Source-linked AI summary
Anonymous Hedonic Game for Task Allocation in a Large-Scale Multiple Agent System
Inmo Jang, Hyo-Sang Shin, Antonios Tsourdos
TL;DR
Large swarms require task allocation that cannot depend on direct supervision and must accommodate local, asynchronous coordination. The paper proposes a decentralized game-theoretic framework for self-interested agents, and reports convergence to stable assignments under stated conditions. Its conclusion also identifies solution-quality analysis as an area needing further research.
Problem
Large swarms make direct human supervision infeasible, while decentralized decision-making must accommodate local information and asynchronous operation.
Source: §I · p. 1Method
The paper models task allocation as a hedonic game and uses a decentralized algorithm executable over a strongly connected network, including asynchronously.
Source: §I · p. 1, E. Decentralized Algorithm · p. 6Results
Agents with social inhibition can converge to a Nash stable partition, and the framework is reported to be scalable, adaptable, and robust.
Source: §VII · p. 13Takeaways & Limitations
The framework’s reported scope includes local interactions in asynchronous settings and agents with differing interests when their preferences satisfy SPAO.
Source: §VII · p. 13Further research is needed to analyze the quality of Nash stable partitions in terms of min-max.
Source: §VII · p. 14
Abstract
from arXiv · showhide
This paper proposes a novel game-theoretical autonomous decision-making framework to address a task allocation problem for a swarm of multiple agents. We consider cooperation of self-interested agents, and show that our proposed decentralized algorithm guarantees convergence of agents with social inhibition to a Nash stable partition (i.e., social agreement) within polynomial time. The algorithm is simple and executable based on local interactions with neighbor agents under a strongly-connected communication network and even in asynchronous environments. We analytically present a mathematical formulation for computing the lower bound of suboptimality of the solution, and additionally show that 50% of suboptimality can be at least guaranteed if social utilities are non-decreasing functions with respect to the number of co-working agents. The results of numerical experiments confirm that the proposed framework is scalable, fast adaptable against dynamical environments, and robust even in a realistic situation.
I. INTRODUCTION
The paper addresses decentralized task allocation for large swarms, where agents must be divided among tasks without direct human supervision. It adapts hedonic games to this setting and analyzes stability, convergence, and solution quality.
I. INTRODUCTION
- Large swarms make direct supervision infeasible, motivating autonomous partitioning of agents among tasks when each agent can perform at most one task.The framework is expected to meet decentralization, scalability, predictability, adaptability, and asynchronous robustness needs.Source: §I · p. 1
A. Decentralized Coordination of Robotic Swarms
- Self-organized coordination can reduce communication and computation, but top-down models offer predictable aggregate behavior while bottom-up rules can accommodate individual preferences.The paper focuses on self-organized methods because large systems are expected to rely on local information.Source: A. Decentralized Coordination of… · p. 2
B. Hedonic Games
- Hedonic games model self-interested agents forming coalitions, and prior anonymous-game work showed convergence for socially inhibitive agents using a centralized algorithm [44].Anonymous agents evaluate coalition size rather than member identities.Source: B. Hedonic Games · p. 2
C. Main Contributions
- The paper proposes a decentralized hedonic-game framework with O(n^2_a) iterations, a suboptimality analysis, and experiments on scalability, adaptability, and robustness.It also reports a 50% suboptimality guarantee when coalition social utility is non-decreasing in coalition size, and supports differing agent interests when preferences satisfy SPAO.Source: C. Main Contributions · pp. 2–3
III. GROUP AGENT PARTITIONING AND PLACING EVENT
The paper formulates task allocation as assigning agents to tasks to maximize summed individual utility, then recasts the problem as GRAPE, an anonymous hedonic game. Its target outcome is a Nash stable partition requiring no agent to prefer a unilateral move.
A. Problem Formulation
- Each agent is assigned to at most one task, and task utility depends on the task and the number of co-working agents; the objective is to maximize total individual utility.A void task represents doing no task, with zero individual utility.Source: A. Problem Formulation · p. 3
- The formulation assumes homogeneous, limited-capability agents, multi-robot tasks, and a strongly connected communication network.Agents know their own utilities but not those of other agents; task assignments may be learned through communication.Source: A. Problem Formulation · p. 3
B. Proposed Game-theoretical Approach: GRAPE
- GRAPE represents task allocation as an anonymous game in which agents rank task-coalition pairs by preference.A pair specifies a task and participant count; preferences can be derived from individual utility.Source: B. Proposed Game-theoretical Approach:… · pp. 3–4
- A Nash stable partition is one where no agent prefers joining another coalition, providing agreement without requiring negotiation.Agents need only notify others of a changed decision, which can reduce inter-agent communication.Source: B. Proposed Game-theoretical Approach:… · p. 4
C. SPAO Preference: Social Inhibition
The paper’s SPAO condition formalizes social inhibition: agents weakly prefer fewer co-workers for any given task. Under this condition, GRAPE guarantees a Nash stable partition and a polynomial iteration bound, including from a random initial partition.
C. SPAO Preference: Social Inhibition
- SPAO preferences weakly favor smaller coalitions for each task, capturing agents’ social inhibition.The paper notes that diminishing individual utility with more collaborators can arise from interference or shared-resource use, even when social utility increases.Source: C. SPAO Preference: Social Inhibition · p. 4
- SPAO permits agents with different interests to follow their individual preferences while retaining the framework’s convergence guarantees.The paper identifies combinations of swarms from different organizations as a possible use when their preferences satisfy SPAO.Source: C. SPAO Preference: Social Inhibition · p. 4
D. Existence of and Convergence to a Nash Stable Partition
- Under SPAO, a Nash stable partition always exists, and GRAPE reaches one within |A| · (|A| + 1)/2 iterations.The incremental argument bounds re-convergence after adding one agent by the new instance’s agent count.Source: D. Existence of and Convergence to a… · p. 5
- The convergence guarantee also holds when agents start from a random partition.The proof uses the property that an agent that displaces an existing coalition member will not deviate again after another agent joins.Source: D. Existence of and Convergence to a… · pp. 5–6
E. Decentralized Algorithm
The decentralized algorithm lets agents make decisions from local information and update simultaneously over a strongly connected network. A distributed mutex selects a valid partition so the convergence results continue to hold.
E. Decentralized Algorithm
- Each agent locally evaluates whether to join its most-preferred task coalition and shares its partition with neighbors.An agent joins only when it strongly prefers the new coalition to its current one.Source: E. Decentralized Algorithm · p. 6
- The distributed mutex selects the most valid locally known partition using iteration counts and random timestamps.Agents reconsider their choices when they learn of a more valid partition.Source: E. Decentralized Algorithm · p. 6
- Under a strongly connected network, only one partition eventually survives, and the agents can find a Nash stable partition in polynomial time.The mutex also permits distributed selection when multiple Nash stable partitions are reached.Source: E. Decentralized Algorithm · pp. 6–7
IV. ANALYSIS
The analysis bounds the algorithm’s convergence and per-agent running time in terms of network diameter, agent count, and task count. It also describes a trade-off between communication overhead and running time.
A. Algorithmic Complexity (Scalability)
- O(d_G n_a^2) time steps suffice to find a Nash stable partition, and an agent’s running time is bounded by O(d_G n_t n_a^2).For a fully connected network, the time-step bound is O(n_a^2); practical running time may be lower than the conservative bound.Source: A. Algorithmic Complexity (Scalability) · p. 7
- The analysis’s iteration bound is conservative because agents need not find every Nash stable partition for every subset of agents.The paper notes that unnecessary iterations may therefore be reduced in practice.Source: A. Algorithmic Complexity (Scalability) · p. 7
- O(|N_i| · n_a) is the communication overhead per agent per iteration, and it could be quadratic when neighbor count grows with agent count.The paper says that growth is rare in practice because of spatial distribution and physical communication limits.Source: A. Algorithmic Complexity (Scalability) · p. 7
- Limiting each agent to one transaction per iteration reduces communication overhead to O(n_a), but may raise convergence time to O(n_a^3).The convergence guarantees remain valid when the union of communication graphs over time intervals becomes connected.Source: A. Algorithmic Complexity (Scalability) · p. 7
B. Suboptimality
The paper defines suboptimality as the global utility of GRAPE’s Nash stable partition relative to the optimum and derives a lower bound. Under stated conditions, that bound is at least 50%.
B. Suboptimality
- Suboptimality α is the ratio of GRAPE’s global utility to the optimal global-utility value.The paper denotes these values J_GRAPE and J_OPT, respectively.Source: B. Suboptimality · p. 7
- The general-case lower bound can be determined when a Nash stable partition and the agents’ individual utility functions are given.The theorem expresses the bound in terms of global utility.Source: B. Suboptimality · pp. 7–8
- GRAPE guarantees at least 50% of the optimal global utility when coalition utility is non-decreasing in participant count and individual utilities derive SPAO preferences.The guarantee concerns a Nash stable partition obtained by GRAPE.Source: B. Suboptimality · pp. 7–8
C. Adaptability
The framework can adapt to unexpected additions or losses of agents or tasks by establishing a new agreed task assignment within polynomial time.
C. Adaptability
- After a new agent joins an ongoing mission with an assignment already determined, convergence to a new Nash stable partition takes at most the total number of agents’ iterations.The paper attributes adaptability to fast convergence and states that environmental changes can be handled within polynomial time.Source: C. Adaptability · p. 9
D. Robustness in Asynchronous Environments
Asynchronous operation may add dummy iterations, but does not affect the existence, convergence, or suboptimality of a Nash stable partition.
D. Robustness in Asynchronous Environments
- Temporary disconnection or agent non-operation may add dummy iterations without affecting a Nash stable partition’s existence, convergence, or suboptimality.Each agent needs only local information from one neighbor to decide and send an updated partition to some neighbors.Source: D. Robustness in Asynchronous… · p. 9
V. GRAPE WITH MINIMUM REQUIREMENTS
For tasks with minimum staffing requirements, the framework modifies utilities for understaffed tasks so that its preference-based procedure can still find an assignment meeting those requirements. Under the stated conditions, it obtains a Nash stable partition in polynomial time.
V. GRAPE WITH MINIMUM REQUIREMENTS
- For tasks requiring R_j agents, the original utility cannot directly yield SPAO preferences, so the method defines auxiliary utilities for p ≤ R_j.The dummy utility is also used at p = R_j to support assignments meeting Equation (24).Source: §V · p. 9
- When |A| ≥ Σ_j R_j, a Nash stable partition satisfies the minimum staffing requirements if Condition 1 holds.The condition makes agents prefer a task below its minimum requirement over tasks whose requirements are already met.Source: §V · p. 9
- Under the proposition’s monotonicity and utility-setting conditions, GRAPE finds a Nash stable partition in polynomial time that satisfies the staffing requirement.The result uses auxiliary utilities satisfying Condition 1 and SPAO.Source: §V · p. 9
- After finding the partition, the objective value is computed with original utilities rather than the auxiliary utilities used to obtain it.The auxiliary utilities are used to construct the preference relations and find the partition.Source: §V · pp. 9–10
- The paper gives a suboptimality bound for a partition obtained using the auxiliary utilities, with δ defined as the difference between auxiliary- and original-utility objective values.A 100-agent example with three tasks illustrates setting dummy utilities when one task requires at least five agents and has much lower individual utility than the other tasks.Source: §V · pp. 9–10
˜JGRAP E
The section relates the auxiliary-utility objective to the original objective and describes the task-reward functions used in simulations. The experiments use peaked-reward and submodular-reward tasks.
˜JOP T
- Auxiliary utilities are at least as large as original utilities, so the corresponding GRAPE and optimal objective values are also at least as large.The section rewrites the suboptimality bound using δ, the difference between auxiliary- and original-utility GRAPE objective values.Source: ˜JOP T · p. 10
VI. SIMULATION AND RESULTS
- The numerical experiments evaluate scalability, suboptimality, adaptability in dynamic environments, and robustness in asynchronous environments.The supplied passage states the evaluation scope, not the experimental outcomes.Source: §VI · p. 10
1) Utility functions:
- The simulations use peaked-reward tasks, whose reward rises as the number of agents approaches a desired count, and submodular-reward tasks, whose reward rises with diminishing marginal gain.Individual utility is the equally shared reward minus the agent’s task cost; costs are based on distance to the task.Source: 1) Utility functions · p. 10
2) Parameters generation:
The simulations randomize agent and task locations and generate task parameters for the two reward types.
2) Parameters generation:
- Each run places tasks and agents uniformly at random in a 1000 m × 1000 m arena and a 250 m × 250 m arena within it.The task and agent counts are denoted n_t and n_a.Source: 2) Parameters generation · p. 11
- For submodular-reward tasks, ε_j is set to 2 and rmin_j is generated uniformly using a formula dependent on n_a/n_t.The supplied formula also depends on log base ε_j and the quantity n_a/n_t + 1.Source: 2) Parameters generation · p. 11
3) Communication network:
The experiments examine convergence under strongly-connected and fully-connected networks, solution suboptimality, and recovery after agents or tasks change. Results show near-optimal solutions in almost all tested cases, while convergence effort and recovery vary with network structure and mission changes.
3) Communication network:
- The strongly-connected network is modeled as a bidirectional minimum spanning tree based on agent positions; it is sustained during missions except in robustness tests.Experiments also compare fully-connected networks.Source: 3) Communication network · p. 11
B. Scalability
- Required iterations grow quadratically with agent count; the coefficient is much smaller than the theorem’s 1/2 bound and can reach 5 × 10^-4 under a fully-connected network.The number of required iterations decreases as task count increases; strongly-connected networks have steeper trends than fully-connected networks because they are more sensitive to agent conflicts.Source: B. Scalability · p. 11
- Strongly-connected networks require about 3–4 dummy iterations per normal iteration, whereas fully-connected networks need none.In the tested strongly-connected cases, dummy iterations are much fewer than the graph-diameter bound.Source: B. Scalability · p. 11, Figure 2 · p. 11
C. Suboptimality
- The framework provides near-optimal solutions in almost all cases, and each Nash stable partition’s suboptimality is enclosed by the corresponding lower bound.Across the large-size problems, the lower bound averages above 60–70%; the communication network does not affect that bound, while utility type can influence it.Source: C. Suboptimality · p. 12
D. Adaptability
- After mission changes, additional iterations remain at the same order as the number of agents, and increase as the agent-to-task ratio rises.Adding agents requires more iterations, while losing agents shows no apparent relation to iteration count; eliminating tasks requires more iterations than adding tasks.Source: D. Adaptability · pp. 12–13
E. Robustness in Asynchronous Environments
With randomly non-operating agents, the framework still finds Nash stable partitions, although required time steps increase. The conclusion reports convergence guarantees while identifying comparison and min-max partition quality as open questions.
E. Robustness in Asynchronous Environments
- As non-operating agents increase, normal iteration counts and Theorem 3’s suboptimality lower bound remain unaffected, while required time steps increase.Communication discontinuity can increase waiting and information dissemination time, adding dummy iterations even though the framework still finds a Nash stable partition.Source: E. Robustness in Asynchronous… · p. 13, Figure 6 · p. 13
F. Visualization
- Across the circle, skewed-circle, and square scenarios, allocations appear reasonable given task rewards and agent-to-task distance costs.Figure 7 encodes task positions and rewards with squares, agents and communication links with circles and lines, and assignments through agent colors.Source: F. Visualization · p. 13, Figure 7 · p. 14
- 1,355, 1,380, and 1,295 iterations find Nash stable partitions in Scenarios #1, #2, and #3, respectively.Dummy iterations are 20–30% of normal iterations, fewer than in Figure 2 because these networks are more connected.Source: F. Visualization · p. 13
VII. CONCLUSION
- Self-interested agents with SPAO preferences can reach a Nash stable partition using the decentralized algorithm under a strongly connected network, including asynchronously.The conclusion also reports scalability, guaranteed suboptimality, adaptability, robustness, and potential accommodation of different agent interests.Source: §VII · p. 13
- Comparisons with existing methods and analysis of Nash-partition quality in terms of min-max remain future work.The authors note that fair comparison scenarios require care and that experiments showed balanced individual utilities.Source: §VII · pp. 13–14