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Computing Envy-Free up to Any Good (EFX) Allocations via Local Search

Simina Brânzei

TL;DR

This work addresses the open problem of computing exact EFX (envy-freeness up to any good) allocations for additive valuations by proposing a simple local search method based on simulated annealing. The algorithm operates over the space of allocations using a single-item-transfer neighborhood and optimizes the objective f(A), the total number of EFX violations, ultimately finding allocations with f(A)=0 in thousands of random instances and at scales with hundreds of items and agents. A theoretical contribution provides a potential function Φ for identical valuations, showing that any strict-descent procedure under the single-transfer neighborhood strictly decreases Φ whenever an allocation is not EFX, guaranteeing termination at an EFX allocation and yielding an alternative existence proof for identical valuations. Together, the empirical results establish a practical, scalable benchmark for EFX computation and highlight the promise of local-search dynamics for fair division in additive settings.

Abstract

We present a simple local search algorithm for computing EFX (envy-free up to any good) allocations of $m$ indivisible goods among $n$ agents with additive valuations. EFX is a compelling fairness notion, and whether such allocations always exist remains a major open question in fair division. Our algorithm employs simulated annealing with the total number of EFX violations as an objective function together with a single-transfer neighborhood structure to move through the space of allocations. It found an EFX allocation in all the instances tested, which included thousands of randomly generated inputs, and scaled to settings with hundreds of agents and/or thousands of items. The algorithm's simplicity, along with its strong empirical performance makes it a simple benchmark for evaluating future approaches. On the theoretical side, we provide a potential function for identical additive valuations, which ensures that any strict-descent procedure under the single-transfer neighborhood ends at an EFX allocation. This represents an alternative proof of existence for identical valuations.

Computing Envy-Free up to Any Good (EFX) Allocations via Local Search

TL;DR

This work addresses the open problem of computing exact EFX (envy-freeness up to any good) allocations for additive valuations by proposing a simple local search method based on simulated annealing. The algorithm operates over the space of allocations using a single-item-transfer neighborhood and optimizes the objective f(A), the total number of EFX violations, ultimately finding allocations with f(A)=0 in thousands of random instances and at scales with hundreds of items and agents. A theoretical contribution provides a potential function Φ for identical valuations, showing that any strict-descent procedure under the single-transfer neighborhood strictly decreases Φ whenever an allocation is not EFX, guaranteeing termination at an EFX allocation and yielding an alternative existence proof for identical valuations. Together, the empirical results establish a practical, scalable benchmark for EFX computation and highlight the promise of local-search dynamics for fair division in additive settings.

Abstract

We present a simple local search algorithm for computing EFX (envy-free up to any good) allocations of indivisible goods among agents with additive valuations. EFX is a compelling fairness notion, and whether such allocations always exist remains a major open question in fair division. Our algorithm employs simulated annealing with the total number of EFX violations as an objective function together with a single-transfer neighborhood structure to move through the space of allocations. It found an EFX allocation in all the instances tested, which included thousands of randomly generated inputs, and scaled to settings with hundreds of agents and/or thousands of items. The algorithm's simplicity, along with its strong empirical performance makes it a simple benchmark for evaluating future approaches. On the theoretical side, we provide a potential function for identical additive valuations, which ensures that any strict-descent procedure under the single-transfer neighborhood ends at an EFX allocation. This represents an alternative proof of existence for identical valuations.

Paper Structure

This paper contains 25 sections, 1 theorem, 5 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction.
  2. Our Contribution.
  3. Related Work.
  4. EFX existence.
  5. Approximate EFX.
  6. EF1 and proportionality.
  7. Competitive equilibrium, share-based notions, and stochastic settings.
  8. Local search in other domains.
  9. Definitions.
  10. Algorithm.
  11. Pseudocode.
  12. Experimental Results.
  13. Experimental environment.
  14. Scaling with the Number of Items (Fixed Agents)
  15. Effect of the Item-to-Agent Ratio
  16. ...and 10 more sections

Key Result

Proposition 1

Consider an instance with identical valuations, where each good $g \in M$ has a common value $w_g > 0$. For an allocation $A = (A_1, \ldots, A_n)$, define Then every strict descent procedure on $\Phi$ using the single-good transfer neighborhood from def:neighbor is guaranteed to terminate at an EFX allocation.

Figures (5)

  • Figure 1: Average runtime (in minutes) vs. the item-to-agent ratio ($m/n$) for finding an EFX allocation with $n=15$ agents. Each data point shows the average runtime over 100 independent instances where agent valuations were drawn uniformly at random. The shaded region indicates one standard deviation. The figure presents three panels: a close-up of the region $m/n \in [0, 13]$, a second close-up of the region $m/n \in [13, 50]$, and a full-range view.
  • Figure 2: Performance of the algorithm with $m = 10,000$ items as the number of agents grows from $n=4$ to $n=100$. For each value of $n$ displayed, the data point is obtained by taking the average of 100 runs with uniformly random valuations. The left panel shows the average runtime and standard deviation (in minutes), while the right panel shows the average and standard deviation in steps.
  • Figure 3: Runtime (in minutes) for finding an EFX allocation with $n=8$ agents and $m=160$ items as the correlation strength grows from 0 to 1. For each correlation strength, the results are averaged over 100 trials. The left figure shows the mean and the right figure shows the median.
  • Figure 4: Runtime (in minutes) for finding an EFX allocation with $n=15$ agents and $m=52$ items as the correlation strength grows from 0 to 1. For each correlation strength, the results are averaged over 100 trials. The left figure shows the mean and the right figure shows the median.
  • Figure 5: Average number of steps vs. the item-to-agent ratio ($m/n$) for finding an EFX allocation with $n=15$ agents. A step counts a neighboring allocation of the current one that was generated by the algorithm. Each data point shows the average number of steps taken by the algorithm over 100 independent instances where agent valuations were drawn uniformly at random. The shaded region indicates one standard deviation. The figure presents three panels: a close-up of the region $m/n \in [0, 13]$, a second close-up of the region $m/n \in [13, 50]$, and a full-range view.

Theorems & Definitions (5)

  • Definition 1: EFX violation
  • Definition 2: EFX violation count
  • Definition 3: Single-transfer neighborhood
  • Proposition 1
  • proof