FAIRFORMER: A transformer architecture for discrete fair division

TL;DR

FairFormer tackles the problem of allocating indivisible goods under additive valuations without money by learning a permutation-equivariant transformer that assigns items to agents in a single forward pass. The two-tower architecture uses within-set self-attention and item-to-agent cross-attention to produce a fractional allocation optimized for expected Nash welfare, followed by row-wise discretization and EF1-Quick-Repair to enforce fairness with minimal welfare loss. Empirical results show near-optimal performance on Nash welfare (~96.6–96.8%) and utilitarian welfare (~95.6–95.9%), outperforming RR, ECE, and MU+EF1 across diverse distributions and problem sizes, with favorable runtime characteristics. The approach demonstrates strong cross-size generalization, robustness to scarcity, and practical scalability, offering a compelling combination of fairness guarantees and high efficiency in discrete fair division.

Abstract

We propose a deep neural network-based solution to the problem of allocating indivisible goods under additive subjective valuations without monetary transfers, trading off economic efficiency with envy-based fairness. We introduce FairFormer, an amortized, permutation-equivariant two-tower transformer that encodes items and agents as unordered token sets, applies self-attention within each set, and uses item-to-agent cross-attention to produce per-item assignment distributions in a single forward pass. FairFormer is trained end-to-end to maximize expected log-Nash welfare on sampled instances, requiring no solver supervision, unrolled allocation procedures, or fairness labels. At test time, we discretize by row-wise $\arg\max$ and apply a lightweight post-processing routine that transfers items to eliminate violations of envy-freeness up to one item while prioritizing improvements in Nash welfare. Our approach generalizes beyond its training regime and achieves near-optimal welfare (e.g., for uniformly sampled valuations, $96$--$97\%$ for Nash welfare; $95$--$96\%$ for utilitarian welfare), outperforming strong baselines in solution quality and/or runtime.

Figures (6)

• Figure 1: FairFormer architecture: Agents and items are embedded separately, interact via cross-attention, and produce per-item assignment distributions through a temperature-scaled softmax.
• Figure 2: Nash welfare as a function of the items-per-agent ratio $m/n$. Learned models remain near-optimal across scarcity regimes, while classical EF1 procedures exhibit larger degradation, especially ECE.
• Figure 3: Distribution of welfare ratios across problem configurations. Boxplots visualize stability: learned models are tightly concentrated, while ECE exhibits heavy tails and large variance.
• Figure 4: Nash welfare versus total problem size $(n+m)$. Learned models are robust across the evaluated scale range, while ECE shows pronounced instability.
• Figure 5: Nash welfare heatmaps for the three FairFormer training strategies across $(n,m)$, highlighting cross-size generalization. Color indicates Nash welfare as a percentage of the per-instance optimum.

Theorems & Definitions (2)

• Proposition 1.1
• proof