Optimal Pricing in Social Networks with Incomplete Information
In revenue maximization of selling a digital product in a social network, the utility of an agent is often considered to have two parts: a private valuation, and linearly additive influences from other agents. We study the incomplete information case where agents know a common distribution about others’ private valuations, and make decisions simultaneously. The “rational behavior” of agents in this case is captured by the well-known Bayesian Nash equilibrium.
Two challenging questions arise: how to compute an equilibrium and how to optimize a pricing strategy accordingly to maximize the revenue assuming agents follow the equilibrium? In this paper, we mainly focus on the natural model where the private valuation of each agent is sampled from a uniform distribution, which turns out to be already challenging.
Our main result is a polynomial-time algorithm that can exactly compute the equilibrium and the optimal price, when pairwise influences are non-negative. If negative influences are allowed, computing any equilibrium even approximately is PPAD-hard. Our algorithm can also be used to design an FPTAS for optimizing discriminative price profile.
KeywordsNash Equilibrium Probability Vector Optimal Price Uniform Price Congestion Game
Unable to display preview. Download preview PDF.
- 3.Bloch, F., Quérou, N.: Pricing with local network externalities. Technical report (July 2009)Google Scholar
- 5.Chen, W., Wang, Y., Yang, S.: Efficient influence maximization in social networks. In: SIGKDD 2009, pp. 199–208 (2009)Google Scholar
- 6.Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure nash equilibria. In: STOC 2004, pp. 604–612 (2004)Google Scholar
- 9.Hartline, J., Mirrokni, V., Sundararajan, M.: Optimal marketing strategies over social networks. In: WWW 2008, pp. 189–198 (2008)Google Scholar
- 10.Hartline, J.D., McGrew, R.: From optimal limited to unlimited supply auctions. In: ACM-EC 2005, pp. 175–182 (2005)Google Scholar
- 11.Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: SIGKDD 2003, pp. 137–146 (2003)Google Scholar
- 14.Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V.V.: Algorithmic game theory. Cambridge University Press (2007)Google Scholar
- 15.Sundararajan, A.: Local network effects and complex network structure. The B.E. Journal of Theoretical Economics 7(1) (2008)Google Scholar
- 18.Zhu, Z.A.: Two Topics on Nash Equilibrium in Algorithmic Game Theory. Bach- elor’s thesis. Tsinghua University (June 2010)Google Scholar