Optimal Growth of a Two-Sided Platform with Heterogeneous Agents

We consider the dynamics of a two-sided platform, where the agent population on both sides experiences growth over time with heterogeneous retention and new adoption. The compatibility between buyers and sellers is captured by a bipartite network. The platform strategically sets commissions to optimize its total profit over a horizon of T periods, considering the trade-off between short-term profit and growth as well as the spatial imbalances in supply and demand. We design an asymptotically optimal commission policy with the profit loss upper-bounded by a constant independent of T, in contrast with a myopic policy that is shown to be arbitrarily bad even if the platform is a monopoly intermediary in the market. To obtain this policy, we first develop a single-period benchmark problem that captures the optimal steady state of the platform, then delicately control the growth of the agent types with the lowest relative population level compared with the single-period benchmark in each period over time. We further examine the impact of the growth potential and the market network structure on agents' payment/income, and the platform's optimal commissions and optimal profit. To achieve that, we introduce innovative metrics to quantify the long-run growth potential of each agent type. Using these metrics, we first show that for each agent type, higher ratios of their compatible counterparts' long-term growth potential to their own cause lower payments (higher income) at the optimal steady state. In addition, the impact of the relative long-run growth potential on the platform's optimal commissions in a submarket depends on the convexity/concavity of the value distribution function of agents. Finally, we show that a ``balanced" network, where the relative long-run growth potential between sellers and buyers for all submarkets is the same as that for the entire market, allows the platform to achieve maximum profitability. The extent of network imbalance determines the lower bound for the achievable long-run average profit