An Infinite Horizon Differential Game of Optimal CLV-Based Strategies with Non-atomic Firms

  • Gerasimos Lianos
  • Igor SloevEmail author
Part of the Static & Dynamic Game Theory: Foundations & Applications book series (SDGTFA)


We study the structure of optimal customer acquisition and customer retention strategies as a differential game over an infinite horizon in an industry with a large number of non-atomic firms. The optimal retention effort is constant over time and the optimal acquisition effort is proportional to the size of potential customer base. Greater customer profitability leads to higher per- capita acquisition and retention efforts, larger size of firms, and lower churn rate. A greater discount rate leads to lower per-capita acquisition and retention efforts, smaller firm size, and a greater churn rate. Tougher competition lowers the firms’ acquisition and retention expenditures and it does not affect per-capita values. Both the churn rate and the share of acquisition expenditures in the total marketing budget decrease as firms grow over time. We revisit the concepts of the customer lifetime value (CLV) and the value of the firm in the dynamic equilibrium of an industry with a large number of players and demonstrate the equivalence between maximization of the value of the firm and maximization of a firm’s individual CLV.


Differential games Non-atomic games Dynamic competition Customer acquisition Customer retention 

Subject Classification:

90B60 90B50 91A80 90A13 



We are grateful to the organizers and participants of 2015 European Meeting on Game Theory (SING11-GTM 2015) in St. Petersburg for their comments. We also thank organizers participants and referees of 2014 AMS conference at Indianapolis, USA, as well as seminar participants at the HSE, SWUFE, and EUSP for comments and suggestions on early drafts. Especially we thank Chakravarthi Narasimhan and Philip Dybvig for their encouragement and comments. Finally, we thank editors and anonymous referee for their careful consideration of our paper and valuable comments and suggestions that have helped us improve it.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Economics, School of Business AdministrationUniversity of MiamiCoral GablesUSA
  2. 2.National Research University Higher School of EconomicsMoscowRussia

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