# A Multi-Agent Min-Cost Flow problem with Controllable Capacities - Complexity of Finding a Maximum-flow Nash Equilibrium

### Nadia Chaabane Fakhfakh, Cyril Briand, Marie-José Huguet

#### Abstract

A Multi-Agent Minimum-Cost Flow problem is addressed in this paper. It can be seen as a basic multi-agent transportation problem where every agent can control the capacities of a set of elementary routes (modeled as arcs inside a network), each agent incurring a cost proportional to the chosen capacity. We assume that a customer is interesting in transshipping a product flow from a source to a sink node through the transportation network. It offers a reward that is proportional to the flow that the agents manage to provide. The reward is shared among the agents according to a pre-established policy. This problem can be seen as a non-cooperative game where every agent aims at maximizing its individual profit. We take interest in finding stable strategies (i.e., Nash Equilibrium) such that no agent has any incentive to modify its behavior. We show how such equilibrium can be characterized by means of augmenting or decreasing path in a reduced network. We also focus on the problem of finding a Nash equilibrium that maximizes the flow value and prove its NP-hardness.

#### References

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#### Paper Citation

#### in Harvard Style

Chaabane Fakhfakh N., Briand C. and Huguet M. (2014). **A Multi-Agent Min-Cost Flow problem with Controllable Capacities - Complexity of Finding a Maximum-flow Nash Equilibrium** . In *Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-017-8, pages 27-34. DOI: 10.5220/0004765500270034

#### in Bibtex Style

@conference{icores14,

author={Nadia Chaabane Fakhfakh and Cyril Briand and Marie-José Huguet},

title={A Multi-Agent Min-Cost Flow problem with Controllable Capacities - Complexity of Finding a Maximum-flow Nash Equilibrium},

booktitle={Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2014},

pages={27-34},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004765500270034},

isbn={978-989-758-017-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - A Multi-Agent Min-Cost Flow problem with Controllable Capacities - Complexity of Finding a Maximum-flow Nash Equilibrium

SN - 978-989-758-017-8

AU - Chaabane Fakhfakh N.

AU - Briand C.

AU - Huguet M.

PY - 2014

SP - 27

EP - 34

DO - 10.5220/0004765500270034