Bargaining order and delays in multilateral bargaining with heterogeneous sellers

We consider a complete-information multilateral bargaining game in which a single buyer negotiates with two heterogeneous sellers selling perfect complementary units. While bilateral negotiations take place through a sequence of offers and counteroffers, the bargaining order is exogenously given. We solve for the conditions under which (a) the buyer prefers to negotiate with the lower-valuation seller first and (b) efficient (inefficient) outcomes emerge for the two bargaining orders. We find that the buyer prefers to negotiate with the lower-valuation seller first whenever the players are relatively impatient or the sellers are sufficiently heterogeneous. We show that there exists a unique efficient outcome when the buyer negotiates first with the lower-valuation seller and the sellers are sufficiently heterogeneous; however, significant delay in reaching agreements may arise when they are not. In case the buyer bargains with the higher-valuation seller first, an inefficient outcome is shown to exist even when players are extremely impatient.