Pareto-Optimal K-Proportional Division

• The paper introduces Pareto-optimal K-proportional division as an interpolation between classical proportionality (k = n) and envy-freeness (k = 2), establishing graded fairness criteria.
• It rigorously defines k-proportionality and demonstrates that stricter fairness (lower k) leads to structural impossibilities in connected and equitable divisions.
• The work highlights algorithmic challenges and practical implications in resource allocation, suggesting trade-offs and avenues for future research in multi-dimensional settings.

Pareto-optimal K-proportional division is a conceptual and algorithmic framework in fair division theory that interpolates between classical proportional distribution, where each agent secures at least $1/n$ of the aggregate value, and envy-freeness, where no agent prefers another agent’s allocation over their own. The notion of $k$-proportionality, and its interplay with Pareto-optimality, establishes a scale of fairness guarantees indexed by $k$ ($2 \leq k \leq n$) and explicitly characterizes the feasibility/infeasibility landscape for connected, equitable, or efficient divisions (Chèze, 16 Sep 2025). This framework is technically formalized for divisible goods, cake-cutting on continuous spaces, and can be adapted to piecewise or discrete settings. Below, the principal theoretical structures, impossibility phenomena, and algorithmic implications are presented.

1. Formal Definition of k-Proportionality

A division of a homogeneous divisible good $X$ among $n$ agents is $k$-proportional if, for every subset $J \subseteq \{1, \ldots, n\}$ with $|J| = k$, and any agent $i \in J$, it holds that

$$\mu_i(X_i) \geq \frac{1}{k} \sum_{j \in J} \mu_i(X_j)$$

where $\mu_i$ is agent $i$’s (absolutely continuous) measure on $X$ and $X = X_1 \sqcup \cdots \sqcup X_n$ the allocation partition. For $k = n$, the condition reduces to classical proportionality, requiring

$\mu_i(X_i) \geq \mu_i(X)/n$

for each agent $i$. For $k=2$, the definition reduces to envy-freeness:

$\mu_i(X_i) \geq \mu_i(X_j) \;\; \forall j \neq i$

Notably, $k$-proportionality interpolates between proportionality and envy-freeness: higher $k$ corresponds to weaker fairness, with the spectrum sharpest for connected pieces or partitions on the real line or $S^1$.

2. Relationships among k-Proportionality, Proportionality, and Envy-Freeness

The $k$-proportional criterion acts as a scale:

• Proportional division $(k = n)$: Each agent receives at least $1/n$ of the total value.
• Envy-free division $(k = 2)$: Each agent receives at least as much as any other agent’s allocation, as valued by their own measure.

There are equivalence reformulations for $k$-proportionality. Specifically, a division is $k$-proportional if and only if, for every $i$ and subset $J' \subseteq N \setminus \{i\}$ with $|J'| = k-1$:

$$\mu_i(X_i) \geq \frac{1}{k-1} \sum_{j \in J'} \mu_i(X_j)$$

This captures the condition that an agent does not envy the average share of any group of $k-1$ others.

Furthermore, a monotonicity result holds: If a division is $k$-proportional, then it is also $(k+1)$-proportional. Thus, higher $k$-proportionality is strictly weaker.

3. Impossibility Theorems: Pareto-Optimal and Equitable k-Proportional Division

The critical structural insight is that tightening fairness to $k \leq n-1$—even short of strict envy-freeness—incurs strong impossibility effects in standard models:

• Pareto-optimal $k$-proportionality with connected pieces: For $k \leq n-1$, there exist measures such that no partition of $X$ into connected pieces satisfies both $k$-proportionality and Pareto-optimality (Chèze, 16 Sep 2025).
• Equitable $k$-proportionality: For $k \leq n-1$ and $n \geq 5$, for certain measures on $S^1$, no connected division is both $k$-proportional and equitable, where equitability requires every $\mu_i(X_i)$ to be equal.

Previously, these impossibility results were known only in the case $k = 2$ (envy-freeness), but the paper extends such theorems to all $k \leq n-1$.

A mathematical illustration extracts the forced structure: requiring $(n-1)$-proportionality in connected settings might determine, for agents $i \geq 2$, piece lengths $lg(X_i) = (1-x_1)/(n-1)$, but in aggregate $x_1 = 1/n$ is forced, which leads to dominance by a strictly improving partition—violating Pareto-optimality.

The paper shows that the threshold for these impossibility phenomena is precisely at $k = n-1$: the first step beyond classical proportionality triggers the necessity of structural incompatibilities.

4. Strong k-Proportionality and Existence Criteria

The concept of strong k-proportionality is defined by strict inequality:

$\mu_i(X_i) > \frac{1}{k} \sum_{j \in J} \mu_i(X_j)$

for every agent $i$ in every subset $J$ of $k$ agents.

A sharp existence criterion is proved: A strong $k$-proportional division exists for measures $\mu_1, \ldots, \mu_n$ if and only if every subset $S \subseteq \{\mu_1, \ldots, \mu_n\}$ of size $k$ contains at least two distinct measures.

For $k = n$, this generalizes the Dubins–Spanier theorem (strong proportional division exists iff at least two measures differ), whereas for $k = 2$, it coincides with strong envy-free division.

5. Algorithmic and Practical Implications

k-proportionality provides graded fairness guarantees bridging the gap between proportionality and envy-freeness. Its technical role is significant in situations where one cannot feasibly guarantee envy-freeness or full equitability but requires greater robustness than pure proportionality. For example:

• Resource allocation tasks (land, timeslots, budget division) may admit proportional division but not $k$-proportional or envy-free division when connectivity or equitability is imposed.
• Algorithmic complexity scales with increased fairness: known algorithms for proportional division (e.g., Even–Paz) do not directly generalize to the $k$-proportional regime for $k < n$, where impossibility may arise.

In practical design, one can adjust $k$ (the “fairness parameter”) to balance stricter fairness with algorithmic and structural feasibility; this interpolation clarifies the threshold where impossibility sets in and suggests designing less-demanding division protocols for greater tractability.

6. Directions for Further Research

Open problems and potential areas of development highlighted include:

• Systematic paper of the algorithmic complexity of $k$-proportional division for intermediate values of $k$.
• Investigation into whether thresholds for impossibility can vary with domain, agent preferences, multi-dimensional resources, or relaxed connectivity constraints.
• Analysis of adaptation of $k$-proportionality to alternative models and non-static resource division scenarios.
• Application of matrix characterizations for strong fairness criteria in more complex allocation settings.

The existence and construction of Pareto-optimal $k$-proportional divisions in discrete, multi-dimensional, and preference-correlated environments remain areas where further mathematical and algorithmic work is needed.

Table: k-Proportionality and Division Notions

$k$	Fairness Notion	Existence (connected pieces)
$n$	Proportionality	Always exists
$n-1$	$(n-1)$-proportional	May fail (impossibility for some measures)
$2$	Envy-freeness	May fail (impossibility for some measures)

This table summarizes the critical thresholds for existence, as proved in (Chèze, 16 Sep 2025).

Summary

Pareto-optimal $k$-proportional division provides a nuanced continuum of fairness, spanning the spectrum from proportionality to envy-freeness. While classical proportionality is always attainable, strengthening requirements by decreasing $k$ rapidly exposes regions of impossibility for connected, equitable, or efficient allocations. The framework substantiates that the difficulties inherent in strict fair division manifest immediately beyond proportionality and enables precise tuning of fairness standards in both theoretical and applied contexts.