A duopoly model with heterogeneous congestion-sensitive customers

Michel Mandjes, Judith Timmer

Research output: Contribution to journalArticleAcademicpeer-review

5 Citations (Scopus)


This paper analyzes a model with two firms (providers), and two classes of customers. These customers classes are characterized by their attitude towards ‘congestion’ (caused by other customers using the same resources); a firm is selected on the basis of both the prices charged by the firms, and the ‘congestion levels’. The model can be represented by a two-stage game: in the first providers set their prices, whereas in the second the customers choose the provider (or to not use any service at all) for given prices. We explicitly allow the providers to split their resources, in order to serve more than just one market segment. This enables us to further analyze the Paris metro pricing (Pmp) proposal for service differentiation in the Internet. We prove that the stage-2 game (the customers’ behavior for given prices, and a given division of the providers’ resources) has a unique equilibrium. Insight is gained into the structural properties of the equilibrium. We also show that the objective functions in the stage-1 game are continuous (in the providers’ decision variables), thus enabling an efficient search for its equilibrium. We comment on the viability of the Pmp proposal.
Original languageEnglish
Pages (from-to)445-467
Number of pages23
JournalEuropean journal of operational research
Issue number1
Publication statusPublished - Jan 2007


  • MSC-60K25
  • MSC-60K30
  • MSC-91A10
  • MSC-91A20
  • MSC-90B18
  • Pricing
  • Duopoly
  • Game theory
  • Congestion
  • Quality-of-service differentiation

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