Team-Justified Envy-Freeness in Allocations

• The paper introduces team-justified envy-freeness, where envy counts only if validated by a sufficient number of peers, forming a continuum from classical to unanimous envy-freeness.
• It analyzes various allocation settings—indivisible goods, group allocations, and many-to-one matchings—highlighting theoretical bounds and NP-hardness challenges.
• The study formulates algorithmic solutions, including mixed-integer programming and discrepancy theory, to achieve practical approximations of fair, group-justified allocations.

Searching arXiv for the cited work and closely related notions to ground the article in the literature. arXiv search query: team justified envy freeness approval envy fair in the eyes of others (Shams et al., 2019) Team-justified envy-freeness is a family of fairness notions in which envy is not treated as decisive merely because it is subjectively felt by an individual. Instead, envy matters only when it is supported by a relevant collective judgement. In allocations of indivisible goods, this appears as $k$-justified envy, or $k$-approval envy: an agent’s envy of another agent counts only if sufficiently many agents agree that the envy is warranted (Shams et al., 2019). In allocations to groups, the same idea is expressed through approximate guarantees such as EF$c$, under which every member of a team can eliminate envy toward any other team by removing at most $c$ goods from the other team’s bundle; the exposition explicitly interprets this as a form of team-justified envy-freeness (Manurangsi et al., 2021). In many-to-one matching, the notion becomes team-justified EF1 (TJEF1), where a team compares its bundle only after disregarding participants whose own preferences make the comparison unjustified, and then removing at most one further participant (Igarashi et al., 28 Sep 2025). The unifying theme is the distinction between purely subjective envy and envy that is validated by peers, teammates, or the receiving side of a match.

1. Formal scope and core definitions

The literature uses closely related formulations in several models. In each case, the central move is to weaken classical envy-freeness by incorporating a justificatory filter.

Setting	Formal object	Team-justified condition
Indivisible goods	Allocation $A=(A_1,\dots,A_n)$	No ordered pair $(a_i,a_j)$ exhibits $k$-justified envy
Groups of agents	Ordered partition $(A_1,\dots,A_k)$ of goods	Every agent’s envy can be removed by deleting at most $c$ goods
Many-to-one matching	Matching $\mu:P\to T\cup\{\emptyset\}$	For each teams $k$0, $k$1 for some $k$2

In the additive indivisible-goods model, let $k$3 be a set of agents and $k$4 a set of indivisible goods. An allocation $k$5 is a partition of $k$6 among the agents, and each agent $k$7 has an additive utility function $k$8. Classically, agent $k$9 envies $c$0 under $c$1 if $c$2. For an integer $c$3, $c$4, agent $c$5 experiences $c$6-justified envy toward $c$7 in $c$8 if and only if $c$9 and there exists a subset $c$0 of size $c$1 with $c$2 such that $c$3. Equivalently, at least $c$4 agents, including $c$5 herself, agree that $c$6 should envy $c$7 (Shams et al., 2019).

In the group-allocation model, the goods are allocated to $c$8 groups of agents. If the $c$9 agents are partitioned into groups of sizes $A=(A_1,\dots,A_n)$0, and each agent $A=(A_1,\dots,A_n)$1 in group $A=(A_1,\dots,A_n)$2 has additive utility $A=(A_1,\dots,A_n)$3, then an allocation $A=(A_1,\dots,A_n)$4 is EF if every agent weakly prefers its own group’s bundle to every other group’s bundle. It is EF$A=(A_1,\dots,A_n)$5 if each agent’s envy toward any other group can be eliminated by removing at most $A=(A_1,\dots,A_n)$6 goods from the envied bundle. The exposition states that in many applications one wants that every member of each team “justifies” the allocation by feeling at most $A=(A_1,\dots,A_n)$7 bad goods missing from any other team, and therefore EF$A=(A_1,\dots,A_n)$8 certifies that no individual in any team has more than $A=(A_1,\dots,A_n)$9 “unjustified” envy toward any other team (Manurangsi et al., 2021).

In the many-to-one matching model, let $(a_i,a_j)$0 be the set of teams and $(a_i,a_j)$1 the set of participants. Each team has a weakly transitive preference over participants and $(a_i,a_j)$2, and each participant has a weakly transitive preference over teams and $(a_i,a_j)$3. A matching $(a_i,a_j)$4 induces a bundle $(a_i,a_j)$5 for each team $(a_i,a_j)$6. To compare bundles, the model uses the stochastic-dominance relation $(a_i,a_j)$7 over subsets of $(a_i,a_j)$8: for any two bundles $(a_i,a_j)$9, $k$0 iff there is an injection $k$1 such that for every $k$2 one has $k$3. Given distinct teams $k$4, let

$k$5

be the set of members of team $k$6 who do not prefer $k$7 to $k$8. A matching is TJEF1 if for every ordered pair of distinct teams $k$9 there exists a deletion set $(A_1,\dots,A_k)$0 with $(A_1,\dots,A_k)$1 such that $(A_1,\dots,A_k)$2 (Igarashi et al., 28 Sep 2025).

2. $(A_1,\dots,A_k)$3-approval envy as peer-validated fairness

The $(A_1,\dots,A_k)$4-approval formulation creates a continuum of fairness notions ranging from classical envy-freeness to unanimous-envy-freeness. An allocation is called $(A_1,\dots,A_k)$5-justified-envy-free, or $(A_1,\dots,A_k)$6-approval-envy-free, if for no ordered pair $(A_1,\dots,A_k)$7 does $(A_1,\dots,A_k)$8 experience $(A_1,\dots,A_k)$9-justified envy toward $c$0. In particular, $c$1 coincides with classical envy-freeness, while $c$2 coincides with “unanimous envy-freeness.” By varying $c$3 from $c$4 to $c$5, one obtains a continuum of fairness notions ranging from EF to unanimous-EF (Shams et al., 2019).

Several structural properties are established. If an allocation is $c$6-approval-envy-free, then it is also $c$7-approval-envy-free, but not vice versa in general for $c$8; the hierarchy is strict for $c$9. By contrast, $\mu:P\to T\cup\{\emptyset\}$0 collapses to EF: any $\mu:P\to T\cup\{\emptyset\}$1-approval-envy)-free allocation can be transformed, via weakly improving swaps, into a purely envy-free allocation. For two agents, absence of EF implies unanimous envy, so $\mu:P\to T\cup\{\emptyset\}$2 and $\mu:P\to T\cup\{\emptyset\}$3 coincide.

Existence is not guaranteed. Unlike EF1 or EFx, $\mu:P\to T\cup\{\emptyset\}$4-approval-envy-free allocations need not exist for arbitrary $\mu:P\to T\cup\{\emptyset\}$5. A specific obstruction is given: if all agents rank the same good first and its value exceeds the sum of the rest, then any allocation gives unanimous envy toward its holder, so no $\mu:P\to T\cup\{\emptyset\}$6-approval-envy-free allocation exists for any $\mu:P\to T\cup\{\emptyset\}$7. This places the notion between a relaxation of EF and a criterion that can still fail outright.

The formulation is motivated by the observation that envy is inherently subjective, yet subjective envy may lack an objective basis. The proposed response is to treat the judgement of the other agents as a proxy for objectivity. This suggests an interpretation of $\mu:P\to T\cup\{\emptyset\}$8 as an approval threshold: low $\mu:P\to T\cup\{\emptyset\}$9 tracks stringent anti-envy requirements, while high $k$00 screens out envy that lacks broad intersubjective support.

3. Optimization, tractability, and empirical behavior

The optimization problem associated with $k$01-approval envy asks for the minimum threshold $k$02 such that an instance admits a $k$03-approval-envy-free allocation. The mixed-integer programming formulation introduces binary variables $k$04, $k$05, and $k$06, together with an integer variable $k$07 to be minimized. Here $k$08 iff good $k$09, $k$10 iff agent $k$11 judges that $k$12 envies $k$13, and $k$14 iff $k$15 envies $k$16. With a large constant $k$17, the constraints enforce that every good is allocated once, that $k$18 captures the sign of $k$19, that self-envy implies $k$20, and that any active envy relation receives at most $k$21 approvals. An optimal solution returns the minimum $k$22 for which the induced allocation is $k$23-approval-envy-free; infeasibility corresponds precisely to instances exhibiting unanimous envy (Shams et al., 2019).

The general optimization problem inherits classical hardness. Deciding whether there exists an envy-free allocation is NP-complete, and since $k$24-approval-envy-free with $k$25 is EF, minimizing $k$26 inherits this hardness. Even deciding whether there exists some $k$27, $k$28, for which a $k$29-approval-envy-free allocation exists is NP-complete. At the same time, a polynomial-time special case is identified: in house allocation, where there are $k$30 agents and $k$31 goods and each agent receives exactly one good, one can compute in $k$32 an allocation minimizing $k$33.

The experimental study uses a MIP solved via Gurobi with timeout $k$34 min, and a house-allocation algorithm in C++. Benchmarks include Spliddit real instances with $k$35 up to $k$36 and $k$37 up to $k$38, random add-MARA with utilities uniform in $k$39, house allocation with $k$40 up to $k$41, and cardinal Mallows (Von Mises–Fisher) with varying concentration. On uniform tests restricted to non-EF instances, the reported frequencies are: for $k$42, all instances solved, mean $k$43, and $k$44 unanimous envy; for $k$45, mean $k$46 and $k$47 strict-majority EF exists; for $k$48, the solver struggles beyond $k$49 and mean $k$50 stabilizes near $k$51. In house allocation, the polynomial algorithm runs in $k$52 s for $k$53 up to $k$54, unanimous envy occurrences drop rapidly with $k$55 with an $k$56 bound, and optimal $k$57 almost independently of $k$58. Under correlated preferences in the cardinal Mallows model, as concentration $k$59 and preferences become identical, unanimous envy becomes universal; for medium concentration, small $k$60 suffices and majority approval-EF often exists. Overall, the experiments report that when pure envy-freeness fails, one typically finds allocations with $k$61 or even strict-majority approval-EF.

4. Approximate team justification for group allocations

For groups of agents receiving indivisible goods, the principal fairness relaxation is envy-freeness up to $k$62 goods. If $k$63 is the set of goods and the $k$64 agents are partitioned into $k$65 groups of sizes $k$66, then $k$67 denotes the smallest $k$68 that can be guaranteed for every instance with groups of sizes $k$69. The main theorem states that when the number of groups is fixed,

$k$70

In the balanced case $k$71, this simplifies to

$k$72

Thus, when the number of groups is constant and the $k$73 agents are divided into groups arbitrarily, there exists an allocation that is envy-free up to $k$74 goods, and this bound is tight (Manurangsi et al., 2021).

The proof is discrepancy-theoretic. The instance is encoded by an $k$75 matrix $k$76 whose rows are utility vectors. One then applies a multi-color discrepancy theorem, obtaining a coloring $k$77, viewed as an allocation, such that for every row $k$78 and every color $k$79,

$k$80

A counting-and-removal argument converts this fractional utility balancing into EF$k$81 with $k$82. The same reduction is algorithmic: recent work yields a deterministic polynomial-time algorithm that, on input $k$83 and integer $k$84, outputs a coloring with

$k$85

in time $k$86. The high-level algorithm invokes the multi-color discrepancy algorithm and returns the corresponding bundles.

The lower bound is also algorithmic. For any fixed $k$87 there exists $k$88 such that, given an instance with equal-sized groups, it is NP-hard to distinguish the case where there is an EF0 allocation from the case where no allocation is EF$k$89. The reduction starts from NP-hardness of multi-color discrepancy and converts matrix rows into agents’ utilities over goods. The resulting interpretation is explicit: EF$k$90 suffices to certify that no individual in any team has more than $k$91 “unjustified” envy toward any other team. A plausible implication is that, in group settings, team-justified envy-freeness is best understood not as a single exact predicate but as an approximation regime indexed by the number of goods that must be removed to neutralize envy.

5. Many-to-one matching and two-sided team-justified fairness

In many-to-one matching, team-justified fairness is integrated with a second, participant-side fairness condition. A participant $k$92 envies another participant $k$93 if $k$94, $k$95, and $k$96. This envy is justified if in addition $k$97, meaning that team $k$98 would rather swap $k$99 for $c$00. A matching is participant-justified envy-free (PJEF) if no such pair exists. TJEF1 complements PJEF by comparing team bundles through stochastic dominance after excluding members of the other team whose own preference for their current team makes the team comparison unjustified (Igarashi et al., 28 Sep 2025).

The algorithmic construction generalizes both round-robin and Gale–Shapley. For each team $c$01 with quota $c$02, one creates $c$03 slots. Each participant maintains an eligibility set $c$04, initially empty. In each round, every unmatched participant adds the next-most-preferred team or teams not already in $c$05, and the algorithm computes, among all matchings consistent with these eligibility sets, one that maximizes team-side priorities in round-robin slot order and breaks ties lexicographically in favor of smaller-indexed participants. The implementation can be realized through a sequence of max-flow/min-cut or bipartite-matching subproblems, and each of the $c$06 expansions of $c$07 triggers one bipartite-matching routine, yielding an overall polynomial-time procedure.

The main theorem states that the algorithm returns a matching satisfying simultaneously: TJEF1, PJEF, balancedness, Pareto-optimality, and group-strategyproofness for participants. Balancedness is given by $c$08, and Pareto-optimality implies swap-stability. The proof of TJEF1 relies on the fact that no slot envies any later slot by more than one eligible participant whose envy would be justified, since otherwise the matching could be improved in the slot-lex ordering. The proof of PJEF uses a monotonicity lemma under which each slot is assigned a weakly better participant over successive iterations. Group-strategyproofness is obtained by strengthening tie-breaking to a unique lexicographically maximal matching and excluding blocking paths.

The model also extends to quotas and incomplete lists. For quotas, each team simply receives the corresponding number of slots, and the TJEF1 comparison is adapted so that when comparing $c$09 to $c$10, any subset $c$11 of size $c$12 must satisfy $c$13 for some $c$14. For incomplete lists and unassigned agents, the construction introduces a dummy team $c$15 with large capacity and dummy participants representing being unassigned, ties them at the bottom of each real party’s list, runs the algorithm on the augmented instance, and drops all dummy matches at the end. The resulting properties include stability, extended TJEF1, Pareto-optimality, and group-strategyproofness.

6. Relation to justified envy in coalition formation

A related, but distinct, development appears in 3-dimensional additively separable hedonic games, where feasible coalitions are exactly triples. In this model, there are $c$16 agents, each ordered pair $c$17 has a valuation $c$18, and agent $c$19’s utility for a coalition $c$20 containing $c$21 is

$c$22

Given a partition $c$23 into triples, agent $c$24 envies agent $c$25 if replacing $c$26 by $c$27 in $c$28’s triple would strictly increase $c$29’s utility. This envy is justified if every other member $c$30 of $c$31’s triple strictly prefers $c$32 to $c$33, that is,

$c$34

A partition is justified-envy-free (JEF) if no pair $c$35 exhibits such justified envy (McKay et al., 2022).

The existence and complexity landscape is sharply stratified by preference domain. Under binary valuations $c$36, a JEF partition into triples always exists; in fact, JEF-Exists is in P and a JEF partition can be found in $c$37. Under ternary valuations $c$38, or under symmetric valuations with values up to $c$39, JEF may fail and deciding existence is NP-complete. In the binary case, the polynomial-time method constructs a symmetric counterpart via

$c$40

runs a known polynomial-time algorithm for a core-stable partition in symmetric-binary ASHGs, and returns the resulting triple partition; core-stability implies JEF.

This line of work is not identical to team-justified envy-freeness, because the objects being compared are coalition positions rather than bundles of goods or participants. Nevertheless, the structural analogy is direct: envy is prohibited only when it survives the preferences of the agents whose coalition would have to accept the swap. This suggests a broader family of “justified envy” notions in which fairness constraints are filtered through the consent or endorsement of affected third parties rather than evaluated solely from the claimant’s viewpoint.