This is a list of questions and topics I would like to research. If you find any of them interesting and would like to collaborate, drop me an email.
Open Questions
I would very much like to know the answer to the following questions.

Two agents with different incomes have to divide a set of indivisible items. Does a competitive equilibrium exist for almost all incomevectors? This question was raised by Babaioff and Nisan and TalgamCohen. My AAMAS 2018 paper solves some related problems, but the case of two additive agents is still open.

How many cuts are required for cakecutting when the agents have different entitlements? This working paper has some preliminary results. It is a combinatorics question that may be related to measure theory.

Is it possible to find in bounded time an envyfree and proportional cakedivision with connected pieces? The Wikipedia page on envyfree cakecutting gives some background. Our TALG 2016 paper “Waste Makes Haste” presents the status as of 8/2016. This is mainly an algorithmic question.

Given a collection of points in the plane, what is the largest possible collection of pairwisedisjoint wetsquares? This draft presents the question formally and gives some directions. Our fairandsquare papers (extended abstract and full preprint) present the economic motivation. This is mainly a geometric question.

Does there exist a connected envyfree cakecutting among n agents with mixed positive and negative valuations? My AAMAS 2018 paper “Fairly Dividing a Cake after Some Parts were Burnt in the Oven” shows that the answer is yes when n=3. Frédéric Meunier and Shira Zerbib show that the answer is yes when n=4 or n is a prime number. It is still open whether the answer is yes for all n. The proofs combine combinatorics and algebraic topology.

Some indivisible items should be divided among two groups of agents. What is the largest fraction of agents in each group, that can be guaranteed at least one of their two favorite items? Our IJCAI 2018 paper “Democratic fair allocation of indivisible goods” proves that the answer is between 3/5 and 2/3. The exact fraction is still a mystery (the paper contains other related open questions).

When a cake is redivided, what is the largest fraction of proportionality compatible with democratic ownership? The IJCAI 2018 paper “Redividing the Cake” proves that the fraction is at least 1/3 for onedimensional intervals, at least 1/4 for rectangles, and at least 1/5 for twodimensional convex objects. The exact fraction in these and other natural settings is still open. The problem combines geometry and combinatorics.
Reasearch projects

Experiments in fair division. I have some simulation results on fair division of land, and an online game for experiments with humans on fair cakecutting. I need partners to advance.

Implementing fair division algorithms in realestate projects. Currently, apartments are divided by lottery and/or protection… There are better ways and I would like to check how they can be implemented.