Participant-Justified Envy-Freeness

• Participant-justified envy-freeness is a fairness concept that filters envy to only those cases where a swap would not violate an agent’s reservation utility.
• The framework guarantees that feasible swaps align with individual rationality and Pareto optimality through competitive equilibria with price-dependent incomes.
• Algorithmic approaches and complexity analyses across models like probabilistic allocations, team assignments, and hedonic games demonstrate its versatile application and robustness.

Searching arXiv for the cited paper and related terminology. Participant-justified envy-freeness designates a family of fairness requirements that weaken classical envy-freeness by filtering envy through constraints imposed by the affected participants. In the probabilistic-allocation setting with participation constraints, the central formulation is “no justified envy” (NJE): agent $i$’s envy of agent $j$ is ruled out only when the obvious remedy—swapping their bundles—would not violate $j$’s reservation utility (Echenique et al., 2019). Closely related work studies justified envy in two-sided team assignment (Igarashi et al., 2022), justified envy-freeness and weakly justified envy-freeness in triple-based hedonic games (McKay et al., 2022), and approval envy—also described in the supplied summary as participant-justified envy-freeness—where an envy claim is counted only if sufficiently many agents agree with it (Shams et al., 2019). Across these formulations, the common objective is to distinguish envy that is normatively actionable from envy that cannot be remedied without undermining feasibility, stability, or participants’ own judgments.

1. Core formulation under participation constraints

In the model of probabilistic allocations with participation constraints, agents are $I=\{1,\dots,N\}$, objects are $O=\{1,\dots,L\}$ with capacities $Q=(q_\ell)_{\ell\in O}$, and each agent $i$ has consumption space

$$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$

A feasible allocation is a vector $x=(x^i)_{i\in I}$ such that $x^i\in C^i$ for all $j$0 and $j$1 for all $j$2. Each agent has a continuous, monotone, quasi-concave utility $j$3 and a reservation utility $j$4 (Echenique et al., 2019).

Individual rationality requires

$j$5

Ordinary envy is defined in the standard way: agent $j$6 envies $j$7 if

$j$8

Participant-justified envy, or justified envy in this setting, adds a second condition. Agent $j$9 has justified envy toward $j$0 at allocation $j$1 iff

$j$2

The second clause states that exchanging the two bundles would not push $j$3 below $j$4’s participation threshold. An allocation satisfies no justified envy (NJE) if no such pair exists; the paper also defines “strong” justified envy when $j$5 (Echenique et al., 2019).

This definition is explicitly motivated by the incompatibility between envy-freeness and reservation utilities. An agent may envy another allocation even though any remedy would violate the other agent’s participation constraint. The NJE criterion therefore narrows attention to envy that is compatible with the affected participant’s minimal guarantee (Echenique et al., 2019).

2. Relation to classical envy-freeness and to individual rationality

The participant-justified formulation preserves the standard envy comparison $j$6 but qualifies its normative force. Classical envy-freeness forbids all such inequalities. NJE forbids only those envies for which the swap-based remedy is feasible with respect to the envied agent’s reservation utility (Echenique et al., 2019).

This makes NJE structurally dependent on individual rationality. The relevant question is not merely whether $j$7 prefers $j$8’s bundle, but whether $j$9 could accept $I=\{1,\dots,N\}$0’s bundle without violating $I=\{1,\dots,N\}$1. In that sense, participant-justified envy-freeness is weaker than envy-freeness but better aligned with environments in which agents arrive with outside options, property rights, or protected fallback claims (Echenique et al., 2019).

A related formulation appears in two-sided team assignment. There, an allocation assigns each player to exactly one team; players have ordinal preferences over teams, and teams have additive valuations over players. Player $I=\{1,\dots,N\}$2 has justified envy toward player $I=\{1,\dots,N\}$3 if $I=\{1,\dots,N\}$4 and team $I=\{1,\dots,N\}$5 would strictly prefer $I=\{1,\dots,N\}$6 in place of $I=\{1,\dots,N\}$7, namely $I=\{1,\dots,N\}$8. Justified envy-freeness then requires that no such pair exists (Igarashi et al., 2022). The same general pattern is present: envy becomes actionable only if the “receiving side” would accept the replacement.

An analogous idea appears in 3D hedonic games. With partitions into triples, justified envy-freeness requires that whenever $I=\{1,\dots,N\}$9 would strictly prefer occupying $O=\{1,\dots,L\}$0’s position in $O=\{1,\dots,L\}$1’s triple, each of the other two members of that triple must strictly prefer $O=\{1,\dots,L\}$2 to $O=\{1,\dots,L\}$3. Weakly justified envy-freeness relaxes the strict inequalities to weak acceptance (McKay et al., 2022). Here too, the concept filters envy through the preferences of the participants who would bear the consequences of the swap.

3. Competitive-equilibrium support and welfare properties

The principal existence result for NJE under participation constraints is formulated through a competitive equilibrium with price-dependent incomes. For prices $O=\{1,\dots,L\}$4, the expenditure function is

$O=\{1,\dots,L\}$5

and $O=\{1,\dots,L\}$6 (Echenique et al., 2019).

The construction defines incomes $O=\{1,\dots,L\}$7 using a common scalar $O=\{1,\dots,L\}$8 satisfying

$O=\{1,\dots,L\}$9

where

$Q=(q_\ell)_{\ell\in O}$0

One then sets $Q=(q_\ell)_{\ell\in O}$1, so that $Q=(q_\ell)_{\ell\in O}$2 and $Q=(q_\ell)_{\ell\in O}$3 (Echenique et al., 2019).

Given these incomes, each agent solves

$Q=(q_\ell)_{\ell\in O}$4

and equilibrium additionally requires market clearing: $Q=(q_\ell)_{\ell\in O}$5

Under the assumptions that each $Q=(q_\ell)_{\ell\in O}$6 is continuous, monotone, and quasi-concave, together with one of the paper’s standard technical conditions, there exists a competitive equilibrium $Q=(q_\ell)_{\ell\in O}$7 with price-dependent incomes such that three properties hold simultaneously: individual rationality, Pareto optimality, and no justified envy (Echenique et al., 2019). The theorem is due to Echenique–Miralles–Zhang as stated in the supplied summary.

This coexistence result is the main conceptual departure from classical impossibility intuitions surrounding envy-freeness with participation constraints. The paper’s construction does not insist on eliminating all envy; instead, it rules out only envy with an individually rational remedy, thereby allowing fairness, efficiency, and participation guarantees to coexist (Echenique et al., 2019).

4. Algorithmic structure and illustrative equilibrium

The proof of existence in the probabilistic setting uses fixed-point arguments, but the supplied summary presents a tâtonnement-style heuristic. Starting from an interior price vector $Q=(q_\ell)_{\ell\in O}$8, one repeatedly computes incomes $Q=(q_\ell)_{\ell\in O}$9 via the median construction, solves each agent’s demand problem under budget $i$0, computes excess demand $i$1, and updates prices by

$i$2

followed by normalization back into $i$3 (Echenique et al., 2019). The summary presents this as an algorithmic sketch rather than as a proved convergence theorem.

The same source gives a two-agent, two-good example with $i$4, unit-demand constraints $i$5, linear utilities

$i$6

$i$7

At the symmetric price vector $i$8, the relevant expenditure values all equal $i$9, so the median construction yields $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$0. Agent 1 then demands $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$1, agent 2 demands $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$2, markets clear, and the allocation satisfies both individual rationality and no envy, hence also NJE (Echenique et al., 2019).

This example is simple but structurally informative. It exhibits the intended role of the income rule: each agent receives enough purchasing power to reach the reservation utility, no more than needed for satiation, and total income equals the value of aggregate endowment. A plausible implication is that the NJE framework is best interpreted not as a local swap rule alone, but as a market-supported fairness concept whose content depends on how reservation utilities are embedded into budget formation.

5. Variants based on other participants’ acceptance or endorsement

The team-assignment model in “Fair Division with Two-Sided Preferences” generalizes the justified-envy idea to two-sided markets. There, the fairness constraint is imposed on players, while teams evaluate swaps additively. Player-side justified envy requires both a strict improvement in the player’s ordinal ranking of teams and a strict improvement for the destination team from replacing one incumbent with the envying player (Igarashi et al., 2022). This is not phrased through reservation utilities, but it is still participant-justified in the sense that the envied side must endorse the deviation.

The paper establishes that EF1 for teams and justified envy-freeness for players can conflict. Proposition 4.1 gives a counterexample with two teams and nonnegative valuations showing that no allocation may satisfy both conditions. The decision problem is NP-complete in general, although for $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$3 teams there is an $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$4 algorithm, and with identical nonnegative valuations a discrete cut-and-choose algorithm guarantees existence in polynomial time (Igarashi et al., 2022).

In 3D hedonic games, the participant-justification idea is even more explicit. If $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$5 envies $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$6, justified envy-freeness asks whether every other member of $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$7’s triple strictly prefers $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$8 to $$C^i=\Bigl\{x^i\in \mathbb R^L_+:\sum_{\ell=1}^L x^i_\ell \le c^i\Bigr\}.$$9; weakly justified envy-freeness asks for weak preference instead (McKay et al., 2022). The paper identifies a monotone trend: as the fairness notion weakens from envy-freeness to weakly justified envy-freeness to justified envy-freeness, existence and polynomial-time solvability hold under successively weaker restrictions. In particular, JEF always exists and can be found in polynomial time for binary valuations without symmetry, while ternary preferences or symmetric valuations up to $x=(x^i)_{i\in I}$0 already yield NP-completeness for JEF existence (McKay et al., 2022).

A different variant is approval envy. For indivisible-object allocation with additive utilities, agent $x=(x^i)_{i\in I}$1 $x=(x^i)_{i\in I}$2-approval envies $x=(x^i)_{i\in I}$3 if $x=(x^i)_{i\in I}$4 envies $x=(x^i)_{i\in I}$5 and at least $x=(x^i)_{i\in I}$6 agents, including $x=(x^i)_{i\in I}$7, also judge from their own perspectives that $x=(x^i)_{i\in I}$8’s bundle is better than $x=(x^i)_{i\in I}$9’s. An allocation is $x^i\in C^i$0-approval-envy-free if no such pair exists (Shams et al., 2019). The supplied summary states that approval envy is also called participant-justified envy-freeness. In this version, “justification” comes not from reservation utilities or swap acceptance, but from intersubjective endorsement by sufficiently many participants.

6. Complexity landscape and structural trade-offs

The computational profile of participant-justified envy-freeness depends sharply on the surrounding allocation model.

In probabilistic allocation with participation constraints, the key emphasis is existence and welfare compatibility rather than hardness classification. Under the stated assumptions, a competitive equilibrium with price-dependent incomes yields individual rationality, Pareto optimality, and NJE simultaneously (Echenique et al., 2019).

In two-sided team assignment, combining player-side justified envy-freeness with EF1 for teams is computationally difficult in general. The existence problem is NP-complete even with nonnegative integer valuations and strict player preferences, but becomes polynomial-time solvable for two teams, with a further unconditional existence guarantee under identical nonnegative valuations (Igarashi et al., 2022).

In triple-based hedonic games, the complexity frontier is stratified by valuation restrictions. For binary nonsymmetric preferences, JEF always exists and is computable in $x^i\in C^i$1. For ternary preferences, JEF may fail and existence is NP-complete; for symmetric valuations up to $x^i\in C^i$2, existence is again NP-complete (McKay et al., 2022). The same paper shows that stronger notions such as EF and WJEF become hard under milder density assumptions, reinforcing the interpretation of JEF as the weakest of the three fairness notions considered there.

In approval envy, computing the minimum threshold $x^i\in C^i$3 such that an $x^i\in C^i$4-approval-envy-free allocation exists is NP-hard in general additive settings, since envy-freeness appears as the special case $x^i\in C^i$5. Deciding existence of unanimous-envy-free allocations is also NP-complete. The house-allocation special case with $x^i\in C^i$6, however, admits a polynomial $x^i\in C^i$7 algorithm based on a bipartite graph and binary search on $x^i\in C^i$8 (Shams et al., 2019).

Taken together, these results indicate that weakening envy-freeness through participant-centered justification does not eliminate computational difficulty in general. Rather, it relocates tractable structure. Reservation-utility models recover strong welfare guarantees through market equilibrium; two-team settings admit constructive algorithms; binary hedonic valuations yield polynomial JEF; and house allocation admits efficient threshold minimization (Echenique et al., 2019, Igarashi et al., 2022, McKay et al., 2022, Shams et al., 2019).

7. Conceptual significance and common points of confusion

A common misconception is to treat justified envy-freeness as merely “envy-freeness with exceptions.” The supplied papers support a more precise interpretation. In each setting, the notion encodes a theory of when an envy complaint is socially or institutionally admissible. In the participation-constraint model, admissibility is determined by whether the envied agent’s reservation utility would remain satisfied after a swap (Echenique et al., 2019). In two-sided assignments and hedonic games, admissibility is determined by whether the receiving side would prefer or at least accept the replacement (Igarashi et al., 2022, McKay et al., 2022). In approval envy, admissibility depends on agreement by other participants (Shams et al., 2019).

Another potential confusion concerns the relation between these notions and efficiency. The probabilistic-allocation paper shows that no justified envy can coexist with Pareto optimality and individual rationality through a competitive-equilibrium construction (Echenique et al., 2019). By contrast, the team-assignment paper shows that justified envy-freeness for players and EF1 for teams do not necessarily coexist, and deciding whether they do is NP-complete (Igarashi et al., 2022). This suggests that participant-justified envy-freeness is not a single theorem but a design pattern whose consequences depend on which side’s constraints define the justification test.

A final point is terminological. The supplied materials use “justified envy,” “participant-justified envy,” “no justified envy,” “weakly justified envy-freeness,” and “approval envy” in related but non-identical senses (Echenique et al., 2019, McKay et al., 2022, Shams et al., 2019). The common denominator is a departure from purely subjective envy toward a relational criterion grounded in other participants’ rights, acceptance, or judgments. This suggests that participant-justified envy-freeness is best understood as a broader family of fairness concepts centered on actionable envy rather than as a single uniform axiom across all fair-division models.