Technical Note—Revenue Volatility Under Uncertain Network Effects

We study the revenue volatility of a monopolist selling a divisible good to consumers in the presence of local network externalities among consumers. The utility of consumers depends on their consumption level as well as those of their neighbors in a network through network externalities. In the eye of the seller, there exist uncertainties in the network externalities, which may be the result of unanticipated shocks or a lack of exact knowledge of the externalities. However, the seller has to commit to prices ex ante. We quantify the magnitude of revenue volatility under the optimal pricing in the presence of those random externalities. We consider both a given uncertainty set (from a robust optimization perspective) and a known uncertainty distribution (from a stochastic optimization perspective) and carry out the analyses separately. For a given uncertainty set, we show that the worst case of revenue fluctuation is determined by the largest eigenvalue of the matrix that represents the underlying network. Our results indicate that in networks with a smaller largest eigenvalue, the monopolist has a less volatile revenue. For the known uncertainty, we model the random noise in the form of a Wigner matrix and investigate large networks such as social networks. For such networks, we establish that the expected revenue is the sum of the revenue associated with the underlying expected network externalities and a term that depends on the noise variance and the weighted sum of all walks of different lengths in the expected network. We demonstrate that in a less connected network, the revenue is less volatile to uncertainties, and perhaps counterintuitively, the expected revenue increases with the level of uncertainty in the network. We show that a seller in the two settings favors the opposite type of network. In particular, if the underlying network is such that all the edge weights equal 1, the seller in the robust optimization setting prefers more asymmetry and the seller in the stochastic optimization setting prefers less asymmetry in the underlying network; by contrast, if the underlying network is such that the sum of all the edge weights is fixed, the seller in the robust optimization setting prefers less symmetry and the seller in the stochastic optimization setting prefers more asymmetry.