Temporal Envy-Freeness up to One Good

• TEF1 is a dynamic fairness concept that generalizes static EF1 by ensuring each agent’s cumulative allocation is envied by at most one good over time.
• The framework applies to online fair division and repeated matching, with specific algorithms and buffer strategies enabling fair allocations under varied valuation models.
• Key results highlight computational trade-offs and feasibility limits, including NP-hard cases for general valuations and efficient solutions for symmetric or binary settings.

Temporal Envy-Freeness Up to One Good (TEF1) designates a temporally cumulative fairness criterion for the online or repeated allocation of indivisible goods or assignments. TEF1 requires that for every pair of agents $i, j$ and every time step $t$, agent $i$ does not envy agent $j$'s cumulative bundle up to time $t$ by more than a single good; explicitly, the removal of any single item from $j$'s bundle suffices to eliminate envy. TEF1 generalizes the static “envy-freeness up to one good” (EF1) property to temporal and dynamic environments, including online allocation, repeated matching in two-sided markets, and temporally extended fair division with or without scheduling. TEF1 operates as a central criterion for feasibility and fairness in dynamic resource allocation, striving to guarantee that no agent accumulates substantial envy as goods or matches are dispensed sequentially and irrevocably.

1. Formal Definitions and Model Foundations

TEF1 is defined for both online fair division of indivisible goods and for repeated allocation (e.g., bipartite matching) settings. In the canonical model for indivisible goods (Elkind et al., 2024), at each discrete round $t$, one or more items arrive and are allocated to agents from a set $N$ according to additive valuations $v_i$. The cumulative bundle $A_i^t$ of agent $i$ is the set of items allocated to them up to round $t$; an allocation is TEF1 if for all $t$ and all agent pairs $i,j$, there exists $g \in A_j^t \cup \{\emptyset\}$ such that: $v_i(A_i^t) \ge v_i(A_j^t \setminus \{g\}).$

In repeated matching in two-sided markets (Gollapudi et al., 2020), agents are partitioned into two sets $N$ and $M$, and at each time $t$, a matching $x^t$ is formed. The role of “one good” is replaced by the removal of a single match from a cumulative bundle. Valuations are often required to be additive, symmetric, and binary for positive results. For all $i, j \in N$ and every $t$, the definition is: $$v_i^t(X_i(t)) < v_i^t(X_j(t)) \implies \exists\, k \in X_j(t)\ \textrm{with}\ v_i^t(X_i(t)) \ge v_i^t(X_j(t)) - v_i^t(\{k\}).$$

The definition equally applies by side-symmetry for $M$. This cumulative perspective distinguishes TEF1 from static EF1, demanding no growing envy over any time prefix.

2. Existence and Impossibility Results

TEF1 existence and nonexistence are sharply delineated by setting, valuation domain, scheduling, and agent or item structure.

• For standard online fair division without scheduling, TEF1 always exists for two agents, for two item types with symmetric valuations, and for generalized binary valuations (where each agent values each item as either $0$ or a fixed $p_j > 0$). However, for general valuations and $n \geq 3$, TEF1 allocations may not exist, and deciding existence is NP-hard (Elkind et al., 2024).
• In the house allocation setting (exactly one item per agent per round), with identical days and three rounds, a TEF1 allocation is guaranteed and efficiently computable (Choi et al., 19 Jan 2026).
• For repeated matching in symmetric, binary, additive settings (with possibly dynamic valuations), there always exists a temporal EF1-over-rounds allocation that is also maximum-weight (efficient), and can be computed in polynomial time (Gollapudi et al., 2020).
• Impossibility arises for repeated matching with asymmetric or non-binary valuations: either the simultaneous guarantee of maximum-weight per round and TEF1 fails, or even TEF1 alone may be impossible when valuations are not symmetric (Gollapudi et al., 2020).
• With scheduling buffers, existence is improved: for identical days and buffer size at least $n/2$, TEF1 is always achievable in polynomial time (Choi et al., 19 Jan 2026). For buffer size $r \geq T/2$, TEF1 can be obtained by grouping and reallocating, but the buffer threshold is tight in many cases.

3. Algorithms and Complexity

Polynomial-time algorithms for TEF1 allocation have been established in several restricted domains.

• For two agents, a variant of the envy-cycle-elimination procedure suffices: allocate each arriving item to the less-advantaged agent, swapping bundles if mutual envy arises. Using a pointer $s$ to the last envy-free state, bundles accrued since $s$ are periodically swapped. Complexity is $O(nm)$ (Elkind et al., 2024).
• For generalized binary valuations, a greedy allocation to the current least-valued agent among those who positively value the arriving item ensures TEF1. Each item requires $O(n)$ time to determine the recipient; total complexity is $O(nm)$.
• For the house allocation setting with three rounds, an algorithm combines forward round-robin, assignment via matching in bipartite graphs, and reverse round-robin to preserve cumulative TEF1. This routine operates in polynomial time (Choi et al., 19 Jan 2026).
• With scheduling buffers (buffer $r \geq n/2$), the algorithm partitions rounds into blocks, delays items, and then performs block-wise round-robin allocations, leading to TEF1. Each stage is polynomial; overall, at most $O(\mathrm{poly}(n,T))$ (Choi et al., 19 Jan 2026).
• In repeated matching, the EF1-Matching algorithm starts with a maximum-weight matching and iteratively swaps pairs to eliminate excess envy, while maintaining efficiency. Each round admits $O(n^{2.5})$ total computational effort (Gollapudi et al., 2020).
• For all other general settings, particularly with three or more agents under arbitrary additive valuations, the decision problem for TEF1 existence is NP-hard (Elkind et al., 2024).

4. Structural Barriers and Incompatibilities

Several impossibility and incompatibility results highlight the structural limitations of TEF1.

• In general, for $n \geq 3$ and arbitrary additive valuations, TEF1 may fail—a result traced via reductions from 1-in-3-SAT and Partition (Elkind et al., 2024).
• Even with scheduling, the buffer thresholds for TEF1 are tight: for identical days, smaller buffers than $n/2$ can lead to persistent violations of TEF1, as shown through adversarial round constructions (Choi et al., 19 Jan 2026).
• Stronger timewise fairness notions—such as Temporal Envy-Freeness up to Any Good (TEFX) or Temporal Maximin Share (TMMS)—are infeasible even with larger buffer capacities or restricted domains (Choi et al., 19 Jan 2026).
• Simultaneous satisfaction of TEF1 and Pareto-optimality is often impossible: there exist explicit instances with two agents or two item types for which any TEF1 allocation is strictly Pareto-dominated. Maximizing standard welfare goals under a TEF1 constraint is NP-hard, as any welfare maximizer must also satisfy Pareto-optimality (Elkind et al., 2024).
• For two-sided matching, dropping symmetry or binarity in valuations permits the existence of excess envy-unbounded executions or incompatibility with efficient matchings, establishing that efficiency and cumulative fairness can only be guaranteed simultaneously in the symmetric, binary case (Gollapudi et al., 2020).

5. Scheduling and Buffering: Trade-offs and Techniques

The introduction of scheduling buffers transforms the landscape for achievable temporal fairness.

• With bufferless allocation (immediate and irrevocable assignment), TEF1 is generally unachievable beyond two agents or three rounds, even when days are identical (Choi et al., 19 Jan 2026).
• A scheduling buffer (the ability to defer assignment) of size at least $n/2$ suffices to guarantee the existence and recovery of TEF1 by batch grouping and blockwise fair division, typically via round-robin or matching methods. The “mid-block” trick ensures that goods can be equitably grouped and distributed so that cumulative envy stays within one good at each prefix (Choi et al., 19 Jan 2026).
• For buffer sizes $r \geq T/2$, grouping rounds and allocating in batches realizes TEF1 by emulating a static fair division within each batch.
• These trade-offs quantify the “price of waiting” for fairness: without sufficient latency allowance (buffering), strict temporal fairness remains out of reach, whereas moderate buffering robustly restores feasibility.

6. Extensions, Open Questions, and Context

Current research continues to refine the understanding and scope of TEF1:

• TEF1 has been extended to mixed manna settings (goods and chores), where for two agents, TEF1 always exists by reduction, but for larger $n$ the question remains open (Elkind et al., 2024).
• Alternative fairness principles such as temporal proportionality up to one good (TPROP1), group-level TEF1, and parameterized relaxations (e.g., $\alpha$-TEFX) are under investigation to assess whether weaker or group-wise fairness can be achieved where TEF1 is impossible.
• For generalized matching and allocation settings, further delineation is required: open questions include the existence of TEF1 for house allocation with more than three rounds, and whether the buffer threshold for TEF1 in the scheduling context is intrinsic or can be lowered.
• For repeated matching, the primary question remains to what extent symmetry or binarity can be relaxed while still achieving some temporal EF1 (over time or rounds) (Gollapudi et al., 2020).
• TEF1 is significant for applications in sequential equipment assignment, online task allocation, and dynamic marketplaces where fairness at all temporal prefixes is required to maintain agent satisfaction.

7. Comparative Summary of Settings and Results

Setting	TEF1 Existence	Efficient Computation	Notes
Two agents	Yes	Yes	Special procedures exist (Elkind et al., 2024)
General valuations (n≥3)	No (in general)	NP-hard	Impossibility; hard to decide
Generalized binary	Yes	Yes	Greedy assignment suffices
Unimodal preferences	Yes	Yes	Allocation to smallest bundle
Scheduling (r≥n/2)	Yes (identical days)	Yes	Blockwise round-robin (Choi et al., 19 Jan 2026)
House allocation (T=3)	Yes (identical days)	Yes	Specialized round-robin
Two-sided matching, symmetric binary	Yes	Yes (MW matching+swaps)	Fails outside symmetric binary (Gollapudi et al., 2020)

TEF1 thus establishes a central paradigm for temporal fairness in the allocation of indivisible goods and repeated matching, with exact boundaries determined by valuation type, scheduling flexibility, matching structure, and the number of agents. The property illuminates the interaction between time, combinatorial structure, fairness, and computational tractability in multi-agent resource assignment.