The size of the core in assignment markets

Assignment markets involve matching with transfers, as in labor markets and housing markets. We consider a two-sided assignment market with agent types and stochastic structure similar to models used in empirical studies, and characterize the size of the core in such markets. Each agent has a randomly drawn productivity with respect to each type of agent on the other side. The value generated from a match between a pair of agents is the sum of the two productivity terms, each of which depends only on the type but not the identity of one of the agents, and a third deterministic term driven by the pair of types. We allow the number of agents to grow, keeping the number of agent types fixed. Let $n$ be the number of agents and $K$ be the number of types on the side of the market with more types. We find, under reasonable assumptions, that the relative variation in utility per agent over core outcomes is bounded as $O^*(1/n^{1/K})$, where polylogarithmic factors have been suppressed. Further, we show that this bound is tight in worst case. We also provide a tighter bound under more restrictive assumptions. Our results provide partial justification for the typical assumption of a unique core outcome in empirical studies

Which Random Matching Markets Exhibit a Stark Effect of Competition?

We revisit the popular random matching market model introduced by Knuth (1976) and Pittel (1989), and shown by Ashlagi, Kanoria and Leshno (2013) to exhibit a "stark effect of competition", i.e., with any difference in the number of agents on the two sides, the short side agents obtain substantially better outcomes. We generalize the model to allow "partially connected" markets with each agent having an average degree $d$ in a random (undirected) graph. Each agent has a (uniformly random) preference ranking over only their neighbors in the graph. We characterize stable matchings in large markets and find that the short side enjoys a significant advantage only for $d$ exceeding $\log^2 n$ where $n$ is the number of agents on one side: For moderately connected markets with $d=o(\log^2 n)$, we find that there is no stark effect of competition, with agents on both sides getting a $\sqrt{d}$-ranked partner on average. Notably, this regime extends far beyond the connectivity threshold of $d= \Theta(\log n)$. In contrast, for densely connected markets with $d = \omega(\log^2 n)$, we find that the short side agents get $\log n$-ranked partner on average, while the long side agents get a partner of (much larger) rank $d/\log n$ on average. Numerical simulations of our model confirm and sharpen our theoretical predictions. Since preference list lengths in most real-world matching markets are much below $\log^2 n$, our findings may help explain why available datasets do not exhibit a strong effect of competition

Dynamic Assignment Control of a Closed Queueing Network under Complete Resource Pooling

We study the design of dynamic assignment control in networks with a fixed number of circulating resources (supply units). Each time a demand arises, the controller has (limited) flexibility in choosing the node from which to assign a supply unit. If no supply units are available at any compatible node, the demand is lost. If the demand is served, this causes to the supply unit to relocate to the "destination" of the demand. We study how to minimize the proportion of lost requests in steady state (or over a finite horizon) via a large deviations analysis. We propose a family of simple state-dependent policies called Scaled MaxWeight (SMW) policies that dynamically manage the distribution of supply in the network. We prove that under a complete resource pooling condition (analogous to the condition in Hall's marriage theorem), any SMW policy leads to exponential decay of demand-loss probability as the number of supply units scales to infinity. Further, there is an SMW policy that achieves the $\textbf{optimal}$ loss exponent among all assignment policies, and we analytically specify this policy in terms of the demand arrival rates for all origin-destination pairs. The optimal SMW policy maintains high supply levels adjacent to structurally under-supplied nodes. We discuss two applications: (i) Shared transportation platforms (like ride-hailing and bikesharing): We incorporate travel delays in our model and show that SMW policies for assignment control continue to have exponentially small loss. Simulations of ride-hailing based on the NYC taxi dataset demonstrate excellent performance. (ii) Service provider selection in scrip systems (like for babysitting or for kidney exchange): With only cosmetic modifications to the setup, our results translate fully to a model of scrip systems and lead to strong performance guarantees for a "Scaled Minimum Scrip" service provider selection rule

The Fault in Our Recommendations: On the Perils of Optimizing the Measurable

Recommendation systems are widespread, and through customized recommendations, promise to match users with options they will like. To that end, data on engagement is collected and used. Most recommendation systems are ranking-based, where they rank and recommend items based on their predicted engagement. However, the engagement signals are often only a crude proxy for utility, as data on the latter is rarely collected or available. This paper explores the following question: By optimizing for measurable proxies, are recommendation systems at risk of significantly under-delivering on utility? If so, how can one improve utility which is seldom measured? To study these questions, we introduce a model of repeated user consumption in which, at each interaction, users select between an outside option and the best option from a recommendation set. Our model accounts for user heterogeneity, with the majority preferring ``popular'' content, and a minority favoring ``niche'' content. The system initially lacks knowledge of individual user preferences but can learn them through observations of users' choices over time. Our theoretical and numerical analysis demonstrate that optimizing for engagement can lead to significant utility losses. Instead, we propose a utility-aware policy that initially recommends a mix of popular and niche content. As the platform becomes more forward-looking, our utility-aware policy achieves the best of both worlds: near-optimal utility and near-optimal engagement simultaneously. Our study elucidates an important feature of recommendation systems; given the ability to suggest multiple items, one can perform significant exploration without incurring significant reductions in engagement. By recommending high-risk, high-reward items alongside popular items, systems can enhance discovery of high utility items without significantly affecting engagement

The set of solutions of random XORSAT formulae

The XOR-satisfiability (XORSAT) problem requires finding an assignment of $n$ Boolean variables that satisfy $m$ exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing $n$ variables and $m$ clauses of size $k$. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as $k$-satisfiability ($k$-SAT), hypergraph bicoloring and graph coloring. For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random $k$-XORSAT. In particular, we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is intimately tied to the construction of specific subgraphs of the hypergraph associated with an instance of $k$-XORSAT. In order to study such subgraphs, we establish novel local weak convergence results for them.Comment: Published at http://dx.doi.org/10.1214/14-AAP1060 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A dynamic model of barter exchange

We consider the problem of efficient operation of a barter exchange platform for indivisible goods. We introduce a dynamic model of barter exchange where in each period one agent arrives with a single item she wants to exchange for a different item. We study a homogeneous and stochastic environment: an agent is interested in the item possessed by another agent with probability p, independently for all pairs of agents. We consider two settings with respect to the types of allowed exchanges: a) Only two-way cycles, in which two agents swap their items, b) Two or three-way cycles. The goal of the platform is to minimize the average waiting time of an agent. Somewhat surprisingly, we find that in each of these settings, a policy that conducts exchanges in a greedy fashion is near optimal, among a large class of policies that includes batching policies. Further, we find that for small p, allowing three-cycles can greatly improve the waiting time over the two-cycles only setting. Specifically, we find that a greedy policy achieves an average waiting time of Θ(1/p2) in setting a), and Θ(1/p3/2) in setting b). Thus, a platform can achieve the smallest waiting times by using a greedy policy, and by facilitating three cycles, if possible. Our findings are consistent with empirical and computational observations which compare batching policies in the context of kidney exchange programs

Dynamic Spatial Matching

Motivated by a variety of online matching platforms, we consider demand and supply units which are located i.i.d. in $[0,1]^d$, and each demand unit needs to be matched with a supply unit. The goal is to minimize the expected average distance between matched pairs (the "cost"). We model dynamic arrivals of one or both of demand and supply with uncertain locations of future arrivals, and characterize the scaling behavior of the achievable cost in terms of system size (number of supply units), as a function of the dimension $d$. Our achievability results are backed by concrete matching algorithms. Across cases, we find that the platform can achieve cost (nearly) as low as that achievable if the locations of future arrivals had been known beforehand. Furthermore, in all cases except one, cost nearly as low as the expected distance to the nearest neighboring supply unit is achievable, i.e., the matching constraint does not cause an increase in cost either. The aberrant case is where only demand arrivals are dynamic, and $d=1$; excess supply significantly reduces cost in this case