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Minimizing Cumulative Envy in Allocating a Sequence of Items

Paul W. Goldberg, Isaac Robinson, Nicholas Teh

arXiv:2610.09843v1cs.GTcs.MA

TL;DR

The paper asks how to minimize unfairness throughout sequential allocation when valuations and future arrivals are known, rather than evaluating only the final allocation. It defines cumulative maximum envy as the sum of round-wise maximum envy and studies its complexity and algorithms.

  • Problem

    In sequential fair division, fairness is often evaluated only after the last good, although agents experience the allocation process as it unfolds.

    Source: §1 · p. 2

  • Method

    The paper optimizes the sum of maximum envy over allocation rounds and analyzes the problem using a dynamic program for restricted settings.

    Source: §1 · p. 3, §4 · p. 11

  • Results

    The fixed-order decision problem is strongly NP-complete even with identical valuations and admits no constant-factor approximation unless P = NP; a dynamic program gives pseudopolynomial-time solvability in its stated setting.

    Source: §3 · p. 8, §4 · p. 11

  • Takeaways & Limitations

    The cumulative objective permits short-term imbalances when they are compensated later, while penalizing imbalances that persist.

    Source: §2 · p. 7

    The objective need not distribute envy evenly across agents; directly minimizing the largest or total individual cumulative envy is left for future work.

    Source: §1 · p. 3

Abstract

from arXiv · show

We study temporal fair division with indivisible goods that arrive sequentially and must be allocated irrevocably. In contrast to the usual online model, we assume that valuations and future arrivals are known in advance, and ask how unfairness evolves during the process. We introduce \emph{cumulative maximum envy}: the sum, over all rounds, of the maximum pairwise envy at that round. Equivalently, this is the area under the worst-envy curve, and it captures both the magnitude and the duration of envy. For a fixed arrival order, we show that the corresponding decision problem is strongly NP-complete and that minimizing this objective admits no constant-factor approximation unless P = NP, even under identical valuations and even under binary valuations. We complement these hardness results with a dynamic program that gives pseudopolynomial-time solvability for a constant number of agents, polynomial-time algorithms in further restricted settings, and an FPTAS for fixed $n$ under identical integer valuations. We then study a sequencing variant where the algorithm may choose the arrival order. This variant remains NP-complete even for two agents with identical valuations; however, a simple greedy algorithm achieves a $3/2$-approximation for $n=2$ agents, an $n/(n-1)$-approximation for any number of agents, and an additive guarantee depending on the maximum value of any good.

1 Introduction

The introduction frames fair division as a temporal problem: agents experience allocations as goods arrive, so fairness through the process matters alongside the final allocation. It introduces cumulative maximum envy as an optimization objective and previews hardness, algorithms, and a variant where the arrival order can be chosen.

1 Introduction

  • The objective sums maximum pairwise envy over rounds, measuring both its magnitude and how long it persists.It is the area under the worst-envy curve and remains meaningful when strong fairness at every prefix is impossible.Source: §1 · p. 3

1.1 Our Contributions

  • For a fixed arrival order, minimizing cumulative maximum envy is strongly NP-complete and has no constant-factor approximation unless P = NP, even with identical or binary valuations.Source: §1.1 · p. 4
  • A dynamic program gives pseudopolynomial solvability for a constant number of agents; further restricted cases admit polynomial-time algorithms, and identical integer valuations admit an FPTAS for fixed n.Source: §1.1 · p. 4
  • When the algorithm may choose the arrival order, the problem remains NP-complete for two agents with identical valuations, while the paper gives approximation guarantees for identical valuations.The guarantees include 3/2 for two agents, n/(n −1) for general n, and an additive guarantee depending on the maximum good value.Source: §1.1 · p. 4

1.2 Related Work

  • Unlike prior temporal work focused on maintaining fairness constraints at every prefix, this paper optimizes cumulative envy, which remains meaningful when such constraints are infeasible.Earlier work includes temporal EF1 and related prefix-based guarantees [25, 29]; envy minimization also has a history in offline fair division.Source: §1 · p. 2, §1.2 · p. 6

2 Preliminaries

The paper models sequential allocation with known valuations and arrival order, then evaluates fairness across the allocation trajectory rather than only at the end. Its cumulative maximum envy objective sums the worst pairwise envy across rounds, so both the size and persistence of imbalances matter.

2 Preliminaries

  • Goods arrive in an order π, and each must be allocated as it arrives; the paper considers both fixed and algorithm-chosen orders.Valuations are additive, and allocations can be represented as sequences assigning each arriving good to an agent.Source: §2 · pp. 6–7
  • Cumulative maximum envy sums the maximum pairwise envy at each round, penalizing imbalances that are large or long-lasting.Short-term imbalances may be acceptable when compensated later.Source: §2 · p. 7
  • Minimizing only final-round maximum envy can be badly misaligned with minimizing cumulative envy, with a linear separation shown even under identical valuations.The example contrasts an allocation with final cost 1 but cumulative cost growing linearly in m against one with a much smaller cumulative cost.Source: §2 · pp. 7–8

3 Intractability Results

For a fixed arrival order, minimizing cumulative maximum envy is computationally intractable even under identical or binary valuations. The paper also rules out constant-factor approximation in both restricted settings unless P = NP.

3 Intractability Results

  • Temporal-EnvyMin is strongly NP-complete and has no constant-factor approximation unless P = NP, even with identical valuations.The hardness remains when the maximum value of any good is polynomially bounded in n.Source: §3 · pp. 8, 10
  • The reduction’s strong NP-hardness follows from 3-Partition, whose hardness persists when T is polynomially bounded.The construction assigns goods valued using the 3-Partition instance and uses zero-valued goods after the positive-valued goods.Source: §3 · pp. 8, 10
  • The identical-valuation reduction from 3-Partition creates a threshold gap: yes-instances attain cost at most 10T(3k + 1), while no-instances exceed it.The proof uses a round where zero envy would force a 3-Partition solution; appended zero-valued goods then preserve residual envy and amplify the gap.Source: §3 · p. 10
  • Temporal-EnvyMin is NP-complete and has no constant-factor approximation unless P = NP, even with binary valuations.Binary valuations restrict each agent-good value to 0 or 1.Source: §3 · p. 11

4 Tractability in Restricted Settings

The section identifies tractable restrictions on temporal envy minimization, using an exact dynamic program and specialized algorithms. It also gives an FPTAS for fixed numbers of agents under identical integer valuations, while leaving a broader polynomial-time question open.

Binary identical valuations

  • For binary identical valuations, assigning each arriving good to an agent with the lowest total bundle value minimizes cost and maximum envy at every round.The resulting optimal sequence can be computed in polynomial time.Source: §4 · p. 11

Dynamic programming

  • The dynamic program returns an optimal allocation sequence in O((m+1)(Vmax m^2+1)(mVmax+1)^n^2·n^3) time.For fixed n, this is pseudopolynomial; polynomially bounded integer values give polynomial time.Source: §4 · pp. 12, 14

Binary valuations

  • Under binary valuations, Temporal-EnvyMin is XP with respect to the number of agents.With binary and identical valuations, an optimal sequence can be computed in polynomial time.Source: §4 · pp. 11, 14

Open question

  • Whether polynomial-time solvability holds for constant n with identical valuation functions remains open.The authors conjecture that the answer is no.Source: §4 · p. 14

Identical valuations

  • For fixed n and identical valuations with integer-valued items, a (1+ε)-approximation runs in time polynomial in m and 1/ε.The FPTAS rounds values and solves the rounded instance by dynamic programming, using a lower bound to control accumulated rounding error.Source: §4 · p. 14

5 Choosing the Arrival Sequence

The authors study a variant in which the algorithm chooses the goods’ arrival order while valuations are fixed. This flexibility does not remove computational hardness, but a simple greedy algorithm has provable approximation guarantees under identical valuations.

Order sensitivity

  • When one good dominates in value, placing it early can force cumulative cost Θ(m) times larger than placing it last.In the example, allocating the smaller goods first keeps their prefix envy small before the dominant good arrives.Source: §5 · p. 15

Hardness

  • Choosing the arrival order remains NP-complete even for two agents with identical valuations.The problem asks whether some order and allocation sequence achieve cost at most a given threshold.Source: §5 · p. 16

Approximation guarantees

  • For identical valuations, Identical-Lowest-First achieves a 3/2-approximation for n=2 and an n/(n−1)-approximation for all n≥2.It sorts goods by increasing value, then assigns each to an agent with the currently smallest bundle value.Source: §5 · pp. 16–17

Additive guarantee

  • The greedy algorithm also provides an additive approximation guarantee stated in terms of the maximum value of any good.The supplied theorem passage does not show the additive bound’s expression.Source: §5 · p. 17

6 Conclusion

The paper frames temporal fair division as an optimization problem and establishes both hardness and algorithmic results. It closes by identifying discounted cumulative envy as a direction for future study.

6 Conclusion

  • For fixed arrival order, minimizing cumulative envy is intractable and admits no constant-factor approximation unless P=NP.The paper also gives dynamic programming algorithms for optimal solutions in several special cases.Source: §6 · p. 18
  • Choosing the arrival order remains NP-complete for two agents with identical valuations, alongside efficient approximation algorithms for the sequencing variant.Source: §6 · p. 18
  • Discounted cumulative envy, where recent unfairness matters more than past inequities, is identified as a future research direction.The conclusion notes related discounted-model work in an online setting.Source: §6 · p. 18

A.1 Proof of Theorem 2

The proof reduces Binary-EF to the fixed-order temporal problem by appending zero-valued goods. Persistent envy across those rounds separates yes- and no-instances and supports the inapproximability result.

A.1 Proof of Theorem 2

  • The reduction makes deciding whether cost is at most m^3−1 equivalent to deciding whether the Binary-EF instance has an envy-free allocation.The constructed instance preserves the original binary goods and appends m^3 zero-valued goods.Source: §A.1 · p. 23
  • In a yes-instance, allocating the original goods envy-free makes later zero-valued rounds envy-free; in a no-instance, envy of at least 1 persists through those rounds.The resulting costs are at most m^3−1 in yes-instances and at least m^3+1 in no-instances.Source: §A.1 · pp. 23–24
  • Appending additional zero-valued goods extends the gap, so an α-approximation would decide Binary-EF by comparing its cost with αB.The proof concludes this is impossible unless P=NP.Source: §A.1 · p. 24

B.1 Proof of Proposition 1

For identical binary valuations, allocating each arriving good to a currently least-valued bundle achieves the minimum cumulative envy. For fixed n, scaling valuations and optimizing the scaled instance yields a (1 + ε)-approximation in time polynomial in m and 1/ε.

B.1 Proof of Proposition 1

  • At each round, the greedy algorithm assigns the arriving good to an agent with minimum current bundle value.Source: §B.1 · p. 25
  • The greedy rule keeps the maximum-minus-minimum bundle value at most 1, attaining the unavoidable gap of 0 when q_t is divisible by n and 1 otherwise.Summing these per-round minimum gaps shows the rule minimizes cumulative envy.Source: §B.1 · p. 25

B.2 Proof of Theorem 4

  • For fixed n, optimizing a scaled integer-valuation instance takes time polynomial in m and 1/ε.The proof outputs the scaled-instance optimum as an allocation for the original instance.Source: §B.2 · p. 26
  • The resulting allocation is a (1 + ε)-approximation for the original instance.Source: §B.2 · p. 27

C.1 Proof of Theorem 5

The proof reduces Partition to the two-agent, identical-valuation sequencing problem using paired goods whose values encode the input integers. A low-cost order exists precisely when the Partition instance has a solution, establishing hardness; membership in NP gives NP-completeness.

C.1 Proof of Theorem 5

  • The decision problem is in NP because a proposed order and allocation sequence permit computation of all partial bundles.Source: §C.1 · p. 27
  • The reduction uses 2q goods in pairs valued M and M + x_i, with M = T + 1, encoding each Partition number in a good's value.Source: §C.1 · p. 28
  • A Partition subset summing to T/2 yields an order and greedy allocation whose pair contributions sum to the reduction threshold.The construction orders pairs according to whether their index lies in the subset or its complement.Source: §C.1 · p. 29
  • Conversely, any allocation and order meeting the threshold force the encoded x_i values to split evenly between odd and even positions, yielding a Partition solution.Source: §C.1 · p. 30

C.2 Proof of Lemma 1

For identical valuations, every allocation sequence has cumulative maximum envy at least (1 − 1/n) times the total value of the goods. Round-robin allocation attains this bound when the number of unit-valued goods is a multiple of n.

C.2 Proof of Lemma 1

  • OPT(I) is at least (1 − 1/n)v(G) for identical valuations.Source: §C.2 · p. 31
  • With kn unit-valued goods, round-robin allocation incurs cost k(n − 1), exactly (1 − 1/n)v(G).Envy is 1 at rounds not divisible by n and 0 at rounds divisible by n.Source: §C.2 · p. 31

C.3 Proof of Theorem 6

The proofs analyze a greedy allocation after sorting goods by nondecreasing value and assigning each good to a least-valued bundle. They establish an n/(n − 1)-approximation for the general identical-valuation setting.

C.3 Proof of Theorem 6

  • For two agents in nondecreasing item-value order, the current envy gap is at most the arriving item's value, and assigning it to the smaller bundle makes the new gap equal to that value minus the previous gap.Source: §C.3 · p. 32
  • For any order and allocation with two identical-valuation agents, OPT is at least the value of the maximum-valued good.Source: §C.3 · p. 33

C.4 Proof of Theorem 7

  • Algorithm 2 is an n/(n − 1)-approximation for identical valuations.Source: §C.4 · p. 35

C.5 Tightness of Identical-Lowest-First Analysis for n = 2.

Examples show that Identical-Lowest-First can attain its analyzed approximation bounds, while the proof for Theorem 8 bounds each round’s envy by the value of the arriving good.

C.5 Tightness of Identical-Lowest-First Analysis for n = 2.

  • 3/2: with two agents and goods valued {1, 1, 2}, Identical-Lowest-First costs 3 while an optimal sequence costs 2.The optimal cost is certified by the lower bound in Lemma 1.Source: §C.5 · p. 35

C.6 Approximation Ratio Lower Bound of 2n−1

  • For n ≥ 3, the example gives Identical-Lowest-First cost 2n−1 and an optimal cost of 2n−2.The greedy order sorts the goods as (1, 1, 2, …, 2); the optimal order places the two value-1 goods at the beginning and end.Source: §C.6 · pp. 35–36

C.7 Proof of Theorem 8

  • With identical valuations and goods sorted by value, the proof bounds each round’s envy by the arriving good’s value, so the final round’s envy is at most Vmax.The proof relates each arriving value to that round’s envy plus the increase in the minimum bundle value, then sums across rounds.Source: §C.7 · pp. 36–37

C.8 Tightness of the Additive Approximation Guarantee.

A constructed instance shows that the additive Vmax term in the guarantee cannot be improved in general.

C.8 Tightness of the Additive Approximation Guarantee.

  • Under Identical-Lowest-First, each block of n goods valued k contributes (n−1)k to cumulative cost.The first n−1 arrivals in the block have gap k; the nth has gap 0.Source: §C.8 · p. 38
  • The construction’s descending optimal-order blocks begin with k followed by k+1 repeated k times, for k from n−1 down to 1.The greedy allocation ends with all agents at equal bundle value L1 = n + (n−1) + ··· + 2.Source: §C.8 · pp. 38–39
  • The additive Vmax term is tight in general, as shown by an instance with identical valuations and n²−1 goods.For each k from 1 to n−1 there are n goods valued k, and there are n−1 goods valued n.Source: §C.8 · pp. 37, 38, 40

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