Bilateral trade with the sealed bid k-double auction: Existence and efficiency

For k in the unit interval, the k-double auction determines the terms of trade when a buyer and a seller negotiate transfer of an item. The buyer submits a bid b and the seller submits an offer s. Trade occurs if b exceeds s, at price kb + (1 − k) s. We model trade as a Bayesian game in which each trader privately knows his reservation value, but only has beliefs about the other trader's value. Existence of a multiplicity of equilibria is proven for a class of trader's beliefs. For generic beliefs, however, these equilibria are shown to be ex ante inefficient.