Equal-Endowment No Envy

• Equal-endowment no envy is defined as requiring that agents with identical endowments weakly prefer their own assignments over others’, ensuring fairness via first-order stochastic dominance.
• It is achieved through two primary routes: one using market-clearing prices and capacity-feasible assignments, and the other via partitioning schemes that maintain assignment invariance even with indivisible goods.
• When strict envy-freeness fails due to indivisibility or strategic constraints, approximate fairness concepts and monetary transfers (subsidies) are employed to balance efficiency, stability, and incentive compatibility.

Searching arXiv for recent and relevant papers on equal-endowment no-envy and related fairness notions. Equal-endowment no envy denotes a family of fairness requirements in which agents with symmetric claims, budgets, capacities, or endowments should not prefer one another’s final assignments. In its most explicit formulation, the generalized housing market with fractional endowments defines equal-endowment no envy (EENE) by requiring that whenever two agents have identical endowments, each weakly first-order-stochastically prefers her own assignment to the other’s assignment (Yu et al., 15 Aug 2025). In adjacent literatures, the same idea is implemented through equal prices in rental harmony, equal capacity constraints in capacitated allocation, equal bundle counts or common budgets in indivisible-goods allocation, and proportional or equal-liability guarantees in chore division (Azrieli et al., 2014, Cohen et al., 2010, Sanpui, 2023).

1. Formal content of the notion

In the fractional-endowment housing model, agents $I=\{1,\dots,n\}$ own endowment vectors $\w_i=(\w_{i,o})_{o\in O}$ over objects $O=\{o_1,\dots,o_n\}$, with $\sum_o \w_{i,o}=1$, and allocations are doubly stochastic matrices $p=(p_{i,o})$. Preferences over assignments are extended by first-order stochastic dominance: $p_i \succsim_i^{sd} p'_i$ iff for every cutoff object $o$, $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$. EENE is then the requirement that for all $i,j$ with $\w_i=\w_j$, $\w_i=(\w_{i,o})_{o\in O}$0 and $\w_i=(\w_{i,o})_{o\in O}$1 (Yu et al., 15 Aug 2025).

In divisible bads, the equal-endowment component is typically encoded by proportionality. In chore cutting on $\w_i=(\w_{i,o})_{o\in O}$2, agent $\w_i=(\w_{i,o})_{o\in O}$3’s disutility for piece $\w_i=(\w_{i,o})_{o\in O}$4 is $\w_i=(\w_{i,o})_{o\in O}$5, and a proportional allocation satisfies $\w_i=(\w_{i,o})_{o\in O}$6. Envy-freeness requires $\w_i=(\w_{i,o})_{o\in O}$7 for all $\w_i=(\w_{i,o})_{o\in O}$8. For entire allocations, any envy-free allocation is automatically proportional, and for two agents, envy-freeness is equivalent to proportionality. In this sense, equal-endowment no envy for chores is effectively envy-freeness under full allocation of the bad (Sanpui, 2023).

In the purely ordinal two-agent model of a heterogeneous good, the equal-endowment candidate is the equal split $\w_i=(\w_{i,o})_{o\in O}$9. The paper identifies, for each agent $O=\{o_1,\dots,o_n\}$0, a unique balanced cut $O=\{o_1,\dots,o_n\}$1 such that $O=\{o_1,\dots,o_n\}$2, and proves that a contiguous partition $O=\{o_1,\dots,o_n\}$3 is Pareto efficient and envy-free iff $O=\{o_1,\dots,o_n\}$4 lies in the balanced region $O=\{o_1,\dots,o_n\}$5. Hence the equal split is Pareto efficient and envy-free iff $O=\{o_1,\dots,o_n\}$6 lies in that region (Bhattacharya et al., 4 Jun 2026).

2. Exact existence under symmetric entitlements

A central exact existence theorem appears in rental harmony with roommates. Agents choose among rooms with capacities $O=\{o_1,\dots,o_n\}$7, prices $O=\{o_1,\dots,o_n\}$8, and demand correspondences $O=\{o_1,\dots,o_n\}$9. Under assumptions (A1) non-emptiness, (A2) “free rooms are liked,” and (A3) closed graph, there exists $\sum_o \w_{i,o}=1$0 and a feasible assignment $\sum_o \w_{i,o}=1$1 such that $\sum_o \w_{i,o}=1$2 for every agent. The proof combines Shapley’s K-K-M-S theorem with Hall’s marriage lemma with capacities. Because all agents face the same price vector and each receives one unit of room capacity, the theorem is an existence result for envy-free allocations under symmetric entitlements; the paper also transfers the framework to a cake-cutting application and to a discrete exchange economy with indivisible goods and money (Azrieli et al., 2014).

For indivisible goods, exact symmetry can be imposed on the partition rather than on the assignment. A partition $\sum_o \w_{i,o}=1$3 is symmetrically envy free up to one good (symEF1) if for all agents $\sum_o \w_{i,o}=1$4 and all bundles $\sum_o \w_{i,o}=1$5,

$\sum_o \w_{i,o}=1$6

where $\sum_o \w_{i,o}=1$7 is the maximum-valued item of $\sum_o \w_{i,o}=1$8 for $\sum_o \w_{i,o}=1$9. The condition is assignment-invariant: any assignment of the bundles is EF1. The paper proves that a symEF1 allocation exists if the vertices of a related item graph can be partitioned into as many independent sets as there are agents; the sufficient condition always holds for two agents, and for agents that have identical, disjoint, or binary valuations (Johnston et al., 2024).

These results isolate two exact routes to equal-endowment no envy. One route uses market-clearing prices and capacity-feasible assignments; the other builds partitions whose fairness survives arbitrary reassignment. This suggests that symmetry can be encoded either through common prices and capacities or through bundle-level invariance.

3. Indivisible goods, equal budgets, and approximate no-envy

When exact envy-freeness is blocked by indivisibility, recent work studies approximate no-envy under symmetric structures. In ordered instances of additive indivisible-goods allocation, there exists a complete allocation that is both EFX and $p=(p_{i,o})$0-out-of-$p=(p_{i,o})$1 MMS, and there also exists a complete allocation that is EF1 and $p=(p_{i,o})$2-out-of-$p=(p_{i,o})$3 MMS. In top-$p=(p_{i,o})$4 instances, there exists a partial allocation that is $p=(p_{i,o})$5-out-of-$p=(p_{i,o})$6 MMS and EFX, and a complete allocation that is $p=(p_{i,o})$7-out-of-$p=(p_{i,o})$8 MMS and EF1. These guarantees show that strong envy-based constraints and strong ordinal fair-share guarantees are compatible in symmetric preference domains (Akrami et al., 17 Feb 2026).

Budget-feasible allocation gives a direct endowment interpretation: each agent has a budget $p=(p_{i,o})$9 on total size, and the charity holds all unallocated items. For identical additive valuations, the paper proves that a $p_i \succsim_i^{sd} p'_i$0-EF1 allocation can be computed in polynomial time for arbitrary budgets. In the uniform-budget case, $p_i \succsim_i^{sd} p'_i$1 for all agents, a polynomial-time algorithm computes an exact EF1 allocation; for two agents, exact EF1 is also computable in polynomial time for arbitrary budgets. In the large-budget regime, with $p_i \succsim_i^{sd} p'_i$2, the same algorithm achieves $p_i \succsim_i^{sd} p'_i$3-EF1, and any NSW-maximizing allocation is $p_i \succsim_i^{sd} p'_i$4-EF1 (Gan et al., 2021).

Approximate fairness is also studied through swap-based metrics. In equal-number environments with $p_i \succsim_i^{sd} p'_i$5, draft mechanisms under responsive preferences produce allocations that are both EF1 and swapEF. In multi-dimensional environments, a TTC+SD mechanism under responsive and directionally separable preferences yields allocations that are EF1 and swap bounded envy (swapBE), a weaker but more robust swap-based relaxation than swapEF (Echenique et al., 12 Aug 2025).

Taken together, these results recast equal-endowment no envy as a spectrum. Exact EF is often unavailable, but EF1, EFX, symEF1, swapEF, and swapBE preserve the equal-entitlement intuition while tolerating bounded indivisibility effects.

4. Strategy-proofness, stability, and implementation barriers

The strongest negative results arise once strategic reporting or coalition stability is imposed. In chore cutting with piecewise-constant disutilities, there exists no deterministic truthful envy-free mechanism satisfying the connected piece assumption, even for two agents; the same impossibility persists for two hungry agents, for deterministic non-wasteful truthful envy-free mechanisms, and for deterministic truthful envy-free dictatorship mechanisms. Since entire envy-free chore allocations already imply proportional equal-liability outcomes, these theorems show that equal-endowment no envy cannot, in general, be implemented by deterministic dominant-strategy mechanisms under these structural constraints (Sanpui, 2023).

A parallel impossibility holds in the purely ordinal two-agent model of contiguous bundles of a heterogeneous good. There, Pareto efficiency and envy-freeness characterize the balanced region of cuts, but no rule can simultaneously satisfy Pareto efficiency, envy-freeness, and strategy-proofness. Even when an equal split is envy-free for a given profile, no general strategy-proof rule can always select Pareto-efficient and envy-free outcomes (Bhattacharya et al., 4 Jun 2026).

There is, however, a notable positive special case in mechanism design with equal entitlements. In capacitated allocation games, when capacities are homogeneous, VCG with Clarke pivot payments is envy free; VCG with Clarke pivot payments is always incentive compatible, individually rational, and has no positive transfers. For heterogeneous capacities, no mechanism can simultaneously satisfy efficiency, incentive compatibility, envy-freeness, and no positive transfers. Equal capacities therefore mark a boundary at which truthful equal-endowment no envy remains feasible (Cohen et al., 2010).

Coalitional stability adds a further obstruction. In the housing market model with fractional endowments, the strong core may be empty, the weak core is nonempty and always contains an element satisfying equal treatment of equals, but there exist economies in which every element of the weak core violates EENE. This establishes a direct incompatibility between core-type stability and equal-endowment no envy in the fractional-endowment setting (Yu et al., 15 Aug 2025).

5. Transfers, subsidies, and welfare-preserving repair

A distinct line of work restores envy-freeness by adding money. For additive valuations with marginal value of each item at most one dollar, there always exists an envy-freeable allocation requiring a subsidy of at most one dollar per agent, hence a total subsidy of at most $p_i \succsim_i^{sd} p'_i$6 dollars. For general monotonic valuations, an envy-free allocation always exists with a subsidy of at most $p_i \succsim_i^{sd} p'_i$7 dollars per agent, and the bound is independent of the number of items (Brustle et al., 2019).

Starting from an EF1 allocation, the subsidy bounds can be sharpened in a wider model with absolute marginal value at most one. Given an EF1 allocation, one can compute in polynomial time an envy-free allocation with subsidy of at most $p_i \succsim_i^{sd} p'_i$8 per agent and total subsidy of at most $p_i \succsim_i^{sd} p'_i$9. For monotone valuations with $o$0, there exists an envy-free allocation with subsidies bounded by $o$1 per agent and $o$2 in total (Kawase et al., 2023).

Transfers can also be used to preserve welfare. For general monotone valuations, there always exists an envy-free allocation with transfers whose Nash social welfare is at least an $o$3-fraction of the optimal Nash social welfare. For additive valuations, an envy-free allocation with negligible transfers and whose NSW is within a constant factor of optimal can be found in polynomial time, with $o$4. By contrast, any envy-freeable allocation that achieves a constant fraction of optimal utilitarian welfare requires non-negligible transfers (Narayan et al., 2021).

A complementary exact characterization concerns envy-freeness and equitability simultaneously. For superadditive valuations, if an allocation $o$5 is transfer-stable, then the Knaster payment rule

$o$6

makes the outcome envy-free and equitable; any social-welfare-maximizing allocation is transfer-stable and therefore EFEQ-convertible. For additive utilities, an allocation is EFEQ-convertible if and only if it is transfer-stable (Aziz, 2020).

This body of work shows that when equal-endowment no envy fails combinatorially or strategically, money can serve as a corrective instrument. The corrective strength ranges from small bounded subsidies to balanced transfers that simultaneously recover fairness and welfare.

6. Entitlement-sensitive, social, and probabilistic variants

Recent variants modify either the entitlement baseline or the meaning of envy. Average envy-freeness (AEF) defines envy on the average value of items in bundles and is motivated by collaborative settings in which entitlements reflect contributions and may depend on the allocation itself. Deciding whether an AEF allocation exists is NP-complete, whereas an AEF-1 allocation is guaranteed to exist and can be computed in polynomial time. With quotas, finding an AEF-1 allocation satisfying a quota is NP-hard; for a fixed number of agents, the paper gives polynomial-time algorithms for binary valuation and approximated AEF-1 allocation with a quota for general valuation (Han et al., 2023).

Approval envy replaces purely subjective envy by socially endorsed envy. Agent $o$7 $o$8-approval envies $o$9 if $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$0 envies $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$1 and there exists a subset of $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$2 agents including $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$3 that all rank $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$4’s bundle above $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$5’s bundle from their own perspectives. The hierarchy is strict for $$\sum_{o' \succsim_i o} p_{i,o'} \ge \sum_{o' \succsim_i o} p'_{i,o'}$$6, while 2-approval-envy-freeness collapses to standard envy-freeness. The paper also identifies a polynomial-time algorithm for house allocation based on perfect matching (Shams et al., 2019).

Random matching under priorities yields another extension of no-envy under equal unit capacities. In the general model, weak stability of a generalized deterministic matching is equivalent to individual rationality, non-wastefulness, and no envy. For random matchings, the paper distinguishes ex-ante stability, robust ex-post stability, ex-post stability, fractional stability, and claimwise stability, and gives no envy or claims interpretations via a consumption-process view of random matchings (Aziz et al., 2017).

These variants indicate that equal-endowment no envy is not a single axiom but a design family. The endowment side can be represented by identical endowments, equal prices, equal capacities, equal quotas, bundle-size-normalized entitlements, or random assignment capacities; the no-envy side can be exact, up to one good, swap-bounded, socially endorsed, or stochastic-dominance-based. A plausible implication is that the research frontier is less about a universal definition than about identifying which pairing of entitlement model and envy test remains compatible with existence, computation, welfare, and incentives in a given domain.