Bring a friend! Privately or publicly?

We study the optimal referral strategy of a seller and its relationship with the type of communication channels among consumers. The seller faces a partially uninformed population of consumers, interconnected through a directed social network. In the network, the seller offers rewards to informed consumers (influencers) conditional on inducing purchases by uninformed consumers (influenced). Rewards are needed to bear a communication cost and to induce word of mouth (WOM) either privately (cost per contact) or publicly (fixed cost to inform all friends). From the seller's viewpoint, eliciting Private WOM is more costly than eliciting Public WOM. We investigate (1) the incentives for the seller to move to a denser network, inducing either Private or Public WOM, and (2) the optimal mix between the two types of communication. A denser network is found to be always better not only for information diffusion but also for seller's profits, as long as Private WOM is concerned. Differently, under Public WOM, the seller may prefer an environment with less competition between informed consumers, and the presence of highly connected influencers (hubs) is the main driver to make network density beneficial to profits. When the seller is able to discriminate between Private and Public WOM, the optimal strategy is to cheaply incentivize the more connected people to pass on the information publicly and then offer a high bonus for Private WOM.