Optimal Pricing in Social Networks with Incomplete Information

TL;DR

This paper develops a polynomial-time algorithm to compute equilibrium and optimal pricing strategies in social networks with incomplete information, focusing on uniform valuation distributions and non-negative influences, with complexity results for negative influences.

Contribution

It introduces an exact polynomial-time algorithm for equilibrium and optimal pricing computation under non-negative influences, and proves PPAD-hardness with negative influences.

Findings

Exact equilibrium and optimal price can be computed efficiently for non-negative influences.

Computing equilibrium becomes PPAD-hard with negative influences.

An FPTAS for discriminative pricing is also developed.

Abstract

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.