We study the optimal design of a queueing system when agents' arrival and servicing are governed by a general Markov process. The designer of the system chooses entry and exit rules for agents, their service priority---or queueing discipline---as well as their information, while ensuring that agents have incentives to follow the designer's recommendations not only to join the queue but more importantly to stay in the queue. Under a mild condition, the optimal mechanism has a cutoff structure---agents are induced to enter up to a certain queue length and no agents are to exit the queue once they enter the queue; the agents on the queue are served according to a first-come-first-served (FCFS) rule; and they are given no information throughout the process beyond the recommendations they receive from the designer. FCFS is also necessary for optimality in a rich domain. We identify a novel role for queueing disciplines in regulating agents' beliefs, and their dynamic incentives, thus uncovering a hitherto unrecognized virtue of FCFS in this regard.

We propose a new matching algorithm -- Unpaired kidney exchange -- to tackle the problem of double coincidence of wants without using money. The fundamental idea is that "memory" can serve as a medium of exchange. In a dynamic matching model with heterogeneous agents, we prove that average waiting time under the Unpaired algorithm is close to optimal, substantially less than the standard pairwise and chain exchange algorithms. We evaluate this algorithm using a rich dataset of kidney patients in France. Counterfactual simulations show that the Unpaired algorithm can match 57% of the patients, with an average waiting time of 440 days (state-of-the-art algorithms match about 34% with an average waiting time of 695 days). The optimal algorithm, which is practically infeasible, performs only slightly better: it matches 58% of the patients and leads to an average waiting time of 426 days. The Unpaired algorithm confronts two incentive-related practical challenges. We address those challenges via a modified version of the Unpaired algorithm that employs kidneys from the deceased donors waiting list. It can match 86% of the patients, while reducing the average waiting time to about 155 days.

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogenous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior puts high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it consists of the unique rationalizable action profile. Set valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.