Fair allocation of countably many indivisible goods

The study of fair allocation was initiated during World War II, when Hugo Steinhaus asked his students Stefan Banach and Bronisław Knaster to find a generalisation of divide-and-choose protocol to fairly divide a piece of cake between two people. Today, it is a subject of intense research in mathematics, computer science, economics and political science. In particular, several algorithms and protocols have been devised to fairly divide a set of items between a finite number of agents. So far either a finite number of items have been considered (theory of indivisible items) or uncountably many items (cake-cutting) or a combination of the two. In this work, we consider countably many indivisible items which takes us from the world of finitary combinatorics to the world of infinitary combinatorics, topology, and set theory. We show that results which hold in the finite world do not necessarily hold in the infinitary world. We obtain a series of results for various fair conditions such as Pareto-optimality and envy-freeness, under various set-theoretic assumptions.