022: Possibly a more promising approach is to extend Rebecca Embar's nice bijection to more than three candidates]
HERE IS the article and Maple code
Team leader: Blair Seidler
Team members: Vikrant Ashvinkumar and Daniela Elizondo
Study the amazing Braess paradox, described nicely here p. 889 of pdf (p. 866 of book) in Frank Kelly's wonderful essay. Expand the preliminary work in the Maple package Braess.txt and try to solve the challenge problem (nicely written up by Emily Sergel) raised in Dr. Z.'s lecture.
HERE ARE the Maple Code and article
No takers. Postponed to the future.
Write a Maple package, and use it systematically where there are 2 or more players, but with a large class of pay-off functions, not only the original ones. Write Maple code to find the Nash equilibria, or the sequential version, as well as the "social optimum" and investigate how often the Nash or sequential outcomes are less from the social optimum, and by how much. Also make concrete the very abstract discussion in section 2.1C of Gibbons' book and draw nice pictures for arbitrary U(w,L) and R(L) (satisfying the convexity assumptions)
No takers. Postponed to the future.
For arbitrary game-graphs (inspired by the centipede game (See also here), find how often the "backwards induction" argument is sub-optimal, and systematically find ways to "prune" the game-graph to make it better for everyone.