We consider the online stochastic matching problem proposed by Feldman et al. [4] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins and the other side represents the set of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the number of allocations.We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than 1−1/e were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that our algorithm achieves a competitive ratio of 0.705 when the rates are integral.On the hardness side, we prove that no online algorithm can have a competitive ratio better than 0.823 under the known distribution model (and henceforth under the permutation model). This improves upon the 5 6 hardness result proved by Goel and Mehta [7] for the permutation model.

No abstract

The container relocation problem (CRP) is concerned with finding a sequence of moves of containers that minimizes the number of relocations needed to retrieve all containers, while respecting a given order of retrieval. However, the assumption of knowing the full retrieval order of containers is particularly unrealistic in real operations. This paper studies the stochastic CRP, which relaxes this assumption. A new multistage stochastic model, called the batch model, is introduced, motivated, and compared with an existing model (the online model). The two main contributions are an optimal algorithm called Pruning-Best-First-Search (PBFS) and a randomized approximate algorithm called PBFS-Approximate with a bounded average error. Both algorithms, applicable in the batch and online models, are based on a new family of lower bounds for which we show some theoretical properties. Moreover, we introduce two new heuristics outperforming the best existing heuristics. Algorithms, bounds, and heuristics are tested in an extensive computational section. Finally, based on strong computational evidence, we conjecture the optimality of the “leveling” heuristic in a special “no information” case, where, at any retrieval stage, any of the remaining containers is equally likely to be retrieved next. The online appendix is available at https://doi.org/10.1287/trsc.2018.0828 .

Many matching markets are naturally dynamic. Every year thousands of incompatible patient-donor pairs register to kidney exchange clearinghouses that search periodically to match these pairs. Online platforms (dating, online workplace, etc.), labor markets, and even housing markets can be viewed as dynamic matching markets. The matching policy, which determines when and who to match, plays an important role in the efficiency of the marketplace. A myopic policy, which attempts to match agents upon arrival, may have short run benefits but could harm future arriving agents. So a centralized clearinghouse may wait to thicken the market before identifying matches. In various marketplaces, the matching technology is also instrumental for efficiency. While kidney exchanges were first conducted in 2-way cycles (bilateral matches), most transplants are now conducted through chains initiated by an altruistic donor. This paper is concerned with the effects matching technologies and matching policies have on the efficiency of markets with different thickness levels. We study a stylized infinite horizon model with two types of agents distinguished by their difficulty to match. Every period a single agent arrives to the market whose type is either hard-to-match (H), or easy-to-match (E). So the fraction of H agents joining the market, can be viewed as a measure for the market thickness, with a higher fraction interpreted as a thinner market. Agents in our model prefer to match as early as possible and are indifferent between acceptable matches. We focus on two types of matchings; bilateral and chains, and we assume that only one of these take place. We are interested in the behavior of the average waiting time for small values of p H. Our main contribution is identifying a tight connection between the fraction of H agents joining the market and the effect the matching technology has on efficiency. In particular, we find that when E agents join the market more frequently than H ones, myopic matching policies that use a chain, or just bilateral matches, are both approximately optimal. Otherwise, any bilateral matching policy is highly sub-optimal. Further, we show that no matching policy can reduce the average waiting time of H agents without significantly harming E agents, and thus any attempt to thicken the pool is artificial. Finally, we discuss extensions of our model to include departures, independent arrival rates, two-sided markets, and decentralized markets.

Current kidney exchange pools are of moderate size and thin, as they consist of many highly sensitized patients. Creating a thicker pool can be done by waiting for many pairs to arrive. We analyze a simple class of matching algorithms that search periodically for allocations. We find that if only 2-way cycles are conducted, in order to gain a significant amount of matches over the online scenario (matching each time a new incompatible pair joins the pool) the waiting period should be "very long". If 3-way cycles are also allowed we find regimes in which waiting for a short period also increases the number of matches considerably. Finally, a significant increase of matches can be obtained by using even one non-simultaneous chain while still matching in an online fashion. Our theoretical findings and data-driven computational experiments lead to policy recommendations. . We thank participants in the NBER workshop on market design for valuable suggestions and comments.