Learning Fair and Preferable Allocations through Neural Network

TL;DR

This work addresses learning $EF1$ allocations from implicit expert-like rules in the fair division of indivisible goods. It introduces a differentiable relaxation of Round Robin, $\mathrm{SoftRR}$, and a neural network, $\mathrm{NRR}$, that learns the agent order used by RR while preserving $EF1$ at inference. Key contributions include the first integration of supervised learning to recover implicit EF1 rules from examples, the $\mathrm{SoftRR}$ relaxation enabling backpropagation, and the $\mathrm{NRR}$ architecture that jointly learns agent ordering and allocation. Experiments on synthetic data show $\mathrm{NRR}$ outperforms baselines in allocation proximity and related metrics, demonstrating that implicit fair rules can be recovered and utilized with strict fairness guarantees. The approach offers a scalable way to fuse expert heuristics with differentiable learning for fair division in practical settings.

Abstract

The fair allocation of indivisible resources is a fundamental problem. Existing research has developed various allocation mechanisms or algorithms to satisfy different fairness notions. For example, round robin (RR) was proposed to meet the fairness criterion known as envy-freeness up to one good (EF1). Expert algorithms without mathematical formulations are used in real-world resource allocation problems to find preferable outcomes for users. Therefore, we aim to design mechanisms that strictly satisfy good properties with replicating expert knowledge. However, this problem is challenging because such heuristic rules are often difficult to formalize mathematically, complicating their integration into theoretical frameworks. Additionally, formal algorithms struggle to find preferable outcomes, and directly replicating these implicit rules can result in unfair allocations because human decision-making can introduce biases. In this paper, we aim to learn implicit allocation mechanisms from examples while strictly satisfying fairness constraints, specifically focusing on learning EF1 allocation mechanisms through supervised learning on examples of reported valuations and corresponding allocation outcomes produced by implicit rules. To address this, we developed a neural RR (NRR), a novel neural network that parameterizes RR. NRR is built from a differentiable relaxation of RR and can be trained to learn the agent ordering used for RR. We conducted experiments to learn EF1 allocation mechanisms from examples, demonstrating that our method outperforms baselines in terms of the proximity of predicted allocations and other metrics.

Figures (5)

• Figure 1: The convergence of $\mathrm{SoftRR}_\tau$. The first three figures from the left show the outputs of $\mathrm{SoftRR}_\tau$ for $\tau=1$, $0.05$, and $0.001$, respectively. The rightmost figure is the output of RR.
• Figure 2: The architecture of $\mathrm{NRR}$. (a) Overall architecture: The input valuation $\bm{V}$ as an input, is fed into the network, where a permutation matrix $\hat{\bm{P}}$ is computed. This matrix is then multiplied by $\bm{V}$, and $\mathrm{SoftRR}$ is executed to obtain the allocation result. (b) Sub-network $\pi^{\bm{\theta}}$: This sub-network computes the permutation matrix $\hat{\bm{P}}$ from the input valuation $\bm{V}$.
• Figure 3: Evaluation metrics for varying numbers of agents and goods. (a): Results for $n=15$ agents. (b): Results for $n=30$ agents. The horizontal axis represents the number of goods $m$. The vertical axes in each figure correspond to the following metrics: Hamming distance (leftmost), ratio of EF1 allocations (middle), and utilitarian welfare loss (rightmost). The symbols $\downarrow$ and $\uparrow$ indicate that the metric improves as the value decreases and increases, respectively.
• Figure 4: Kendall's $\tau$ and examples of learned agent orders. (a): The results for $n=15$ agents. (b): The results for $n=30$ agents. For each panel, the first figure shows Kendall's $\tau$ between learned orders and ones calculated by mean valuations. The legend is shared between the two panels. The second figure shows an example of the agent orders of mean valuation and NRR. The point $(x,y)$ means that the agent at rank $x$ in the mean valuation order is placed at rank $y$ in the learned order.
• Figure 5: Examples of allocations. The leftmost heat map represents the valuation profile. The remaining four columns correspond to allocation results by MUW, RR, RR induced by the highest mean valuation order, and pre-trained NRR. In the heatmap, areas with a value of $0$ are represented in black, while areas with a value of $1$ are represented in white.

Theorems & Definitions (8)

• Definition 3.1: EF foley1966resource
• Definition 3.2: EF1 lipton2004approximatelybudish2011combinatorial
• Proposition 5.1: caragiannis2016unreasonable
• Example 5.1
• Definition 5.1: RR induced by permutations
• Proposition 5.2
• Proposition 5.3
• Proposition 5.4