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.

Working papers

These papers need co-authors in order to realize their full potential:

  1. Fairly Dividing a Cake after Some Parts were Burnt in the Oven: Given a cake and n agents with different valuations of its pieces, where the valuations can be positive or negative, does there always exist a connected envy-free division? The paper proves that the answer is yes when n=3. Breaking news: Frédéric Meunier and Shira Zerbib have just proved 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.

  2. How to re-divide a cake fairly? and Price-of-Fairness: Combines computational geometry, combinatorics, economics and social choice. I am looking for experts in any of these fields in order to help verify and improve the results.

Reasearch projects

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

  2. Implementing fair division algorithms in real-estate 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.

Open Questions

I would very much like to know the answer to the following questions.

  1. Two agents with different incomes have to divide a set of indivisible items. Does a competitive equilibrium exist for almost all incomes? This question was raised by Babaioff and Nisan and Talgam-Cohen. I solved some related open problems but the case of two additive agents is still open.

  2. How many cuts are required for cake-cutting when the agents have different entitlements? Mainly a combinatorics question. May be related to measure theory and topology. My MathOverflow question has some preliminary results.

  3. Is it possible to find in bounded time an envy-free and proportional cake-division with connected pieces? Mainly an algorithmic question. The Wikipedia page I wrote on envy-free cake-cutting gives some background. Our waste-makes-haste paper presents the status as of 8/2016.

  4. Given a collection of points in the plane, what is the largest possible collection of pairwise-disjoint wet-squares? Mainly a geometric question. This draft presents the question formally and gives some directions. Our fair-and-square papers (extended abstract and full preprint) present the economic motivation.