arxiv: 2604.26374 · v1 · submitted 2026-04-29 · 💻 cs.RO · cs.MA

Recognition: unknown

Split over n resource sharing problem: Are fewer capable agents better than many simpler ones?

Karthik Soma , Mohamed S. Talamali , Genki Miyauchi , Giovanni Beltrame , Heiko Hamann , Roderich Gross

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:17 UTC · model grok-4.3

classification 💻 cs.RO cs.MA

keywords multi-agent systemsresource sharingcoverageagent speed scalinggroup sizefailure ratesrobotics

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The pith

In coverage tasks where agents share a fixed resource, the initial coverage rate rises with the number of agents, yet overall performance stays flat or favors one agent depending on how speed scales with size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the split over n resource sharing problem to examine whether limited resources should be concentrated in fewer capable agents or spread across many simpler ones. It studies this in a multi-agent coverage setting where each agent's disk-shaped footprint area shrinks exactly as 1/n. Formal analysis shows that the initial coverage rate increases with n, but when speed falls in proportion to radius all group sizes perform the same, while speed falling in proportion to footprint area makes a single agent best. Computer simulations further indicate that greater splitting raises the chance that individual agents fail.

Core claim

In the split over n resource sharing problem for coverage, where n agents equally divide a common resource and each has a disk footprint whose area scales as 1/n, the initial coverage rate grows with n. If agent speed decreases proportionally with radius, groups of every size perform equally well; if speed instead decreases proportionally with footprint area, a single agent performs best. Simulations show that increasing n also raises individual agent failure rates.

What carries the argument

The split over n resource sharing problem applied to coverage with disk footprints whose area scales exactly as 1/n, together with the two alternative speed-scaling rules (linear in radius or linear in area).

If this is right

When speed scales with radius, the choice of n has no effect on overall coverage performance.
When speed scales with footprint area, concentrating all resources in one agent yields the highest coverage.
Increasing n raises the probability that any given agent fails its task.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same speed-scaling logic could be tested in other tasks such as search or transport where resource limits are shared.
Designers could choose n by first determining which speed-scaling rule best matches their hardware.
Failure-rate results suggest adding redundancy or recovery mechanisms when splitting resources widely.

Load-bearing premise

Agent speed decreases proportionally with radius or with footprint area, and the footprint is always a disk whose area is exactly 1/n of the total resource.

What would settle it

Measure coverage performance across different n while holding speed exactly proportional to radius and check whether all group sizes achieve identical overall performance; any systematic difference would falsify the equal-performance claim.

Figures

Figures reproduced from arXiv: 2604.26374 by Genki Miyauchi, Giovanni Beltrame, Heiko Hamann, Karthik Soma, Mohamed S. Talamali, Roderich Gross.

**Figure 1.** Figure 1: Split over n resource sharing problem. A finite resource is equally shared among n robots; What is the optimal n? Different values of n might correspond to different robot platforms, for example, Turtlebot4 [1], e-puck2 [9], and Kilobot [14]. an equal share of a limited resource. The problem addresses a fundamental question in swarm intelligence: given a fixed resource, what is the optimal level of distri… view at source ↗

**Figure 2.** Figure 2: (a) The four velocity profiles as functions of n. (b) Area covered by a diskshaped agent of radius r moving forward with velocity V . per agent is A/n. Each agent has a continuous position in 2-D space and an orientation. It propels by alternating between (i) moving forward with velocity v for δ time steps, and (ii) rotating instantaneously by angle θ (in rad). δ is drawn from a power-law distribution an… view at source ↗

**Figure 3.** Figure 3: (a) Coverage trends when agents instantly relocate to random positions (b) Coverage at 100th simulation step with various velocity profiles. Coverage initially even match ideal rates, corresponding to perfect avoidance of revisiting explored regions (i.e., coverage rate of A per round). However, after a few rounds, the coverage achieved for every group size begins to deviate from the ideal rates, as overl… view at source ↗

**Figure 4.** Figure 4: Coverage trends across velocity profiles. The top row shows collision-free results for 1, 2, 10, 100, 500, and 1000 agents; the bottom row shows results with collisions for 2, 100, and 1000 agents. In each profile, colour intensity encodes group sizes, with darker shades indicating fewer agents. 4.3 Effect of Collisions While the previous sections discarded the effect of collisions, this section quantifies… view at source ↗

**Figure 5.** Figure 5: Coverage trends for different failure rates (left: α = 0.01 and failure rate k < 0.007; right: α = 0.1 and failure rate k ≤ 0.05). α controls the rate at which failure rate approaches its maximum failure rate β = 0.1, where k(n) = β(1 − 1 nα ). rate for n = 1 is 0. Hence, a sole robot would always be assumed fault-free. β > 0, determines the maximum failure rate for n → ∞. α controls how rapidly the failur… view at source ↗

read the original abstract

In multi-agent systems, should limited resources be concentrated into a few capable agents or distributed among many simpler ones? This work formulates the split over $n$ resource sharing problem where a group of $n$ agents equally shares a common resource (e.g., monetary budget, computational resources, physical size). We present a case study in multi-agent coverage where the area of the disk-shaped footprint of agents scales as $1/n$. A formal analysis reveals that the initial coverage rate grows with $n$. However, if the speed of agents decreases proportionally with their radii, groups of all sizes perform equally well, whereas if it decreases proportionally with their footprints, a single agent performs best. We also present computer simulations in which resource splitting increases the failure rates of individual agents. The models and findings help identify optimal distributiveness levels and inform the design of multi-agent systems under resource constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The split-over-n formulation is new and the geometric coverage scaling is clean, but the speed-size rules are introduced without derivation from the resource model.

read the letter

Splitting a fixed resource among more agents improves the initial coverage rate in this coverage task, but that advantage goes away or reverses depending on how agent speed scales with size. The paper does a solid job introducing the split-over-n formulation and showing the geometric scaling for coverage rate under constant speed. The perimeter argument is straightforward and the simulations on failure rates bring in a useful practical angle that pure analysis misses. It also avoids overclaiming by presenting the results conditionally. The main soft spot is that the two speed scaling rules are presented as if conditions without being derived from the resource sharing or any physical model of the agents. They feel like parametric explorations rather than conclusions that follow from the problem setup. The exact 1/n disk footprint is also a strong assumption that might not hold in practice if agents have different shapes or if packing efficiency varies. This is the kind of paper that helps people designing multi-robot systems decide on the right level of distributiveness. It should go to peer review because the core question is important and the analysis provides a starting point, though the authors will likely need to add more motivation for the scaling laws in the full manuscript.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the 'split over n resource sharing problem' for multi-agent systems, using a coverage case study where n agents equally share a resource, causing their disk-shaped footprints to scale in area as 1/n. A formal analysis shows the initial coverage rate increases with n. Conditional on speed scaling with radius, all group sizes perform equally; with footprint, a single agent is optimal. Simulations demonstrate that greater resource splitting raises individual agent failure rates.

Significance. If the results hold under the stated assumptions, this work offers valuable insights for designing multi-agent systems with constrained resources, helping determine optimal levels of distributiveness. Strengths include the formal analysis of coverage rates and the simulation results on failure rates, which provide concrete, quantitative comparisons across different n. These could inform practical applications in robotics under budget or size limits.

major comments (2)

The speed scaling assumptions (v decreases proportionally with radii, or with footprints) are introduced conditionally in the formal analysis without derivation from the resource-sharing premise, agent dynamics, or any physical model. This is load-bearing for the headline claims that 'groups of all sizes perform equally well' or 'a single agent performs best,' as the initial coverage-rate growth with n holds only when speed is independent of size.
The exact disk footprint area scaling as 1/n (and resulting perimeter-driven sqrt(n) coverage growth) is taken as given; the manuscript does not address how deviations in shape, packing efficiency, or non-disk geometries would alter the perimeter term and thus the core scaling result.

minor comments (1)

The abstract and simulation section could clarify the number of Monte Carlo runs, parameter ranges for speed scaling, and whether the reported failure-rate trends are statistically significant.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and assumptions of our work. We address each major comment below with clarifications and indicate proposed revisions to the manuscript.

read point-by-point responses

Referee: The speed scaling assumptions (v decreases proportionally with radii, or with footprints) are introduced conditionally in the formal analysis without derivation from the resource-sharing premise, agent dynamics, or any physical model. This is load-bearing for the headline claims that 'groups of all sizes perform equally well' or 'a single agent performs best,' as the initial coverage-rate growth with n holds only when speed is independent of size.

Authors: We agree that the speed scalings are conditional assumptions rather than derived from first principles. The resource-sharing premise directly yields the 1/n footprint area scaling, but the translation to velocity depends on the specific actuator or power model (e.g., force proportional to area versus linear dimensions). We selected the radius and footprint scalings as two representative cases common in robotics literature. In revision we will expand the introduction and formal analysis sections with a short paragraph motivating these choices via typical robotic dynamics examples, while reiterating their conditional status. This will not alter the mathematical results but will better contextualize the headline claims. revision: partial
Referee: The exact disk footprint area scaling as 1/n (and resulting perimeter-driven sqrt(n) coverage growth) is taken as given; the manuscript does not address how deviations in shape, packing efficiency, or non-disk geometries would alter the perimeter term and thus the core scaling result.

Authors: The circular disk model is chosen to enable an exact closed-form perimeter calculation that produces the sqrt(n) initial coverage growth. We acknowledge that non-circular shapes or packing inefficiencies would modify the effective perimeter-to-area ratio and thus the quantitative scaling. The broader 'split over n' problem formulation itself is geometry-agnostic. We will add a concise limitations paragraph in the discussion section noting that other footprints would require case-specific perimeter analysis, but the qualitative trade-off between coverage rate and failure risk under resource splitting remains applicable. A full generalization to arbitrary shapes lies outside the current scope. revision: partial

Circularity Check

0 steps flagged

No circularity: results follow from explicit geometric and conditional assumptions

full rationale

The paper states its core claims as consequences of two explicit premises: disk area scales exactly as 1/n (hence radius as 1/sqrt(n)) and speed is either constant, proportional to radius, or proportional to area. The initial-coverage-rate growth with n is obtained by multiplying n agents by their individual coverage contributions (speed times perimeter); the equal-performance and single-agent-best cases are obtained by substituting the two speed scalings. These substitutions are algebraic identities under the stated 'if' conditions, not self-definitions, fitted parameters, or self-citations. No load-bearing uniqueness theorem, ansatz smuggled via citation, or renaming of known results appears. The simulation results on failure rates are independent of the analytic scaling derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on geometric scaling assumptions for disk footprints and on two specific functional forms for how speed depends on size; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Agent footprints are disks whose area scales exactly as 1/n when a fixed resource is shared equally.
Explicitly stated as the setup for the coverage case study.
domain assumption Agent speed decreases either proportionally to radius or proportionally to footprint area.
Used to derive the equal-performance and single-agent-best cases.

pith-pipeline@v0.9.0 · 5475 in / 1260 out tokens · 73766 ms · 2026-05-07T13:17:34.933236+00:00 · methodology

discussion (0)

Reference graph

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