Skip to main content
SpringerLink
Log in

Generalized Nash equilibrium and dynamics of popularity of online contents

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper develops a dynamic model of competition for the diffusion of online contents in a two-layer network consisting of content providers and viewers. Each content provider seeks to maximize the profit by determining the optimal views and quality levels. We assume that there is a known and fixed limit to the number of times each viewer can access a content. This requirement generates shared constraints for all the providers. The problem is expressed as a Generalized Nash equilibrium with shared constraints that is then formulated via a variational inequality. We construct the locally projected dynamical system model, which provides a continuous-time evolution of views and quality levels, and whose set of stationary points coincides with the set of solutions to the variational inequality. We discuss some stability conditions using a monotonicity approach, and, finally, we present some numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Altman, E., De Pellegrini, F., Al-Azouzi, R., Miorandi, D., Jimenez, T.: Emergence of equilibria from individual strategies in online content diffusion. In: IEEE INFORCOM NetSciCOmm, pp. 181–186. IEEE (2013)

  2. Altman, E., Jain, A., Shimkin, N., Touati, C.: Dynamic games for analyzing competition in the Internet and in on-line social networks. In: Lasaulce, S., et al. (eds.) Network Games, Control, and Optimization, pp. 11–22. Springer, New York (2007)

    MATH  Google Scholar 

  3. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)

    Article  MathSciNet  Google Scholar 

  4. Aussel, D., Sagratella, S.: Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality. Math Methods Oper Res 85(1), 3–18 (2017)

    Article  MathSciNet  Google Scholar 

  5. Cha, M., Kwak, H., Rodriguez, P., Ahn, Y.-Y., Moon, S.: Analyzing the video popularity characteristics of large-scale user generated content systems. IEEE/ACM Trans Netw 17(5), 1357–1370 (2009)

    Article  Google Scholar 

  6. Cha, M., Kwak, H., Rodriguez, P., Ahn, Y.-Y., Moon, S.: I tube, you tube, everybody tubes: analyzing the world’s largest user generated content video system. In: Proceedings of ACM IMC, San Diego, California, USA, pp. 1–14 (2007)

  7. Cojocaru, M.G., Bauch, C.T., Johnston, M.D.: Dynamics of vaccination strategies via projected dynamical systems. Bull. Math. Biol. 69, 1453–1476 (2007)

    Article  MathSciNet  Google Scholar 

  8. Colajanni, G., Daniele, P., Giuffré, S., Nagurney, A.: Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers. Int. Trans. Oper. Res. 25, 1443–1464 (2018)

    Article  MathSciNet  Google Scholar 

  9. Daniele, P., Scrimali, L.: Strong Nash equilibria for cybersecurity investments with nonlinear budget constraints. In: Daniele, P., Scrimali, L. (eds.) New Trends in Emerging Complex Real Life Problems. AIRO Springer Series, vol. 1, pp. 199–207. Springer, New York (2018)

    Chapter  Google Scholar 

  10. De Pellegrini, F., Reigers, A., Altman, E.: Differential games of competition in online content diffusion. In: 2014 IFIP Networking Conference, pp. 1–9 (2014)

  11. Dupuis, P., Nagurney, A.: Dynamical systems and variational inequalities. Ann. Oper. Res. 44, 9–42 (1993)

    Article  MathSciNet  Google Scholar 

  12. Facchinei, F., Sagratella, S.: On the computation of all solutions of jointly convex generalized Nash equilibrium problems. Optim. Lett. 5(3), 531–547 (2011)

    Article  MathSciNet  Google Scholar 

  13. Facchinei, F., Piccialli, V., Sciandrone, M.: Decomposition algorithms for generalized potential games. Comput. Optim. Appl. 50(2), 237–262 (2011)

    Article  MathSciNet  Google Scholar 

  14. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175(1), 177–211 (2010)

    Article  MathSciNet  Google Scholar 

  15. Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007)

    Article  MathSciNet  Google Scholar 

  16. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)

    MATH  Google Scholar 

  17. Fargetta, G., Scrimali, L.: A game theory model of online content competition. In: Paolucci, M., Sciomachen, A., Uberti, P. (eds.) Advances in Optimization and Decision Science for Society, Services and Enterprises, AIRO Springer Series, vol. 3, pp. 173–184. Springer, New York (2020)

  18. Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)

    Article  Google Scholar 

  19. Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (1996)

    Book  Google Scholar 

  20. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matekon 13, 35–49 (1977)

    Google Scholar 

  21. Kulkarni, A.A., Shanbhag, U.V.: On the variational equilibrium as a refinement of the generalized Nash equilibrium. Automatica 48, 45–55 (2012)

    Article  MathSciNet  Google Scholar 

  22. Lampariello, L., Sagratella, S.: A bridge between bilevel programs and Nash games. J. Optim. Theory Appl. 174(2), 613–635 (2017)

    Article  MathSciNet  Google Scholar 

  23. Luna, J.P.: Decomposition and approximation methods for variational inequalities, with applications to deterministic and stochastic energy markets. PhD Thesis, Instituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro, Brazil (2013)

  24. Maugeri, A., Raciti, F.: On existence theorems for monotone and nonmonotone variational inequalities. J. Convex Anal. 16(3–4), 899–911 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Nabetani, K., Tseng, P., Fukushima, M.: Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints. Comput. Optim. Appl. 48(3), 423–452 (2011)

    Article  MathSciNet  Google Scholar 

  26. Nagurney, A., Li, D.: Competing on Supply Chain Quality: A Network Economics Perspective. Springer, Bern (2016)

    Book  Google Scholar 

  27. Nagurney, A., Li, D., Wolf, T., Saberi, S.: A network economic model of a service-oriented Internet with choices and quality competition. Netnomics 14, 1–15 (2014)

    Article  Google Scholar 

  28. Nagurney, A., Cruz, J., Dong, J.: Global Supply Chain Networks and Risk Management: A Multi-Agent Framework. Springer, Berlin (2006)

    Google Scholar 

  29. Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer, Dordrecht (1996)

    Book  Google Scholar 

  30. Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers, Dordrecht (1993)

    Book  Google Scholar 

  31. Nash, J.F.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  32. Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36, 48–49 (1950)

    Article  MathSciNet  Google Scholar 

  33. Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasivariational inequalities. Discret. Contin. Dyn. Syst. 31, 1383–1396 (2011)

    Article  Google Scholar 

  34. Oggioni, G., Smeers, Y., Allevi, E., Schaible, S.: A generalized Nash equilibrium model of market coupling in the European power system. Netw. Spat. Econ. 12(4), 503–560 (2012)

    Article  MathSciNet  Google Scholar 

  35. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)

    Article  MathSciNet  Google Scholar 

  36. Passacantando, M.: Stability of equilibrium points of projected dynamical systems. In: Qi, L., Teo, K., Yang, X. (eds) Optimization and Control with Applications. Applied Optimization, Vol 96. Springer, Boston (2005)

  37. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (2009)

    MATH  Google Scholar 

  38. Rosen, J.B.: Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33, 520–534 (1965)

    Article  MathSciNet  Google Scholar 

  39. Sagratella, S.: Algorithms for generalized potential games with mixed-integer variables. Comput. Optim. Appl. 68(3), 689–717 (2017)

    Article  MathSciNet  Google Scholar 

  40. Google: What is the YouTube Partner Program? http://support.google.com/youtube/answer/72851?hl=en

Download references

Acknowledgements

The research of the second author was partially supported by the research project PON SCN 00451 CLARA - CLoud plAtform and smart underground imaging for natural Risk Assessment, Smart Cities and Communities and Social Innovation.

Author information

Authors and Affiliations

  1. DMI, University of Catania, Viale Andrea Doria, 6, 95125, Catania, Italy

    Georgia Fargetta & Laura R. M. Scrimali

Authors
  1. Georgia Fargetta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laura R. M. Scrimali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laura R. M. Scrimali.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fargetta, G., Scrimali, L.R.M. Generalized Nash equilibrium and dynamics of popularity of online contents. Optim Lett 15, 1691–1709 (2021). https://doi.org/10.1007/s11590-019-01528-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-019-01528-4

Keywords

Search

Navigation