An EF2X Allocation Protocol for Restricted Additive Valuations

My strongest case is a high-multiplicity mirror of the paper’s restricted-additive allocation problem. The lead anchor is Theorem 7.3; Theorem 5.10 gives a second, more ambitious mirror.

The natural regime is a large population receiving indivisible food or aid packages. There may be millions of agents but only a modest number \(\tau\) of relevant types: for example, vegetarian, halal, nut-allergic, diabetic, or otherwise restricted dietary profiles. A type \(t\) is the complete interest set \(I_t\) of goods that the agent values, together with any other parameter affecting the allocation. Goods retain common inherent values \(v(g)\), and an agent of type \(t\) has valuation \(u_t(g)=v(g)\) when \(g\in I_t\), and \(0\) otherwise.

The population is a distribution \(\mu=(\mu_t)_{t\in T}\). To keep goods indivisible, the supply must scale with the population. Let \(q_g\) be the number of copies of good category \(g\) per unit population. An individual agent still receives an integral bundle: formally, a bundle is \(B\in\mathbb Z_{\geq 0}^{G}\), and \(u_t(B)=\sum_g B_g u_t(g)\). The continuous allocation is a mass distribution \(z_{t,B}\), where \(z_{t,B}\) is the mass of type-\(t\) agents receiving the integral bundle \(B\). It must satisfy \(\sum_B z_{t,B}=\mu_t\) and \(\sum_{t,B}B_gz_{t,B}=q_g\) for every \(g\).

Thus \(z\) is fractional only at the level of population mass. No agent receives a fractional good. This is the appropriate high-multiplicity limit: a finite instance with \(n_t\) agents of type \(t\) and one copy of each good corresponds to \(\mu_t=n_t/n\) and \(q_g=1/n\); conversely, rational \(\mu\) and \(q\) can be expanded into a sufficiently large finite replicated instance.

My lead question is:

Continuous Restricted-Additive \(\mathsf{EFX^+}\)-Pareto Allocation. Given \(G\), the common values \(v(g)\), the finite type set \(T\), the type distribution \(\mu\), and per-capita supplies \(q\), find a complete mass allocation \(z\) such that:

• for every positive-mass type-\(t\) bundle \(B\), every positive-mass bundle \(B'\) received by any agent, and every copy of \(g\) in \(B'\) with \(g\in I_t\), \(u_t(B)\geq u_t(B'-e_g)\); and

• the allocation is Pareto efficient: there is no reassignment of positive-mass agents to integral bundles, using the same supplies, that weakly improves every agent and strictly improves a positive-mass set.

The first condition is exactly the paper’s \(\mathsf{EFX^+}\) condition, applied pointwise to agents and integral bundles rather than to type averages. This distinction matters: replacing each type’s bundles by an average bundle would be a different and weaker problem.

The anchor is Theorem 7.3, proved in this paper. Its full content is: assuming the valuations are restricted additive, there exists a polynomial-time algorithm that returns an \(\mathsf{EFX^+}\) and Pareto-efficient allocation. The paper proves this using the algorithm that sorts goods by non-increasing inherent value and allocates each good to an interested agent with minimum current utility. It establishes non-wastefulness, hence Pareto efficiency, and proves \(\mathsf{EFX^+}\) inductively.

This is a particularly credible continuization because the finite proof already has the right structure for a mass version. Sorting by \(v(g)\) remains meaningful, and “give the good to a least-valued interested agent” becomes a water-filling operation over the lowest-utility mass of interested agents. The result should therefore be tractable—Class A—in a natural high-multiplicity representation, at least when the supply and bundle support admit a compact description. The underlying problem has not been weakened to divisible goods or average fairness: the water-filling process produces mass over integral bundles, and the fairness test remains pairwise.

The second mirror is:

Continuous Complete \(\mathsf{EF2X}\) Allocation. Given the same data, find a complete mass allocation \(z\) such that for every type-\(t\) agent with bundle \(B\), every bundle \(B'\) occurring with positive mass, and every selection \(D\leq B'\) of \(\min(2,\lVert B'\rVert_1)\) individual goods, \(u_t(B)\geq u_t(B'-D)\).

This mirrors Theorem 5.10, proved in the paper: assuming the valuation functions are restricted additive, the paper’s complete-\(\mathsf{EF2X}\) algorithm returns a complete \(\mathsf{EF2X}\) allocation. The theorem is supported by the paper’s configuration, champion-graph, envy-elimination, and potential-function machinery.

I expect this second problem also to be tractable in the intended finite-type regime, though less confidently than the lead problem. The paper’s champion graph can plausibly be lifted from named agents to positive-mass bundle strata, with envy-elimination becoming a mass transfer between integral bundles. The continuous formulation may then be attacked by a finite-support transportation or column-generation method. But I would not claim that Theorem 5.10 itself proves a polynomial algorithm in \((m,\tau,L)\); the paper proves termination of its finite-agent procedure, not a compressed type-level complexity bound.

The paper’s authors should recognize both mirrors. Restricted additive valuations already identify agents by a small amount of information—whether each agent accepts each good—and the proposed regime simply treats repeated dietary or accessibility profiles as types. The goods remain indivisible, the fairness notion remains the same, and Pareto efficiency remains defined against individual improvements. The only continuous object is the population distribution, with supply scaling as required to make allocation meaningful.

The main further questions are whether the \(\mathsf{EFX^+}\) greedy procedure admits a polynomial type-level implementation without enumerating exponentially many bundle patterns; whether the configuration and champion-graph proof of Theorem 5.10 survives the nonatomic limit; and whether Pareto efficiency of the continuous allocation is equivalent to Pareto efficiency in sufficiently large finite replications.

My weakest point is precisely that last representation issue. Theorem 7.3 is polynomial in the explicit number of agents and goods, whereas compressing millions of agents into \(\tau\) types can create many distinct integral bundle states and utility levels. The paper does not establish the required compressed algorithm. Also, if one refuses to scale supplies with the population, a continuum of agents cannot receive a meaningful allocation of finitely many indivisible goods. The mirror therefore depends on the programme’s permitted co-modification: population mass becomes continuous, while goods remain indivisible but are supplied in repeated per-capita quantities.

I would not use the paper’s two-agent impossibility Example 1 as a hardness anchor: it is an impossibility result, not an NP-hardness theorem. The supplied paper contains no named discrete hardness result to transfer. Its positive algorithmic theorems are therefore the appropriate anchors, with Theorem 7.3 providing the cleanest and most defensible continuous question.