Optimal design of a name-your-own price channel

A retailer places a certain product (eg compact rental cars) for sale on the internet. Customers are invited to ‘name–their–own price’ for the product. The retailer will accept a given bid x with probability equal to p(.). It is assumed that customers know the function p(.) and will place bids that maximise their individual expected profits. Knowing that customers will behave this way, the retailer wants to choose the function p(.) that maximises the retailer's expected profit. We demonstrate that there is an explicit ϵ–optimal solution to this problem.