Bargaining and Exclusion With Multiple Buyers

A seller trades with q out of n buyers who have valuations a₁ ≥ a₂ ≥ ⋯ ≥ a_n > 0 via sequential bilateral bargaining. When q < n, buyer payoffs vary across equilibria in the patient limit, but seller payoffs do not, and converge to (Formula presented.) If l^* is the (generically unique) maximizer of this optimization problem, then each buyer i < l^* trades with probability 1 at the fair price a_i/2, while buyers i ≥ l^* are excluded from trade with positive probability. Bargaining with buyers who face the threat of exclusion is driven by a sequential outside option principle: the seller can sequentially exercise the outside option of trading with the extra marginal buyer q + 1, then with the new extra marginal buyer q, and so on, extracting full surplus from each buyer in this sequence and enhancing the outside option at every stage. A seller who can serve all buyers (q = n) may benefit from creating scarcity by committing to exclude some remaining buyers as negotiations proceed. An optimal exclusion commitment, within a general class, excludes a single buyer but maintains flexibility about which buyer is excluded. Results apply symmetrically to a buyer bargaining with multiple sellers.