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Fixed-point conjecture for the EFX allocation map

Let mm agents and nn goods be given, with valuation parameters vℓiv_{\ell i} and V=∑ℓ,ivℓiV=\sum_{\ell,i}v_{\ell i}. Let ϵ>0\epsilon>0 be arbitrarily small and assume V≤ϵV\leq\epsilon. Let

T:[−1,0]mn→[−1+ϵ,0]mnT:[-1,0]^{mn}\to[-1+\epsilon,0]^{mn}

be the continuous map whose coordinates are

T_{kj}(y)=\min\left\\{y_{kj},h(y_k)-A_{kj}(y)\right\\},

for all k,jk,j, where h(yℓ)h(y_{\ell}) and Akj(y)A_{kj}(y) are defined as in the source. Fixed-point conjecture. The map TT admits a fixed point. This fixed-point claim is intended to characterize the existence of an EFX allocation through the preceding reduction: a fixed point of the map corresponds to an EFX allocation. The statement is presented as a conjecture, and no resolution is supplied in the given text.

Sources & referencesView supporting material

Primary source

S. Rasoul Etesami, “A Fixed Point Framework for the Existence of EFX Allocations”, arXiv:2510.04915 (2025).

efx-allocationsfair-divisionfixed-point-theory

Progress summary

Refreshed
Open

The conjecture remains unresolved: numerical tests support it, but no proof or counterexample has been found.

The conjecture asks whether the continuous map TT admits a fixed point, which the preceding reduction makes equivalent to the existence of an EFX allocation. The relevant manuscript labels this as Conjecture 5 and reports that its authors do not know the answer.

Known results

  • Brouwer’s theorem guarantees a fixed point for a perturbed proxy map T~\widetilde T, but the required implication for the original EFX constraints is unproved.
  • Numerical experiments consistently found fixed points of both the original and proxy maps, without yielding a theorem.

2025 manuscript

The manuscript explains that direct application of Brouwer fails because TT maps [−1,0]mn[-1,0]^{mn} into [−1+ϵ,0]mn[-1+\epsilon,0]^{mn} rather than necessarily into its own domain. The supplied scan found no proof, counterexample, or independent verification for this conjecture.

Current status (as of August 2026): the fixed-point conjecture and hence the general EFX existence question remain open; only the proxy-map fixed-point result and supporting numerical evidence are established.

Sources

Solutions 0

No solutions have been posted yet.