Competing for customers in a social network

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July 2014, 1(3): 377-409. doi: 10.3934/jdg.2014.1.377

Competing for customers in a social network

Pradeep Dubey ^1, , Rahul Garg ^2, and Bernard De Meyer ^3,

1.	Center for Game Theory in Economics, Stony Brook University, Stony Brook, NY 11794-4384, United States
2.	Opera Solutions-India, Floor 6, Express Trade Towers 1, Plot No. 15-16, Sector 16A, Noida 201 301, New Delhi, India
3.	PSE-Univesité Paris 1, 112 Boulevard de l'Hôpital, 75013 Paris, France

Received June 2012 Revised October 2013 Published July 2014

Abstract
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Customers' proclivities to buy products often depend heavily on who else is buying the same product. This gives rise to non-cooperative games in which firms sell to customers located in a ``social network''. Nash Equilibrium (NE) in pure strategies exist in general. In the quasi-linear case, NE are unique. If there are no a priori biases between customers and firms, there is a cut-off level above which high cost firms are blockaded at an NE, while the rest compete uniformly throughout the network. Otherwise firms could end up as regional monopolies.
The connectivity of a customer is related to the money firms spend on him. This becomes particularly transparent when externalities are dominant: NE can be characterized in terms of the invariant measures on the recurrent classes of the Markov chain underlying the social network.
When cost functions of firms are convex, instead of just linear, NE need no longer be unique as we show via an example. But uniqueness is restored if there is enough competition between firms or if their valuations of clients are anonymous.
Finally we develop a general model of nonlinear externalities and show that existence of NE remains intact.

Keywords: externalities, noncooperative game, Nash equilibrium., Social network.

Mathematics Subject Classification: Primary: 91D30, 91A0.

Citation: Pradeep Dubey, Rahul Garg, Bernard De Meyer. Competing for customers in a social network. Journal of Dynamics and Games, 2014, 1 (3) : 377-409. doi: 10.3934/jdg.2014.1.377

References:

[1]	A. Banerji and B. Dutta, Local network externalities and market segmentation, International Journal of Industrial Organization, 27 (2009), 605-614. doi: 10.1016/j.ijindorg.2009.02.001.
[2]	F. Bloch and N. Quérou, Pricing in social network, Games and Economic Behavior, 80 (2013), 243-261. doi: 10.1016/j.geb.2013.03.006.
[3]	P. Domingos and M. Richardson, Mining the network value of customers, in Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, 2001, 57-66. doi: 10.1145/502512.502525.
[4]	J. L. Doob, Stochastic Processes, John Wiley & Sons, New York, 1953.
[5]	P. Dubey, R. Garg and B. De Meyer, Competing for customers in a social network: The quasi-linear Case, in Internet and Network Economics: Second International Workshop, Lecture Notes in Computer Science, 4286, Springer, Berlin-Heidelberg, 2006, 162-173. doi: 10.1007/11944874_16.
[6]	J. Hartline, V. Mirrokni and M. Sundarajan, Optimal marketing strategies over social networks, in Proceedings of WWW 2008, Beijing, China, 2008, 189-198. doi: 10.1145/1367497.1367524.
[7]	M. Jackson, The economics of social networks, in Proceedings of the 9th World Congress of the Econometric Society (eds. R. Blundell, W. Newey and T. Persson), Cambridge University Press, 2005.
[8]	B. Julien, Competing in Network Industries: Divide and Conquer, Mimeo, IDEI and GREMAQ, 2001.
[9]	D. Kempe, J. Kleinberg and E. Tardos, Maximizing the spread of influence through a social network, in Proceedings of the 9th International Conference on Knowledge Discovery and Data Mining, 2003, 137-146. doi: 10.1145/956755.956769.
[10]	C. N. Moore, Summability of series, The American Mathematical Monthly, 39 (1932), 62-71. doi: 10.2307/2302048.
[11]	J. Nash, Equilibrium points in $n$-person games, Proceedings of the National Academy of Science, 36 (1950), 48-49. doi: 10.1073/pnas.36.1.48.
[12]	M. Richardson and P. Domingos, Mining knowledge-sharing sites for viral marketing, in Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Edmonton, Alberta, Canada, 2002, 61-70. doi: 10.1145/775056.775057.
[13]	P. Saaskilahti, Monopoly Pricing of Social Goods, MPRA Paper 3526, University Library of Munich, 2007.
[14]	S. Sahi, A note on the resolvent of a nonnegative matrix and its applications, Linear Algebra and Its Applications, 432 (2010), 2524-2528. doi: 10.1016/j.laa.2009.11.004.
[15]	J. Scott, Social Network Analysis: A Handbook, 2nd edition, Sage Publications, London, 2000.
[16]	C. Shapiro and H. R. Varian, Information Rules: A Strategic Guide to the Network Economy, Harvard Business School Press, 1998.
[17]	O. Shy, The Economics of Network Industries, Cambridge University Press, 2001.
[18]	G. Tullock, Efficient rent-seeking, in Toward a Theory of the Rent-Seeking Society (eds. J. M. Buchanan, R. D. Tollison and G. Tullock), College Station: Texas A & M University Press, 1980, 97-112.

show all references

References:

[1]	A. Banerji and B. Dutta, Local network externalities and market segmentation, International Journal of Industrial Organization, 27 (2009), 605-614. doi: 10.1016/j.ijindorg.2009.02.001.
[2]	F. Bloch and N. Quérou, Pricing in social network, Games and Economic Behavior, 80 (2013), 243-261. doi: 10.1016/j.geb.2013.03.006.
[3]	P. Domingos and M. Richardson, Mining the network value of customers, in Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, 2001, 57-66. doi: 10.1145/502512.502525.
[4]	J. L. Doob, Stochastic Processes, John Wiley & Sons, New York, 1953.
[5]	P. Dubey, R. Garg and B. De Meyer, Competing for customers in a social network: The quasi-linear Case, in Internet and Network Economics: Second International Workshop, Lecture Notes in Computer Science, 4286, Springer, Berlin-Heidelberg, 2006, 162-173. doi: 10.1007/11944874_16.
[6]	J. Hartline, V. Mirrokni and M. Sundarajan, Optimal marketing strategies over social networks, in Proceedings of WWW 2008, Beijing, China, 2008, 189-198. doi: 10.1145/1367497.1367524.
[7]	M. Jackson, The economics of social networks, in Proceedings of the 9th World Congress of the Econometric Society (eds. R. Blundell, W. Newey and T. Persson), Cambridge University Press, 2005.
[8]	B. Julien, Competing in Network Industries: Divide and Conquer, Mimeo, IDEI and GREMAQ, 2001.
[9]	D. Kempe, J. Kleinberg and E. Tardos, Maximizing the spread of influence through a social network, in Proceedings of the 9th International Conference on Knowledge Discovery and Data Mining, 2003, 137-146. doi: 10.1145/956755.956769.
[10]	C. N. Moore, Summability of series, The American Mathematical Monthly, 39 (1932), 62-71. doi: 10.2307/2302048.
[11]	J. Nash, Equilibrium points in $n$-person games, Proceedings of the National Academy of Science, 36 (1950), 48-49. doi: 10.1073/pnas.36.1.48.
[12]	M. Richardson and P. Domingos, Mining knowledge-sharing sites for viral marketing, in Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Edmonton, Alberta, Canada, 2002, 61-70. doi: 10.1145/775056.775057.
[13]	P. Saaskilahti, Monopoly Pricing of Social Goods, MPRA Paper 3526, University Library of Munich, 2007.
[14]	S. Sahi, A note on the resolvent of a nonnegative matrix and its applications, Linear Algebra and Its Applications, 432 (2010), 2524-2528. doi: 10.1016/j.laa.2009.11.004.
[15]	J. Scott, Social Network Analysis: A Handbook, 2nd edition, Sage Publications, London, 2000.
[16]	C. Shapiro and H. R. Varian, Information Rules: A Strategic Guide to the Network Economy, Harvard Business School Press, 1998.
[17]	O. Shy, The Economics of Network Industries, Cambridge University Press, 2001.
[18]	G. Tullock, Efficient rent-seeking, in Toward a Theory of the Rent-Seeking Society (eds. J. M. Buchanan, R. D. Tollison and G. Tullock), College Station: Texas A & M University Press, 1980, 97-112.

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