(Almost Full) EFX Exists for Four Agents (and Beyond) Ben Berger * Tel Aviv University benberger1@tauex.tau.ac.il Avi Cohen * Tel Aviv University avicohen2@mail.tau.ac.il Michal Feldman * Tel Aviv University michal.feldman@cs.tau.ac.il Amos Fiat † Tel Aviv University fiat@tau.ac.il March 2, 2021 Abstract The existence of EFX allocations is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and EFX is known to exist for (i) agents with identical valuations, (ii) 2 agents, (iii) 3 agents with additive valuations, (iv) agents with one of two additive valuations and (v) agents with two valued instances. It is also known that EFX exists if one can leave n - 1 items unallocated, where n is the number of agents. We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and, arguably, to simplify proofs of earlier results. Our main results are (i) every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated, (ii) every setting with n additive valuations has an EFX allocation with at most n - 2 unallocated items. Moreover, all of our results extend beyond additive valuations to all nice cancelable valuations (a new class, including additive, unit-demand, budget-additive and multiplicative valuations, among others). Furthermore, using our new techniques, we show that previous results for additive valuations extend to nice cancelable valuations. 1 Introduction The question of justness, fairness and division of resources and commitments dates back to Aristotle [Chr42]. Distributional justice, the “just” allocation of limited resources, is fundamental in the work of Rawls [Raw99]. Some evidence of the great interest in Rawls’ work is that numerous editions of his book have been cited over 100,000 times. The mathematical study of fair division is credited to Hugo Steinhaus, Bronislaw Knaster and Stefan Banach. The tale is that they would meet at the Scottish Caf´ e in Lvov where they wrote a book of open problems — the “Scottish book” — preserved by Steinhaus throughout the war and subsequently translated by Ulam and published in the United States. Subsequent editions of this book were written following the end of the war. In 1944 Hugo Steinhaus proposed the problem of dividing a cake into n pieces so that every agent gets at least a 1/n fraction of her total utility (“proportional division”). Steinhaus was actively working despite then living under German occupation in fear for his life. A solution to Proportional division of a cake was credited to Banach and Knaster by Steinhaus in 1949 [Ste49]. * Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 866132), and by the Israel Science Foundation (grant number 317/17). † Supported by the Israel Science Foundation (grant number 1841/14) and the Blavatnik fund. 1 arXiv:2102.10654v2 [cs.GT] 28 Feb 2021