Swap Bounded Envy in Fair Division

• Swap bounded envy is a fairness concept where envy in indivisible goods allocation is localized by a one-for-one swap evaluated against an acceptable reference bundle.
• It distinguishes itself from EF1 and swapEF by requiring a period-specific, local exchange that preserves fixed-cardinality and multi-dimensional allocation constraints.
• Mechanisms such as draft methods and TTC+SD are shown to achieve swap bounded envy under responsive and directionally separable preferences, offering practical fairness guarantees.

Searching arXiv for the cited papers to ground the article in the relevant literature. Swap bounded envy is a fairness concept for allocating indivisible goods in which envy is controlled by a single exchange operation rather than by removing an item. In the formulation introduced for discrete goods, an allocation has swap bounded envy (swapBE) when, whenever an agent prefers another agent’s bundle, there exists a reference bundle and a one-item swap that witnesses that the envy is localized to a single component of the envied bundle (Echenique et al., 12 Aug 2025). The term also admits related interpretations in adjacent literatures. In cake cutting, bounded envy has been studied through finite-bounded protocols that count guaranteed non-envy relations via the Degree of Guaranteed Envy-Freeness (DGEF), which can be read as a bound on pairwise “swap desires” (0902.0620). In discrete exchange models for indivisible goods, bounded envy appears as envy-freeness up to one good (EF1) reached by a bounded number of bilateral swaps (Yuen et al., 2024). In fair division with divisible goods, bounded envy is further connected to bounded envy-free cake-cutting protocols whose progress is driven by cycle-based exchange operations (Aziz et al., 2016). Across these settings, the common theme is that envy is not eliminated globally in one step but is constrained by local exchange structure, feasibility, or worst-case guarantees.

1. Formal concept in discrete-goods allocation

The most direct definition of swap bounded envy appears in the multi-dimensional allocation model for indivisible goods developed in “Swap Bounded Envy” (Echenique et al., 12 Aug 2025). The setting has agents $N=\{1,\dots,n\}$ and items arranged across periods $t=1,\dots,T$, where each period has item set $O_t$ and a null object $\emptyset_t$. Agents consume tuples $(o_1,\dots,o_T)$ and preferences are assumed monotone. The paper works with responsive preferences, additive preferences as a special case, and the stronger condition of directionally separable preferences in the multi-dimensional results (Echenique et al., 12 Aug 2025).

In this framework, envy is standard: agent $i$ envies agent $j$ if $v_i(A_j)>v_i(A_i)$, or ordinally if $A_j \succ_i A_i$ (Echenique et al., 12 Aug 2025). A one-step swap between agents $i$ and $t=1,\dots,T$0 exchanges one item from each bundle. In the multi-dimensional case, a swap is period-specific: agent $t=1,\dots,T$1 and agent $t=1,\dots,T$2 exchange their period-$t=1,\dots,T$3 items while all other periods remain fixed (Echenique et al., 12 Aug 2025).

Swap bounded envy is defined relative to a reference bundle. A multi-dimensional allocation $t=1,\dots,T$4 has swap bounded envy if, for every pair $t=1,\dots,T$5 with $t=1,\dots,T$6, there exists a reference bundle $t=1,\dots,T$7 and a period $t=1,\dots,T$8 such that $t=1,\dots,T$9 and $O_t$0 (Echenique et al., 12 Aug 2025). The definition does not require that the reference bundle be allocated to any agent. Instead, it requires that agent $O_t$1 regard the reference bundle as no better than her own, and that replacing one coordinate of that reference bundle by the corresponding coordinate from $O_t$2’s bundle suffices to overturn the comparison (Echenique et al., 12 Aug 2025).

This reference-bundle formulation distinguishes swapBE from more familiar one-item relaxations. EF1 removes one item from the envied bundle; swapEF performs a direct one-for-one exchange between the two allocated bundles. SwapBE weakens swapEF by permitting the decisive exchange to be measured against a bundle that is merely acceptable to the envying agent, rather than against her actual allocation (Echenique et al., 12 Aug 2025). The paper explicitly states that $O_t$3, that $O_t$4, and that EF1 and swapEF are incomparable; it also states that EF1 and swapBE are logically unrelated (Echenique et al., 12 Aug 2025).

A central interpretive point is that swapBE is intended for environments in which “dropping” a good is unrealistic. In equal-number allocations, every agent must receive the same cardinality of items; in multi-period environments, every agent may need exactly one item per period. In such cases, the usual EF1 certificate of fairness by deletion can be normatively weak, whereas a one-for-one exchange has direct operational meaning (Echenique et al., 12 Aug 2025).

2. Relationship to adjacent fairness notions

The literature surrounding swap bounded envy involves several nearby notions of approximate envy control. EF1, or envy-freeness up to one good, is defined for additive utilities by the existence, for every envied pair $O_t$5, of some good $O_t$6 such that $O_t$7 (Yuen et al., 2024). The same paper studies the reachability of EF1 from an initial allocation by a sequence of bilateral swaps and shows that this reachability depends fundamentally on the size vector of the allocation (Yuen et al., 2024).

SwapEF is a stricter exchange-based notion than swapBE. In the terminology of (Echenique et al., 12 Aug 2025), if agent $O_t$8 envies agent $O_t$9, swapEF requires the existence of $\emptyset_t$0 and $\emptyset_t$1 such that after swapping $\emptyset_t$2 and $\emptyset_t$3, agent $\emptyset_t$4 weakly prefers her new bundle to agent $\emptyset_t$5’s new bundle. In the multi-dimensional version, the swap is restricted to a single period (Echenique et al., 12 Aug 2025). The implication $\emptyset_t$6 holds because one may take the reference bundle to be $\emptyset_t$7, subject to the caveat noted in the paper that the reference bundle should not use outside options (Echenique et al., 12 Aug 2025).

The discrete-goods literature also distinguishes swapBE from EFX. The paper introducing swapBE notes only that EFX is strictly stronger than EF1 in prior work and that neither implication between EFX and swapBE is established in general there (Echenique et al., 12 Aug 2025). This places swapBE as neither a simple strengthening nor weakening of the dominant one-good relaxations, but as a separate local-exchange criterion.

A useful comparison is with the exchange-based reformability model of (Yuen et al., 2024). There, the objective is not to define a new fairness notion but to ask whether an unfair allocation can be transformed into an EF1 allocation through bilateral swaps only, and if so, how many swaps are needed. This is a different sense of “swap bounded envy”: envy is bounded by the minimal exchange distance to EF1. The paper proves, for example, that balanced size vectors always admit an EF1 allocation of the same size vector, whereas unbalanced size vectors may admit none (Yuen et al., 2024). This suggests a structural distinction: swapBE bounds the local certificate of envy in a final allocation, while the reformability framework bounds the process needed to reach an approximately envy-free allocation.

Another adjacent interpretation appears in ex ante fair division under swap-closed feasibility. “On the Existence of Pareto Efficient and Envy Free Allocations” does not define swap bounded envy, but it shows that when the feasible allocation set is closed under swapping bundles of any two agents, there always exists an ex ante mixed allocation that is both envy-free and Pareto efficient (Cole et al., 2019). There, swaps function as a proof device for eliminating expected-envy cycles rather than as a local certificate in a realized allocation (Cole et al., 2019).

3. Equal-number and multi-dimensional environments

The definition of swapBE is motivated by two environment classes formalized in (Echenique et al., 12 Aug 2025). In the equal-number environment, the total number of items is $\emptyset_t$8, and feasibility requires each agent to receive exactly $\emptyset_t$9 items. This models fixed-cardinality assignments, such as settings where every participant must receive the same number of resources (Echenique et al., 12 Aug 2025). In the multi-dimensional environment, items are partitioned by period, with exactly $(o_1,\dots,o_T)$0 items per period and one item assigned to each agent in each period, possibly with null objects available as outside options (Echenique et al., 12 Aug 2025).

Responsive preferences play a central role in both environments. Under responsiveness, bundle comparisons are induced by an item ranking: if there is a bijection between two equally sized bundles such that each item in one bundle is weakly preferred to its match in the other, then the first bundle is weakly preferred (Echenique et al., 12 Aug 2025). Additive preferences are a strong special case of responsiveness (Echenique et al., 12 Aug 2025). For the multi-dimensional results, the stronger property of directional separability is needed: if the current period’s item is held fixed, the ranking over prior-period bundles must remain unchanged when that same item is appended (Echenique et al., 12 Aug 2025).

These assumptions are not merely technical. The positive existence result for swapBE in multi-dimensional settings depends on preserving earlier bundle comparisons while sequentially assigning later-period items (Echenique et al., 12 Aug 2025). Conversely, the paper reports that draft mechanisms may fail to be EF1 and swapEF in multi-dimensional settings once picks are aligned by period; positive guarantees exist only in some special cases, such as alternating serial dictatorship for $(o_1,\dots,o_T)$1, and for additive utilities when $(o_1,\dots,o_T)$2 in the swapEF sense (Echenique et al., 12 Aug 2025). For larger $(o_1,\dots,o_T)$3, alternating serial dictatorship can fail EF1 for $(o_1,\dots,o_T)$4 and fail swapEF for $(o_1,\dots,o_T)$5 (Echenique et al., 12 Aug 2025).

This contrast between equal-number and multi-dimensional settings is a major structural theme. In equal-number environments, draft mechanisms are comparatively strong under responsive preferences; in multi-dimensional environments, additional machinery is needed to maintain local exchange fairness (Echenique et al., 12 Aug 2025).

4. Mechanisms and algorithmic guarantees

The paper introducing swapBE provides two main positive mechanism results (Echenique et al., 12 Aug 2025). First, in equal-number environments with responsive preferences, any draft mechanism—including serial dictatorship, alternating serial dictatorship, FIFO, and randomized variants—yields an allocation that is EF1 and swapEF (Echenique et al., 12 Aug 2025). The proof compares the order in which each agent chooses across rounds. For any pair of agents $(o_1,\dots,o_T)$6, items chosen by $(o_1,\dots,o_T)$7 in earlier rounds dominate items later chosen by $(o_1,\dots,o_T)$8, and a case analysis on the final-round items shows that either no envy remains or a single swap and a single-item deletion suffice (Echenique et al., 12 Aug 2025). Since swapEF implies swapBE, these draft allocations also satisfy swapBE in that setting.

Second, in the multi-dimensional setting, the paper proposes the TTC+SD mechanism and proves that under responsive and directionally separable preferences it outputs allocations that are EF1 and swapBE (Echenique et al., 12 Aug 2025). TTC+SD operates period by period. At each period $(o_1,\dots,o_T)$9, it first applies Top Trading Cycles (TTC) to the bundles accumulated through period $i$0, clearing cycles and reassigning entire prior bundles. The cycle-clear order induces a priority ordering, and then serial dictatorship is run within period $i$1 according to the reverse cycle-clear order (Echenique et al., 12 Aug 2025).

Two invariance claims underpin the proof. TTC preserves EF1 and swapBE because it reassigns whole bundles and is individually rational: if a deletion or reference-bundle swap previously witnessed bounded envy, the same witness remains valid after the envying agent weakly improves (Echenique et al., 12 Aug 2025). Serial dictatorship in the TTC-induced order then preserves EF1 and swapBE because agents who were more exposed to envy pick earlier in the next period, and directional separability ensures earlier-period comparisons are unaffected by appending the same current-period item (Echenique et al., 12 Aug 2025). The resulting runtime is polynomial, reported as $i$2 with TTC per round $i$3 and serial dictatorship per round $i$4 (Echenique et al., 12 Aug 2025).

The same paper also studies welfare-maximization routes to local exchange fairness. Under additive 0–1 utilities and equal-cardinality constraints, maximizing Nash welfare over equal-cardinality allocations yields EF1, swapEF, and Pareto efficiency within that constrained domain, provided the Nash welfare is positive (Echenique et al., 12 Aug 2025). The proof observes that if one agent envies another by more than one utility unit, then exchanging a 0-valued item from the envying agent with a 1-valued item from the envied agent strictly increases Nash welfare, contradicting optimality (Echenique et al., 12 Aug 2025). In the multi-dimensional setting, however, Nash welfare maximization need not yield swapBE or swapEF, even under additional symmetry assumptions (Echenique et al., 12 Aug 2025).

A different algorithmic perspective is developed in (Yuen et al., 2024), where the issue is how many bilateral exchanges are needed to reform an initial allocation into an EF1 one. For two agents with identical utilities, a greedy algorithm