Bilateral trade with the sealed bid k-double auction: Existence and efficiency, Journal of Economic Theory
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.
Mark Satterthwaite, Steven R. Williams
Satterthwaite, Mark, and Steven R. Williams. 1989. Bilateral trade with the sealed bid k-double auction: Existence and efficiency. Journal of Economic Theory. 48(1): 107-133.