We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain. Players who reach agreement are replaced by new players at the same positions in the network. We show that all equilibria are payoff equivalent. The payoffs and the set of agreement links converge as players become patient. Several new concepts—mutually estranged sets, partners, and shortage ratios—provide insights into the relative strengths of the positions in the network. We develop a procedure to determine the limit equilibrium payoffs for all players. Characterizations of equitable and nondiscriminatory networks are also obtained.

I study intermediation via sequential bilateral bargaining in markets with a network structure. A partition of the network into layers captures intermediation power. Competition governs trade within layers, and hold-ups generate intermediation rents in transactions between layers. Each trader’s intermediation power reflects the competition among intermediation chains as measured by the number of layers separating the trader from buyers. Trade does not maximize welfare or minimize intermediation. The interplay between competition and hold-ups determines the level of inefficiency. Eliminating middlemen or transferring intermediation costs downstream increases seller profits. Downstream and upstream competition respond oppositely to either horizontal or vertical integration.

We study a bargaining game in which a seller can trade with one of two buyers, who have values h and l (h>l). The outside option principle (OOP) predicts that, as players become patient, the seller trades with the high-value buyer with probability converging to 1 at a price converging to max(h/2,l). While this prediction is supported by the Markov perfect equilibrium (MPE), a wide range of trading outcomes may emerge in subgame perfect equilibria (SPEs): in the patient limit, the seller can obtain any price in the interval [h/2,h] (and no other); moreover, allocative inefficiency and costly delay are possible. We propose equilibrium refinements less restrictive than Markov behavior that guarantee trading outcomes consistent with the OOP. One refinement requires that a buyer’s relative probability of trade does not increase dramatically following a failed negotiation with that buyer. Another refinement posits that the seller does not approach a buyer hoping that negotiations fail. SPEs satisfying both refinements conform with the OOP (but need not be MPEs). Our benchmark model features strategic matching by the seller. We provide a parallel analysis for the random matching protocol. Under random matching, prices in SPEs may also rise above and fall below l, but have a narrower range. A refinement particular to this protocol that restores the OOP requires that a random mismatch should not impact the seller excessively.

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.