This dissertation consists of four essays in economic theory. The first essay studies an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain over a unit surplus. Players who reach agreement are replaced by new players at the same positions in the network. We prove that for each discount factor all equilibria are payoff equivalent. The equilibrium payoffs and the set of equilibrium 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 by iteratively applying the following results. Limit payoffs are lowest for the players in the largest mutually estranged set that minimizes the shortage ratio, and highest for the corresponding partners. In equilibrium, for high discount factors, the partners act as an oligopoly for the estranged players. In the limit, surplus within the induced oligopoly subnetwork is divided according to the shortage ratio. We characterize equitable networks, stable networks, and non-discriminatory buyer-seller networks.

In the second essay, co-authored with Dilip Abreu, we consider a model of bargaining in networks where players who reach agreement are removed from the network without replacement. For many networks, all Markov perfect equilibria (MPEs) are asymptotically inefficient as players become patient. The main result is that there exist asymptotically efficient subgame perfect equilibria for every network structure. MPEs are of independent interest. We establish the existence of MPEs, and show that MPE payoffs are not necessarily unique. We provide a method to construct pure strategy MPEs for high discount factors.

The third and fourth essays explore the properties of two mechanisms—random serial dictatorship and deferred acceptance, respectively—which are often used to allocate indivisible objects when monetary transfers are not allowed. The third essay shows that the fraction of preference profiles for which the random serial dictatorship allocation is ordinally efficient vanishes for allocation problems with many object types. We also consider a probabilistic setting where in expectation agents have moderately similar preferences reflecting varying popularity across objects. In this setting we establish that the probability that the random serial dictatorship mechanism is ordinally efficient converges to zero as the number of object types becomes large. We provide results with similarly negative content for allocation problems with many objects of each type.

The fourth essay, co-authored with Fuhito Kojima, provides 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.