Mathematical Methods of Operations Research

, Volume 76, Issue 1, pp 43–65 | Cite as

Nash equilibrium based fairness

  • Hisao KamedaEmail author
  • Eitan Altman
  • Corinne Touati
  • Arnaud Legrand
Original Article


There are several approaches of sharing resources among users. There is a noncooperative approach wherein each user strives to maximize its own utility. The most common optimality notion is then the Nash equilibrium. Nash equilibria are generally Pareto inefficient. On the other hand, we consider a Nash equilibrium to be fair as it is defined in a context of fair competition without coalitions (such as cartels and syndicates). We show a general framework of systems wherein there exists a Pareto optimal allocation that is Pareto superior to an inefficient Nash equilibrium. We consider this Pareto optimum to be ‘Nash equilibrium based fair.’ We further define a ‘Nash proportionately fair’ Pareto optimum. We then provide conditions for the existence of a Pareto-optimal allocation that is, truly or most closely, proportional to a Nash equilibrium. As examples that fit in the above framework, we consider noncooperative flow-control problems in communication networks, for which we show the conditions on the existence of Nash-proportionately fair Pareto optimal allocations.


Nash equilibrium Nash equilibrium based fairness Nash proportionate fairness Flow control Noncooperative game Pareto optimum and inefficiency Power criterion 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Hisao Kameda
    • 1
    Email author
  • Eitan Altman
    • 2
  • Corinne Touati
    • 3
  • Arnaud Legrand
    • 3
  1. 1.Department of Computer ScienceUniversity of TsukubaTsukuba Science City, IbarakiJapan
  2. 2.INRIA Sophia AntipolisSophia Antipolis, CedexFrance
  3. 3.CNRS and INRIA, LIG LaboratoryMontbonnotFrance

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