Local Envy-Freeness in Graph Fairness

• Local envy-freeness is a fairness notion that limits comparisons to a defined local neighborhood in a directed graph.
• It applies to cake cutting, indivisible goods, and ad auctions, where agents compare only with selected neighbors rather than all participants.
• Analyses reveal significant welfare trade-offs and algorithmic challenges, with efficiency hinging on the underlying graph structure.

Searching arXiv for recent and foundational papers on local envy-freeness and related graph-based envy-freeness. Local envy-freeness is a family of fairness notions in which envy constraints are imposed only with respect to a local comparison structure rather than against all other participants. In cake cutting and in allocations of indivisible goods, that locality is represented by a directed graph whose edges specify who compares to whom; in ad exchanges, it is represented by a mediator’s local auction and the bidder bundles that maximize utility at posted prices. Across these settings, local envy-freeness weakens classical global envy-freeness, but it does not trivialize fairness: it induces nontrivial graph-theoretic structure, algorithmic characterizations, and welfare bounds, and it interacts in subtle ways with proportionality, efficiency, and computational complexity (Abebe et al., 2016, Bredereck et al., 2020, Ben-Zwi et al., 2016).

1. Core graph-based definitions

In the cake-cutting formulation, the cake is the interval $[0,1]$, and each agent $i$ has a valuation function $V_i$ mapping finite unions of subintervals to $\mathbb{R}_{\ge 0}$, assumed to be additive, non-atomic, and normalized so that $V_i([0,1])=1$. A feasible allocation $A=(A_1,\dots,A_n)$ is a partition of $[0,1]$ into disjoint pieces $A_i$. The social comparison structure is a directed graph $G=(V,E)$ on the agents, where an edge $(i,j)$ means that agent $i$0 compares her piece to $i$1’s piece. If $i$2 and $i$3, then local envy-freeness and local proportionality are defined by the inequalities (Abebe et al., 2016)

$i$4

and

$i$5

The first condition is local envy-freeness (LEF): no agent envies a neighbor’s allocation. The second is local proportionality (LP): each agent values her own allocation at least as much as the average value of her neighbors’ allocations.

The same graph-based pattern appears in allocations of indivisible goods. Let $i$6 be a set of agents, $i$7 a set of indivisible items, and $i$8 a non-negative utility function for each agent, usually assumed additive. An allocation is a mapping $i$9 with pairwise disjoint bundles. Given a directed social network $V_i$0, graph envy-freeness (GEF) requires

$V_i$1

and strong graph envy-freeness (sGEF) strengthens this to a strict inequality (Bredereck et al., 2020).

These definitions reduce the universal comparison requirement of classical envy-freeness to a local one. On a complete graph they coincide with the classical notion; on sparser graphs they may be strictly weaker, but they remain structurally constrained by the topology of the comparison graph (Abebe et al., 2016, Bredereck et al., 2020).

2. Relation to global envy-freeness and proportionality

Global envy-freeness implies local envy-freeness, because if $V_i$2 for all $V_i$3, then the same inequality holds in particular on every edge $V_i$4. In the graph setting, local envy-freeness implies local proportionality, since pointwise domination over every neighbor immediately implies domination over the average of neighbors’ values (Abebe et al., 2016).

The relation between global proportionality and local proportionality is more delicate. Global proportionality does not imply local proportionality, and local proportionality does not imply global proportionality. The separation is explicit in the four-agent example on the cycle $V_i$5: agents $V_i$6 are uniform on $V_i$7, agent $V_i$8 is piecewise uniform with $V_i$9, $\mathbb{R}_{\ge 0}$0, and zero elsewhere, and the allocation

$\mathbb{R}_{\ge 0}$1

is proportional globally but fails local proportionality on $\mathbb{R}_{\ge 0}$2, because agent $\mathbb{R}_{\ge 0}$3’s value for her own piece is $\mathbb{R}_{\ge 0}$4 while the average value of the neighbors’ pieces is $\mathbb{R}_{\ge 0}$5 (Abebe et al., 2016).

The comparison graph also matters in a stronger sense. For any two distinct connected graphs $\mathbb{R}_{\ge 0}$6 on the same agents, there exist valuations and an allocation that is locally proportional on $\mathbb{R}_{\ge 0}$7 but not on $\mathbb{R}_{\ge 0}$8. This rules out any general containment relation among LP-feasible sets for different graphs. A common misconception is therefore that local fairness merely weakens global fairness in a monotone, graph-independent way; the graph itself is part of the fairness specification, not just a relaxation parameter (Abebe et al., 2016).

In the indivisible-goods setting, the same broad pattern persists. Classical envy-freeness quantifies over all pairs of agents, whereas GEF quantifies only over arcs of the social network. The strong variant sGEF is strictly stronger than GEF, because it requires $\mathbb{R}_{\ge 0}$9 for every outgoing arc $V_i([0,1])=1$0 (Bredereck et al., 2020).

3. Divisible resources: protocol structure in cake cutting

The cake-cutting treatment is formulated in the Robertson–Webb model, where protocols access each valuation $V_i([0,1])=1$1 only through two query types: $V_i([0,1])=1$2, which returns $V_i([0,1])=1$3, and $V_i([0,1])=1$4, which returns a point $V_i([0,1])=1$5 such that $V_i([0,1])=1$6. Query complexity is the worst-case number of such queries needed to produce an allocation (Abebe et al., 2016).

A central algorithmic question is when local envy-freeness can be guaranteed by an oblivious single-cutter protocol, meaning a protocol that designates one agent $V_i([0,1])=1$7 to make all initial cuts based only on $V_i([0,1])=1$8, after which the other agents pick among the resulting pieces. The decisive graph class is the class of subgraphs of a cone of a DAG. If $V_i([0,1])=1$9 is a DAG on $A=(A_1,\dots,A_n)$0, its cone $A=(A_1,\dots,A_n)$1 adds a new apex node $A=(A_1,\dots,A_n)$2 and edges $A=(A_1,\dots,A_n)$3 and $A=(A_1,\dots,A_n)$4 for all $A=(A_1,\dots,A_n)$5 (Abebe et al., 2016).

For graphs of the form $A=(A_1,\dots,A_n)$6, the protocol is explicit. Agent $A=(A_1,\dots,A_n)$7 makes $A=(A_1,\dots,A_n)$8 cuts to partition $A=(A_1,\dots,A_n)$9 into $[0,1]$0 pieces that she values equally, each of value $[0,1]$1. The nodes of $[0,1]$2 are topologically sorted. In that order, each non-cutter agent evaluates the remaining pieces and picks a favorite. The cutter receives the last piece. This yields a locally envy-free allocation on $[0,1]$3, with query complexity

$[0,1]$4

The converse is equally important: if a graph $[0,1]$5 admits an oblivious single-cutter LEF protocol for all valuations, then $[0,1]$6 must be a subgraph of a cone of some DAG. The proof sketch proceeds by contradiction: if $[0,1]$7 is not contained in any cone of a DAG, then for any chosen cutter $[0,1]$8, the subgraph $[0,1]$9 contains a directed cycle, and an adversarial choice of valuations on that cycle forces envy regardless of how the cutter sliced the cake (Abebe et al., 2016).

This characterization isolates the exact topology compatible with bounded $A_i$0 single-cutter procedures. A plausible implication is that local comparison relaxes fairness enough to permit efficient protocols only when the relaxation is aligned with a strong acyclicity condition outside a designated apex.

4. Welfare and the price of local envy-freeness

The price of local envy-freeness compares optimal social welfare without fairness constraints to optimal welfare subject to local envy-freeness. Let $A_i$1 be a welfare-optimal allocation maximizing $A_i$2, and let $A_i$3 be a welfare-optimal allocation subject to LEF on $A_i$4. Then

$A_i$5

For every connected undirected graph $A_i$6 on $A_i$7 agents, there exist valuations such that $A_i$8. This extends the classical lower bound for global envy-freeness to local envy-freeness on arbitrary connected undirected graphs (Abebe et al., 2016).

The proof outline uses $A_i$9, assuming it is integral, and separates the agents into $G=(V,E)$0 “special” agents $G=(V,E)$1 and the remaining agents $G=(V,E)$2. Each special agent concentrates all value on a dedicated interval $G=(V,E)$3 of length $G=(V,E)$4, while each agent in $G=(V,E)$5 is uniform over $G=(V,E)$6. The welfare-optimal allocation gives each $G=(V,E)$7 its dedicated interval, for total welfare approximately $G=(V,E)$8. To show that any LEF allocation has low welfare on an arbitrary connected graph, the argument invokes a graph-theoretic lemma stating that every connected undirected $G=(V,E)$9 has a $(i,j)$0-linked partition with $(i,j)$1. The local envy constraints then propagate along the linking paths and force strong restrictions on what the special agents can receive (Abebe et al., 2016).

The resulting conclusion is that sparse connected graphs do not provide more flexibility with respect to the quality of envy-free allocations. This is one of the more counterintuitive features of the model: localizing comparisons changes existence and algorithmic structure, but it does not, in the worst case, remove the asymptotic welfare loss associated with envy-freeness (Abebe et al., 2016).

5. Indivisible goods: existence, non-existence, and complexity

For indivisible resources, weak graph envy-freeness without an efficiency requirement always admits the trivial empty allocation $(i,j)$2. To avoid that, one studies complete allocations or Pareto-efficient allocations. Under monotonic utilities, complete GEF always exists on acyclic $(i,j)$3: one can pick any source $(i,j)$4 and give all items to $(i,j)$5. By contrast, complete sGEF can never exist on any graph containing a cycle if utilities are identical additive, because strict inequalities around a directed cycle contradict transitivity. Under identical $(i,j)$6 preferences and strongly connected $(i,j)$7, complete GEF exists if and only if the total number of items is divisible by $(i,j)$8; each agent must receive exactly the same number of items (Bredereck et al., 2020).

The computational landscape is mixed. Some restrictions make graph envy-freeness easier than classical envy-freeness, while others make it harder, and the strong variant may differ sharply from the weak one. The following results summarize representative cases (Bredereck et al., 2020).

Setting	Problem	Result
Acyclic $(i,j)$9, monotonic additive utilities	C-GEF	$i$00
General graphs, additive utilities	C-GEF	strongly NP-hard
Acyclic $i$01	C-sGEF	NP-hard
Identical $i$02 utilities, any $i$03	C-GEF and C-sGEF	linear time
Identical additive utilities, general $i$04	C-GEF	NP-hard; W[1]-hard by items or by $i$05 out-degree
General additive utilities on DAGs	W-GEF	NP-hard

Parameterized complexity refines this picture. For C-GEF, identical-utility instances are fixed-parameter tractable for parameter $i$06 (the number of items), and general $i$07-utility instances are fixed-parameter tractable for parameter $i$08 (the number of agents). With combined parameter $i$09, where $i$10 is the maximum out-degree, C-GEF with identical utilities is FPT via a reduction to directed colored Subgraph Isomorphism. For C-sGEF, the problem is FPT in $i$11 for $i$12 or identical utilities, FPT in $i$13 for general utilities, but para-NP-hard and in fact W[1]-hard when parameterized by $i$14 for general additive utilities on DAGs (Bredereck et al., 2020).

These results show that the graph restriction is not merely a softened version of classical envy-freeness. On DAGs, GEF admits simple complete allocations, yet strong graph envy-freeness remains hard even on directed paths. Conversely, for some utility classes, both GEF and sGEF become tractable on arbitrary graphs (Bredereck et al., 2020).

6. Local auctions and three-party equilibrium

In ad exchanges with mediators, local envy-freeness has a different but related meaning. There is a finite set of items $i$15. A mediator $i$16 runs a local auction among bidders $i$17 on a subset $i$18 that she won in the central auction. Each bidder $i$19 has valuation $i$20, and under a posted price vector $i$21, bidder $i$22’s utility for bundle $i$23 is

$i$24

The demand correspondence is

$i$25

An allocation $i$26 of $i$27 and prices $i$28 are locally envy-free if $i$29 for every bidder $i$30 (Ben-Zwi et al., 2016).

This local notion is embedded in a three-party competitive equilibrium involving the exchange, mediators, and advertisers. At the central level, the exchange sets prices $i$31 and allocates bundles $i$32 to mediators in a Walrasian equilibrium, so that each mediator receives a demanded bundle and any positively priced item is allocated. At the local level, each mediator announces a local price vector $i$33 and allocates $i$34 to her bidders so that every bidder receives a demanded bundle. Reserve-price consistency requires $i$35 for every item $i$36. Together, these conditions ensure central envy-freeness, local envy-freeness, and consistency between local and central prices (Ben-Zwi et al., 2016).

For gross-substitute valuations, such a three-party equilibrium always exists. The key steps are: modeling each mediator as an ef-mediator whose demand is computed via a virtual auction with reserve prices; reducing the local problem to a Walrasian equilibrium with reserve prices; proving that, for gross substitutes, the ef-mediator’s demand coincides with the demand of an OR-player; and then invoking the existence of a central Walrasian equilibrium for gross-substitute bidders. The resulting allocation is social-welfare maximizing, and the paper also gives a polynomial-time algorithm. Under the stated oracle assumptions, the overall running time is

$i$37

when bidders are spread evenly among mediators and $i$38 (Ben-Zwi et al., 2016).

This auction-theoretic usage shows that local envy-freeness is not tied to social-network graphs alone. The common principle is that envy constraints are enforced only within a designated local market or comparison neighborhood, while a higher-level allocation mechanism coordinates those local structures.