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.
We study the power and limits of optimal dynamic pricing in combinatorial markets; i.e., dynamic pricing that leads to optimal social welfare. Previous work by demonstrated the existence of optimal dynamic prices for unit-demand buyers, and showed a market with coverage valuations that admits no such prices. However, finding the frontier of markets (i.e., valuation functions) that admit optimal dynamic prices remains an open problem. In this work we establish positive and negative results that narrow the existing gap.On the positive side, we provide tools for handling markets beyond unit-demand valuations. In particular, we characterize all optimal allocations in multi-demand markets. This characterization allows us to partition the items into equivalence classes according to the role they play in achieving optimality. Using these tools, we provide a poly-time optimal dynamic pricing algorithm for up to 3 multi-demand buyers.On the negative side, we establish a maximal domain theorem, showing that for every nongross substitutes valuation, there exist unit-demand valuations such that adding them yields a market that does not admit an optimal dynamic pricing. This result is reminiscent of the seminal maximal domain theorem by Gul and Stacchetti [JET'99] for Walrasian equilibrium. Yang [JET'17] discovered an error in their original proof, and established a different, incomparable version of their maximal domain theorem. En route to our maximal domain theorem for optimal dynamic pricing, we provide the first complete proof of the original theorem by Gul and Stacchetti.
The existence of EFX allocations of goods 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 yet, existence results are known only for very restricted settings, such as: (i) agents with identical valuations, (ii) 2 agents, and (iii) 3 agents with additive valuations. 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 result is that every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated. Beyond our main result, we introduce a new class of valuations, termed nice cancelable, which includes additive, unit-demand, budget-additive and multiplicative valuations, among others. Using our new techniques, we show that both our results and previous results for additive valuations extend to nice cancelable valuations.
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