The timing of consumer arrivals in Edgeworth's duopoly model

In his classic Papers relating to Political Economy (1897), Francis Edgeworth demonstrated that when duopolists have limited productive capacity, there may be no Nash equilibrium in prices. One feature of Edgeworth's model is that consumers are assumed to meet with the duopolists at the same time. ; This paper analyzes a version of the Edgeworth model in which consumers arrive sequentially instead of simultaneously. This departure from Edgeworth's framework should seem reasonable since there are few markets besides auctions in which buyers all meet with sellers at the same time. ; The point of the analysis is to show that when sellers engage in quantity constrained price competition, the timing of consumer arrivals may greatly affect the nature of equilibrium. It turns out that the existence of Nash equilibrium in prices may be restored. It also turns out that the duopolists may be able to maximize joint profits!