A seller trades with q out of n buyers who have valuations a 1 ≥ a 2 ≥ ⋯ ≥ 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 max l ≤ q +1 [( a 1 + a 2 +⋯+ a l −1 )/2+ a l +1 +⋯+ a q +1 ]. 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.

The deferred acceptance algorithm is often used to allocate indivisible objects when monetary transfers are not allowed. We provide two characterizations of agent-proposing deferred acceptance allocation rules. Two new axioms, individually rational monotonicity and weak Maskin monotonicity, are essential to our analysis. An allocation rule is the agent-proposing deferred acceptance rule for some acceptant substitutable priority if and only if it satisfies non-wastefulness and individually rational monotonicity. An alternative characterization is in terms of non-wastefulness, population monotonicity and weak Maskin monotonicity. We also offer an axiomatization of the deferred acceptance rule generated by an exogenously specified priority structure. We apply our results to characterize efficient deferred acceptance rules. 1.