Strategic entering time of a commerce platform

We consider a commerce platform that consists of two queues: one for buyers of an item and the other for sellers of the item. The platform is operated under the first-join-first-trade discipline. Upon a trade, the buyer and the seller gain respective profits but incur the cost of waiting in the platform. To maximize their expected payoffs from trading, both buyers and sellers can choose their arrival times. We characterize the Nash equilibrium in terms of a system of integro-differential equations for arrival time distributions. We show that a unique Nash equilibrium exists by proving that the system of integro-differential equations has a unique solution. Additionally, we compute the price of anarchy and investigate other disciplines that can improve it. Specifically, we show that charging platform usage fees improves the price of anarchy.