What's the name of this pairwise price-setting scheme? Two siblings inherit equal shares of a house. To decide who gets to live in it, they each bid what they think the other half is worth. The person with the higher bid gets it, and pays the other sibling halfway in between the bids. Therefore, everyone gets better than their bid: the seller receives more and the buyer pays less. Does this scheme have a name?
This is known as the "split-the-difference" mechanism, originally due to Samuelson (1985). See Section 6 of Gibbons et al. (1987) for a discussion of this rule. In particular,


this kind of rule only efficiently dissolves the partnership between the two siblings if they have equal (or sufficiently close to equal) shares in the house. Precisely, each sibling should own no more than 57.7\% of the house.
such a mechanism does not typically induce an equilibrium with sincere bidding (i.e., the siblings will report less than their true value for the house).