On Matching and Thickness in Heterogeneous Dynamic Markets

Abstract

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.

Spatial Distribution of Supply and the Role of Market Thickness: Theory and Evidence from Ride Sharing

“…The second set of papers that we relate to is a large, mostly theoretical, literature on the impact of market thickness on the functioning of two-sided platforms in general (such as Akbarpour et al (2017); Ashlagi et al (2019)) and transportation markets in particular (such as Frechette et al (2019); Nikzad (2018)). This literature, to our knowledge, has not examined how the spatial distribution of supply -and its (mis)alignment with that of potential demand-respond to a change in market thickness.…”

Kidney Exchange: An Operations Perspective

“…42 One stream of papers considers models in which compatibilities are based on random graphs, modeling the sparsity due to sensitivity of patients. Several of these papers assume that nodes (pairs) do not depart the pool unless matched (Anderson et al 2017;Ashlagi et al 2018aAshlagi et al , 2019aBlum and Mansour 2020). These papers find that greedy matching is optimal Ashlagi and Roth: Kidney Exchange: An Operations Perspective when minimizing average waiting times.…”

“…Theoretical studies have looked at the benefit of chains and cycles in dynamic settings, where the underlying model is based on random graphs (Anderson et al 2015b, Ashlagi et al 2019a, Blum and Mansour 2020. These papers find that, in sparse pools with many highly sensitized patients, chains result in significantly lower waiting times than relying only on cycles.…”

“…When three-way cycles are also included, greedy is optimal among a large class of policies (Anderson et al 2017). When the fraction of sensitized patients is sufficiently large, chains substantially reduce waiting times (Ashlagi 2019a). 44 The models assume different type structures and different limits regarding the growth of the market (e.g., whether the edge probability remains constant or decreases with the size of the pool).…”

Decentralized Matching in a Probabilistic Environment

“…We focus on such platforms where match compatibility is difficult to determine and can be learned exactly only through matching. A well-known example of such a setting is kidney exchange, which has been studied from a repeated matching perspective [3,4] and a failure-aware perspective [11].…”