Consumer referral in a small world network

We consider a monopolist’s profit-maximization problem when the demand for the product is created by the referrals between consumers who are connected over the social network. The monopolist decides the price of the product and the referral fee which is awarded to consumers whose referrals lead to sales of the product. It is known that the social network in the real world has the small world property that the average distance between any two players in the network is very short. We generate a small world network by applying the rewiring algorithm of Watts and Strogatz and incorporate it into the monopolist’s profit maximization problem. Main results of the paper are (i) that contrary to our anticipation, the small world property of the underlying social network is not inherited to the monopolist’s profit, in other words, the monopolist’s profit smoothly (not rapidly) approaches the profit in the complete network as the rewiring probability increases, i.e., the network becomes denser, (ii) that the per consumer profit inherits the small world property, so that it is saturated to a finite non-zero value, when the rewiring probability is near zero, and (iii) that as a network becomes denser, the optimal referral fee tends to increase when the rewiring probability is small and then decrease for the rest of the rewiring probability values, while the optimal price remains stable above a threshold level of the probability. Overall, optimal values for the price and the referral fee are more significantly affected by the network density, but not by the network size very much.