×

Generalized Nash equilibrium and dynamics of popularity of online contents. (English) Zbl 1470.91056

Summary: 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.

MSC:

91A43 Games involving graphs
91A11 Equilibrium refinements
91B44 Economics of information
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
PDFBibTeX XMLCite
Full Text: DOI

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.; Lasaulce, S., Dynamic games for analyzing competition in the Internet and in on-line social networks, Network Games, Control, and Optimization, 11-22 (2007), New York: Springer, New York · Zbl 1410.91098
[3] Arrow, KJ; Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290 (1954) · Zbl 0055.38007 · doi:10.2307/1907353
[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) · Zbl 1365.49010 · doi:10.1007/s00186-016-0565-x
[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) · doi:10.1109/TNET.2008.2011358
[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, MG; Bauch, CT; Johnston, MD, Dynamics of vaccination strategies via projected dynamical systems, Bull. Math. Biol., 69, 1453-1476 (2007) · Zbl 1298.92097 · doi:10.1007/s11538-006-9173-x
[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) · Zbl 1410.90008 · doi:10.1111/itor.12502
[9] Daniele, P.; Scrimali, L.; Daniele, P.; Scrimali, L., Strong Nash equilibria for cybersecurity investments with nonlinear budget constraints, New Trends in Emerging Complex Real Life Problems, 199-207 (2018), New York: Springer, New York · doi:10.1007/978-3-030-00473-6_22
[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) · Zbl 0785.93044 · doi:10.1007/BF02073589
[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) · Zbl 1259.91009 · doi:10.1007/s11590-010-0218-6
[13] Facchinei, F.; Piccialli, V.; Sciandrone, M., Decomposition algorithms for generalized potential games, Comput. Optim. Appl., 50, 2, 237-262 (2011) · Zbl 1237.91017 · doi:10.1007/s10589-010-9331-9
[14] Facchinei, F.; Kanzow, C., Generalized Nash equilibrium problems, Ann. Oper. Res., 175, 1, 177-211 (2010) · Zbl 1185.91016 · doi:10.1007/s10479-009-0653-x
[15] Facchinei, F.; Fischer, A.; Piccialli, V., On generalized Nash games and variational inequalities, Oper. Res. Lett., 35, 159-164 (2007) · Zbl 1303.91020 · doi:10.1016/j.orl.2006.03.004
[16] Facchinei, F.; Pang, JS, Finite-Dimensional Variational Inequalities and Complementarity Problems (2003), New York: Springer, New York · Zbl 1062.90002
[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) · Zbl 1457.91019
[18] Harker, PT, Generalized Nash games and quasi-variational inequalities, Eur. J. Oper. Res., 54, 81-94 (1991) · Zbl 0754.90070 · doi:10.1016/0377-2217(91)90325-P
[19] Jahn, J., Introduction to the Theory of Nonlinear Optimization (1996), Berlin: Springer, Berlin · Zbl 0855.49001 · doi:10.1007/978-3-662-03271-8
[20] Korpelevich, GM, The extragradient method for finding saddle points and other problems, Matekon, 13, 35-49 (1977)
[21] Kulkarni, AA; Shanbhag, UV, On the variational equilibrium as a refinement of the generalized Nash equilibrium, Automatica, 48, 45-55 (2012) · Zbl 1245.91006 · doi:10.1016/j.automatica.2011.09.042
[22] Lampariello, L.; Sagratella, S., A bridge between bilevel programs and Nash games, J. Optim. Theory Appl., 174, 2, 613-635 (2017) · Zbl 1373.90152 · doi:10.1007/s10957-017-1109-0
[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) · Zbl 1192.47052
[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) · Zbl 1220.90136 · doi:10.1007/s10589-009-9256-3
[26] Nagurney, A.; Li, D., Competing on Supply Chain Quality: A Network Economics Perspective (2016), Bern: Springer, Bern · Zbl 1376.90002 · doi:10.1007/978-3-319-25451-7
[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) · doi:10.1007/s11066-013-9076-6
[28] Nagurney, A.; Cruz, J.; Dong, J., Global Supply Chain Networks and Risk Management: A Multi-Agent Framework (2006), Berlin: Springer, Berlin
[29] Nagurney, A.; Zhang, D., Projected Dynamical Systems and Variational Inequalities with Applications (1996), Dordrecht: Kluwer, Dordrecht · doi:10.1007/978-1-4615-2301-7
[30] Nagurney, A., Network Economics: A Variational Inequality Approach (1993), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0873.90015 · doi:10.1007/978-94-011-2178-1
[31] Nash, JF, Non-cooperative games, Ann. Math., 54, 286-295 (1951) · Zbl 0045.08202 · doi:10.2307/1969529
[32] Nash, JF, Equilibrium points in n-person games, Proc. Natl. Acad. Sci., 36, 48-49 (1950) · Zbl 0036.01104 · doi:10.1073/pnas.36.1.48
[33] Nesterov, Y.; Scrimali, L., Solving strongly monotone variational and quasivariational inequalities, Discret. Contin. Dyn. Syst., 31, 1383-1396 (2011) · Zbl 1238.49019 · doi:10.3934/dcds.2011.31.1383
[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) · Zbl 1332.91082 · doi:10.1007/s11067-011-9166-7
[35] Pang, JS; Fukushima, M., Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Manag. Sci., 2, 21-56 (2005) · Zbl 1115.90059 · doi:10.1007/s10287-004-0010-0
[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) · Zbl 1089.49019
[37] Rockafellar, RT; Wets, RJ-B, Variational Analysis (2009), Berlin: Springer, Berlin · Zbl 0888.49001
[38] Rosen, JB, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica, 33, 520-534 (1965) · Zbl 0142.17603 · doi:10.2307/1911749
[39] Sagratella, S., Algorithms for generalized potential games with mixed-integer variables, Comput. Optim. Appl., 68, 3, 689-717 (2017) · Zbl 1390.91023 · doi:10.1007/s10589-017-9927-4
[40] Google: What is the YouTube Partner Program? http://support.google.com/youtube/answer/72851?hl=en
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.