When is seller price setting with linear fees optimal for intermediaries?

Mechanisms where sellers set the price and are charged a linear commission fee are widely used by real world intermediaries, e.g. by real estate brokers. Empirically these commission fees exhibit very little variance, both across heterogeneous regional markets and over time. So far, there is no theoretical explanation why such seller price setting mechanisms are used and why the linear fees vary so little. In this paper, we first show that in a Bayesian setup seller price setting with linear fees is revenue equivalent to the intermediary optimal direct mechanism derived by Myerson and Satterthwaite (1983) if and only if the seller's cost is drawn from a generalized power distribution. Whenever such a mechanism is optimal, the fee structure is independent of the distribution from which the buyer's valuation is drawn. Second, we derive the intermediary optimal direct mechanism when there are many buyers and possibly many sellers and we show that with one seller any standard auction with linear fees and reserve price setting by the seller (which are used e.g. by eBay) implements this mechanism if the seller's cost is drawn from a power distribution and if buyers' valuations are identically distributed. Third, we show that when the number of buyers approaches infinity while there is still one seller, seller price setting and price setting by the intermediary are equivalent, intermediary optimal mechanisms.