Kyle Lee - DIMACS REU 2025

About Me

Welcome to my website! My name is Kyle Lee, and I am studying Social Choice Theory this summer with Professor Lirong Xia. My mathematical interests include Algebraic Number Theory, Enumerative Combinatorics, and Geometric Group Theory, and I am also interested in the intersection of these fields (and mathematics in general) with applicatory fields such as Computer Science and Economics. This is what makes Social Choice Theory fascinating to me: it is a rigorous and mathematically fulfilling field that has important direct application.

My other interests include literature, space photography, and card magic. I also love to play video games in my free time (my favorites are Disco Elysium and Hollow Knight). I am (as my fellow REU-mates might know) also a board games enthusiast!

As for some background, I am a NJ native that currently attends the University of Michigan pursuing a degree in Mathematics. You can reach me at leekylej@umich.edu and my office for the summer is located in the CoRE Building, Room 440.

Progress Log

Week 1

I spent much of my time in week 1 getting situated and reading through the Handbook of Computational Social Choice: Chapters 1 and 2. After reading Most Equitable Voting Rules, a paper written by my mentor, Dr. Xia, I think I'm getting somewhat enamored by the inherent group symmetries of some of these Social Choice conditions. I still need to figure out what my project should ultimately be about, but this seems like a good start!

Week 2

I started this week by doing a good bit of reading to decide which specific project I was interested in. After speaking to Dr. Farhad Mohsin and briefly considering a machine learning–oriented project, I decided that this was not what I was particularly interested in. The same went for a more probabilistic project, although I found the results of the papers I scanned very interesting. Thus, it seems as if the best project for me is to attempt to expand the work of Professor Xia's group theoretic characterization of ANR Impossibility—perhaps by way of introducing and characterizing some notion of Pareto Efficiency.

Weeks 3-4

We continued background reading, and I read a few economics articles and papers to get a good idea of how Pareto Optimality ought to be defined. I ended up stumbling a bit but did manage to think of a good definition, and now it's our task to try to make some impossibility results based on this.

Weeks 5-6

I wrote some Python code and worked out some very small cases to get a good idea of how ANR + PO Rules work out in general. Not much progress here, but I'm getting a good idea of the math behind it all.

Weeks 6-7

I was also able to reproduce Moulin's 'folklore theorem', or the theorem of ANR + PO for the single-winner ranked choice case, as it is necessary for our progress. I am working on characterizing ANR + PO for M=3, N=2, which I hope will help me gather insight into other cases.

Weeks 8-9

SI was able to produce a result in the ranked choice total ordering case, and am working towards generalizing for non-total-order cases. I also wrote a paper draft, which is in the works, and presented a final presentation with Jasdeep. I hope to continue working on this topic in the future!

Resources & Links

Xia, L. (2023). Most Equitable Voting Rules. arXiv:2205.14838 [cs.GT]
Xia, L. (2021). How Likely Are Large Elections Tied?. arXiv:2011.03791 [cs.GT]
Mohsin, F., Kang, I., Chen, P.-Y., Rossi, F., & Xia, L. (2021). Group Decision Dynamics in NLP and Social Choice. NLP+CSS Workshop at EMNLP.
Kang, I., Mohsin, F., Chen, P.-Y., Rossi, F., & Xia, L. (2022). Learning Individual and Collective Priorities over Moral Dilemmas with the Life Jacket Dataset. MPREF Workshop at AAMAS.
Moulin, H. (1983). The Strategy of Social Choice. North-Holland.
Brandt, F., Conitzer, V., Endriss, U., Lang, J., & Procaccia, A. D. (Eds.). (2016). Handbook of Computational Social Choice. Cambridge University Press.

Random Math Joke

Math jokes/brainrot