Stationary state in a multistage auction model

We consider a game-theoretic multistage bargaining model with incomplete information related with deals between buyers and sellers. A player (buyer or seller) has private information about his reserved price. Reserved prices are random variables with known probability distributions. Each player declares a price which depends on his reserved price. If the bid price is above the ask price, the good is sold for the average of two prices. Otherwise, there is no deal. We investigate model with infinite time horizon and permanent distribution of reserved prices on each stage. Two types of Nash–Bayes equilibrium are derived. One of them is a threshold form, another one is a solution of a system of integro-differential equations.