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On weak Pareto optimality of nonatomic routing networks

Abstract

This paper establishes several sufficient conditions for the weak Pareto optimality of Nash equilibria in nonatomic routing games on single- and multi-commodity networks, where a Nash equilibrium (NE) is weakly Pareto optimal (WPO) if under it no deviation of all players could make everybody better off. The results provide theoretical and technical bases for characterizing graphical structures for nonatomic routing games to admit WPO NEs. We prove that the NE of every nonatomic routing game on a single or multi-commodity network is WPO (regardless of the choices of nonnegative, continuous, nondecreasing latency functions on network links) if the network does not have two links xy and three paths each of which goes from some origin to some destination such that the intersections of the three paths with \(\{x,y\}\) are \(\{x\},\{y\}\) and \(\{x,y\}\), respectively. This sufficient condition leads to an alternative proof of the recent result that the NE of every 2-commodity nonatomic routing game on any undirected cycle is WPO. We verify a general property satisfied by all feasible 2-commodity routings (not necessarily controlled by selfish players) on undirected cycles, which roughly says that no feasible routing can “dominate” another in some sense. The property is crucial for proving the weak Pareto optimality associated to the building blocks of undirected graphs on which all NEs w.r.t. two commodities are WPO.

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Acknowledgements

The authors are indebted to two anonymous referees for their invaluable comments and suggestions. This research is supported in part by National Natural Science Foundation of China under Grants 11531014 and 11901605, Ministry of Science and Technology of China under Grant 2018AAA0101002, and Key Research Program of Frontier Sciences, CAS, under Grant ZDBS-LY-7008.

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Correspondence to Zhuo Diao.

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Chen, X., Diao, Z. & Hu, X. On weak Pareto optimality of nonatomic routing networks. J Comb Optim (2020). https://doi.org/10.1007/s10878-020-00539-7

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Keywords

  • Nonatomic selfish routing
  • Nash equilibrium
  • Weakly Pareto optimal
  • Multi-commodity network