We study the problem of distributing a set of indivisible items among agents with additive valuations in a fair manner. The fairness notion under consideration is Envy-freeness up to any item (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this paper, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture by Caragiannis et al. [9] by showing an instance with three agents for which there is a partial EFX allocation (some items are not allocated) with higher Nash welfare than that of any complete EFX allocation. CCS Concepts: • Theory of computation → Algorithmic game theory.

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The existence of EFX allocations is a fundamental open problem in discrete fair division. Given a set of agents and indivisible goods, the goal is to determine the existence of an allocation where no agent envies another following the removal of any single good from the other agent's bundle. Since the general problem has been illusive, progress is made on two fronts: (i) proving existence when the number of agents is small, (ii) proving existence of relaxations of EFX. In this paper, we improve results on both fronts (and simplify in one of the cases).[CGM20] showed the existence of EFX allocations when there are three agents with additive valuation functions. The proof in [CGM20] is long, requires careful and complex case analysis, and does not extend even when one of the agents has a general monotone valuation function. We prove the existence of EFX allocations with three agents, restricting only one agent to have an additive valuation function (the other agents may have any monotone valuation functions). Our proof technique is significantly simpler and shorter than the proof in [CGM20] and therefore more accessible. In particular, it does not use the concepts of champions, champion-graphs, half-bundles (in contrast to the algorithms in [CKMS21, CGM20, CGM + 21]) and envy-graph (in contrast to most algorithms that prove existence of relaxations of envy-freeness, including weaker relaxations like EF1). Our technique also extends to settings when two agents have any monotone valuation function and one agent has an MMS-feasible valuation function (a strict generalization of nice-cancelable valuation functions [BCFF21] which subsumes additive, budget-additive and unit demand valuation functions). This takes us a step closer to resolving the existence of EFX allocations when all three agents have general monotone valuations.Secondly, we consider relaxations of EFX allocations, namely, approximate-EFX allocations and EFX allocations with few unallocated goods (charity). [CGM + 21] showed the existence of (1 − ε)-EFX allocation with O((n/ε) 4 /5 ) charity by establishing a connection to a problem in extremal combinatorics. We improve the result in [CGM + 21] and prove the existence of (1 − ε)-EFX allocations with O((n/ε) 2 /3 ) charity. In fact, our techniques can be used to prove improved upper-bounds on a problem in zero-sum combinatorics introduced by Alon and Krivelevich [AK21, MS21].

Competitive equilibrium with equal income (CEEI) is considered one of the best mechanisms to allocate a set of items among agents fairly and efficiently. In this paper, we study the computation of CEEI when items are chores that are disliked (negatively valued) by agents, under 1-homogeneous and concave utility functions which includes linear functions as a subcase. It is well-known that, even with linear utilities, the set of CEEI may be non-convex and disconnected, and the problem is PPAD-hard in the more general exchange model. In contrast to these negative results, we design FPTAS: A polynomial-time algorithm to compute ε-approximate CEEI where the running-time depends polynomially on 1 ε . Our algorithm relies on the recent characterization due to Bogomolnaia et al. (2017) of the CEEI set as exactly the KKT points of a non-convex minimization problem that have all coordinates non-zero. Due to this non-zero constraint, naïve gradient-based methods fail to find the desired local minima as they are attracted towards zero. We develop an exterior-point method that alternates between guessing non-zero KKT points and maximizing the objective along supporting hyperplanes at these points. We show that this procedure must converge quickly to an approximate KKT point which then can be mapped to an approximate CEEI; this exterior point method may be of independent interest.When utility functions are linear, we give explicit procedures for finding the exact iterates, and as a result show that a stronger form of approximate CEEI can be found in polynomial time. Finally, we note that our algorithm extends to the setting of un-equal incomes (CE), and to mixed manna with linear utilities where each agent may like (positively value) some items and dislike (negatively value) others.