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In economies without monetary transfers, token systems serve as an alternative to sustain cooperation, alleviate free riding, and increase efficiency. This paper studies whether a token-based economy can be effective in marketplaces with thin exogenous supply. We consider a marketplace in which at each time period one agent requests a service, one agent provides the service, and one token (artificial currency) is used to pay for service provision. The number of tokens each agent has represents the difference between the amount of service provisions and service requests by the agent. We are interested in the behavior of this economy when very few agents are available to provide the requested service. Since balancing the number of tokens across agents is key to sustain cooperation, the agent with the minimum amount of tokens is selected to provide service among the available agents. When exactly one random agent is available to provide service, we show that the token distribution is unstable. However, already when just two random agents are available to provide service, the token distribution is stable, in the sense that agents' token balance is unlikely to deviate much from their initial endowment, and agents return to their initial endowment in finite expected time. Our results mirror the power of two choices paradigm in load balancing problems. Supported by numerical simulations using kidney exchange data, our findings suggest that token systems may generate efficient outcomes in kidney exchange marketplaces by sustaining cooperation between hospitals., Comment: 32 pages

School choice mechanism designers use discrete choice models to understand and predict families' preferences. The most widely-used choice model, the multinomial logit (MNL), is linear in school and/or household attributes. While the model is simple and interpretable, it assumes the ranked preference lists arise from a choice process that is uniform throughout the ranking, from top to bottom. In this work, we introduce two strategies for rank-heterogeneous choice modeling tailored for school choice. First, we adapt a context-dependent random utility model (CDM), considering down-rank choices as occurring in the context of earlier up-rank choices. Second, we consider stratifying the choice modeling by rank, regularizing rank-adjacent models towards one another when appropriate. Using data on household preferences from the San Francisco Unified School District (SFUSD) across multiple years, we show that the contextual models considerably improve our out-of-sample evaluation metrics across all rank positions over the non-contextual models in the literature. Meanwhile, stratifying the model by rank can yield more accurate first-choice predictions while down-rank predictions are relatively unimproved. These models provide performance upgrades that school choice researchers can adopt to improve predictions and counterfactual analyses., Comment: 21 pages, 13 figures

In many centralized labor markets candidates interview with potential employers before matches are formed through a clearinghouse One prominent example is the market for medical residencies and fellowships, which in recent years has had a large increase in the number of interviews. There have been numerous efforts to reduce the cost of interviewing in these markets using a variety of signalling mechanisms, however, the theoretical properties of these mechanisms have not been systematically studied in models with rich preferences. In this paper we give theoretical guarantees for a variety of mechanisms, finding that these mechanisms must properly balance competition. We consider a random market in which agents' latent preferences are based on observed qualities, personal taste and (ex post) interview shocks and assume that following an interview mechanism a final stable match is generated with respect to preferences over interview partners. We study a novel many-to-many interview match mechanism to coordinate interviews and that with relatively few interviews, when suitably designed, the interview match yields desirable properties. We find that under the interview matching mechanism with a limit of $k$ interviews per candidate and per position, the fraction of positions that are unfilled vanishes quickly with $k$. Moreover the ex post efficiency grows rapidly with $k$, and reporting sincere pre-interview preferences to this mechanism is an $\epsilon$-Bayes Nash equilibrium. Finally, we compare the performance of the interview match to other signalling and coordination mechanisms from the literature., Comment: Error in proof of Lemma 1 leaves a step unexplained, proof needs to be corrected and made more rigorous

We study the welfare structure in two-sided large random matching markets. In the model, each agent has a latent personal score for every agent on the other side of the market and her preferences follow a logit model based on these scores. Under a contiguity condition, we provide a tight description of stable outcomes. First, we identify an intrinsic fitness for each agent that represents her relative competitiveness in the market, independent of the realized stable outcome. The intrinsic fitness values correspond to scaling coefficients needed to make a latent mutual matrix bi-stochastic, where the latent scores can be interpreted as a-priori probabilities of a pair being matched. Second, in every stable (or even approximately stable) matching, the welfare or the ranks of the agents on each side of the market, when scaled by their intrinsic fitness, have an approximately exponential empirical distribution. Moreover, the average welfare of agents on one side of the market is sufficient to determine the average on the other side. Overall, each agent's welfare is determined by a global parameter, her intrinsic fitness, and an extrinsic factor with exponential distribution across the population.

This paper considers the problem of offering a scarce object with a common unobserved quality to strategic agents in a priority queue. Each agent has a private signal over the quality of the object and observes the decisions made by other agents. We first show that, under the widely-used first-come-first-served sequential offering mechanism, herding behavior emerges: initial rejections create an information cascade resulting in inefficient waste. To address this issue, we then introduce a class of batching mechanisms. Agents in each batch report whether they would be willing to accept or reject the object based on their private signals and prior information. If the majority opts to accept, the object is randomly allocated within that batch. We prove that suitable batching mechanisms are incentive-compatible and improve efficiency. A key property of the mechanism is the gradual increase of the batch size after each failed allocation; the size is chosen so that it elicits as much information as possible without distorting the incentives of agents to report truthfully. Additionally, from a healthcare policy perspective, our results can shed light on the large wastage in organ allocation. In particular, wastage that arises due to herding may be reduced by applying adaptive simultaneous offering mechanisms., Comment: A preliminary version appeared in the Thirty-fifth Annual Conference on Neural Information Processing Systems (NeurIPS 2021)

Consider a seller that intends to auction some item. The seller can invest money and effort in advertising in different market segments in order to recruit $n$ bidders to the auction. Alternatively, the seller can have a much cheaper and focused marketing operation and recruit the same number of bidders from a single market segment. Which marketing operation should the seller choose? More formally, let $D=\{\mathcal D_1,\ldots, \mathcal D_n\}$ be a set of distributions. Our main result shows that there is always $\mathcal D_i\in D$ such that the revenue that can be extracted from $n$ bidders, where the value of each is independently drawn from $\mathcal D_i$, is at least $\frac 1 2 \cdot (1-\frac 1 e)$ of the revenue that can be obtained by any possible mix of bidders, where the value of each bidder is drawn from some (possibly different) distribution that belongs to $D$. We next consider situations in which the auctioneer cannot use the optimal auction and is required to use a second price auction. We show that there is always $\mathcal D_i\in D$ such that if the value of all bidders is independently drawn from $\mathcal D_i$ then running a second price auction guarantees a constant fraction of the revenue that can be obtained by a second-price auction by any possible mix of bidders. Finally, we show that for any $\varepsilon>0$ there exists a function $f$ that depends only on $\varepsilon$ (in particular, the function does not depend on $n$ or on the set $D$), such that recruiting $n$ bidders which have at most $f(\varepsilon)$ different distributions, all from $D$, guarantees $(1-\varepsilon)$-fraction of the revenue that can be obtained by a second-price auction by any possible mix of bidders.

We study the stable marriage problem in two-sided markets with randomly generated preferences. We consider agents on each side divided into a constant number of "soft tiers", which intuitively indicate the quality of the agent. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the men-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, we show that despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes results of [Pittel 1989] which correspond to uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market, and we give the first explicit calculations of rank beyond uniform markets.

Two-sided matching platforms provide users with menus of match recommendations. To maximize the number of realized matches between the two sides (referred here as customers and suppliers), the platform must balance the inherent tension between recommending customers more potential suppliers to match with and avoiding potential collisions. We introduce a stylized model to study the above trade-off. The platform offers each customer a menu of suppliers, and customers choose, simultaneously and independently, either a supplier from their menu or to remain unmatched. Suppliers then see the set of customers that have selected them, and choose to either match with one of these customers or to remain unmatched. A match occurs if a customer and a supplier choose each other (in sequence). Agents' choices are probabilistic, and proportional to public scores of agents in their menu and a score that is associated with remaining unmatched. The platform's problem is to construct menus for costumers to maximize the number of matches. This problem is shown to be strongly NP-hard via a reduction from 3-partition. We provide an efficient algorithm that achieves a constant-factor approximation to the expected number of matches.

We study the assignment problem of objects to agents with heterogeneous preferences under distributional constraints. Each agent is associated with a publicly known type and has a private ordinal ranking over objects. We are interested in assigning as many agents as possible. Our first contribution is a generalization of the well-known and widely used serial dictatorship. Our mechanism maintains several desirable properties of serial dictatorship, including strategyproofness, Pareto efficiency, and computational tractability while satisfying the distributional constraints with a small error. We also propose a generalization of the probabilistic serial algorithm, which finds an ordinally efficient and envy-free assignment, and also satisfies the distributional constraints with a small error. We show, however, that no ordinally efficient and envy-free mechanism is also weakly strategyproof. Both of our algorithms assign at least the same number of students as the optimum fractional assignment., Comment: 26 pages, conference version published in SODA 2019

We study dynamic matching in exchange markets with easy- and hard-to-match agents. A greedy policy, which attempts to match agents upon arrival, ignores the positive externality that waiting agents generate by facilitating future matchings. We prove that this trade-off between a ``thicker'' market and faster matching vanishes in large markets; A greedy policy leads to shorter waiting times, and more agents matched than any other policy. We empirically confirm these findings in data from the National Kidney Registry. Greedy matching achieves as many transplants as commonly-used policies (1.6\% more than monthly-batching), and shorter patient waiting times.

We study the problem of matching agents who arrive at a marketplace over time and leave after d time periods. Agents can only be matched while they are present in the marketplace. Each pair of agents can yield a different match value, and the planner's goal is to maximize the total value over a finite time horizon. First we study the case in which vertices arrive in an adversarial order. We provide a randomized 0.25-competitive algorithm building on a result by Feldman et al. (2009) and Lehman et al. (2006). We extend the model to the case in which departure times are drawn independently from a distribution with non-decreasing hazard rate, for which we establish a 1/8-competitive algorithm. When the arrival order is chosen uniformly at random, we show that a batching algorithm, which computes a maximum-weighted matching every (d+1) periods, is 0.279-competitive., Comment: arXiv admin note: text overlap with arXiv:1803.01285

We study the problem of matching agents who arrive at a marketplace over time and leave after d time periods. Agents can only be matched while they are present in the marketplace. Each pair of agents can yield a different match value, and the planner's goal is to maximize the total value over a finite time horizon. We study matching algorithms that perform well over any sequence of arrivals when there is no a priori information about the match values or arrival times. Our main contribution is a 1/4-competitive algorithm. The algorithm randomly selects a subset of agents who will wait until right before their departure to get matched, and maintains a maximum-weight matching with respect to the other agents. The primal-dual analysis of the algorithm hinges on a careful comparison between the initial dual value associated with an agent when it first arrives, and the final value after d time steps. It is also shown that no algorithm is 1/2-competitive. We extend the model to the case in which departure times are drawn i.i.d from a distribution with non-decreasing hazard rate, and establish a 1/8-competitive algorithm in this setting. Finally we show on real-world data that a modified version of our algorithm performs well in practice.

We study dynamic matching in an infinite-horizon stochastic market. While all agents are potentially compatible with each other, some are hard-to-match and others are easy-to-match. Agents prefer to be matched as soon as possible and matches are formed either bilaterally or indirectly through chains. We adopt an asymptotic approach and compute tight bounds on the limit of waiting time of agents under myopic policies that differ in matching technology and prioritization. We find that the market composition is a key factor in the desired matching technology and prioritization level. When hard-to-match agents arrive less frequently than easy-to-match ones (i) bilateral matching is almost as efficient as chains (waiting times scale similarly under both, though chains always outperform bilateral matching by a constant factor), and (ii) assigning priorities to hard-to-match agents improves their waiting times. When hard-to-match agents arrive more frequently, chains are much more efficient than bilateral matching and prioritization has no impact. We further conduct comparative statics on arrival rates. Somewhat surprisingly, we find that in a heterogeneous market and under bilateral matching, increasing arrival rate has a non-monotone effect on waiting times, due to the fact that, under some market compositions, there is an adverse effect of competition. Our comparative statics shed light on the impact of merging markets and attracting altruistic agents (that initiate chains) or easy-to-match agents. This work uncovers fundamental differences between heterogeneous and homogeneous dynamic markets, and potentially helps policy makers to generate insights on the operations of matching markets such as kidney exchange programs.

We provide a characterization of revenue-optimal dynamic mechanisms in settings where a monopolist sells k items over k periods to a buyer who realizes his value for item i in the beginning of period i. We require that the mechanism satisfies a strong individual rationality constraint, requiring that the stage utility of each agent be positive during each period. We show that the optimum mechanism can be computed by solving a nested sequence of static (single-period) mechanisms that optimize a tradeoff between the surplus of the allocation and the buyer's utility. We also provide a simple dynamic mechanism that obtains at least half of the optimal revenue. The mechanism either ignores history and posts the optimal monopoly price in each period, or allocates with a probability that is independent of the current report of the agent and is based only on previous reports. Our characterization extends to multi-agent auctions. We also formulate a discounted infinite horizon version of the problem, where we study the performance of "Markov mechanisms."

Many two-sided matching markets, from labor markets to school choice programs, use a clearinghouse based on the applicant-proposing deferred acceptance algorithm, which is well known to be strategy-proof for the applicants. Nonetheless, a growing amount of empirical evidence reveals that applicants misrepresent their preferences when this mechanism is used. This paper shows that no mechanism that implements a stable matching is "obviously strategy-proof" for any side of the market, a stronger incentive property than strategy-proofness that was introduced by Li (2017). A stable mechanism that is obviously strategy-proof for applicants is introduced for the case in which agents on the other side have acyclical preferences.

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.

Correlated equilibrium (Aumann, 1974) generalizes Nash equilibrium to allow correlation devices. Aumann showed an example of a game, and of a correlated equilibrium in this game, in which the agents' surplus (expected sum of payo s) is greater than their surplus in all mixed-strategy equilibria. Following the idea initiated by the price of anarchy literature (Koutsoupias & Papadimitriou, 1999;Papadimitriou, 2001) this suggests the study of two major measures for the value of correlation in a game with non-negative payoffs: 1. The ratio between the maximal surplus obtained in a correlated equilibrium to the maximal surplus obtained in a mixed-strategy equilibrium. We refer to this ratio as the mediation value. 2. The ratio between the maximal surplus to the maximal surplus obtained in a correlated equilibrium. We refer to this ratio as the enforcement value. In this work we initiate the study of the mediation and enforcement values, providing several general results on the value of correlation as captured by these concepts. We also present a set of results for the more specialized case of congestion games (Rosenthal,1973), a class of games that received a lot of attention in the recent literature., Comment: Appears in Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (UAI2005)

We introduce robust learning equilibrium. The idea of learning equilibrium is that learning algorithms in multi-agent systems should themselves be in equilibrium rather than only lead to equilibrium. That is, learning equilibrium is immune to strategic deviations: Every agent is better off using its prescribed learning algorithm, if all other agents follow their algorithms, regardless of the unknown state of the environment. However, a learning equilibrium may not be immune to non strategic mistakes. For example, if for a certain period of time there is a failure in the monitoring devices (e.g., the correct input does not reach the agents), then it may not be in equilibrium to follow the algorithm after the devices are corrected. A robust learning equilibrium is immune also to such non-strategic mistakes. The existence of (robust) learning equilibrium is especially challenging when the monitoring devices are 'weak'. That is, the information available to each agent at each stage is limited. We initiate a study of robust learning equilibrium with general monitoring structure and apply it to the context of auctions. We prove the existence of robust learning equilibrium in repeated first-price auctions, and discuss its properties., Comment: Appears in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence (UAI2006)

It is well known that a stable matching in a many-to-one matching market with couples need not exist. We introduce a new matching algorithm for such markets and show that for a general class of large random markets the algorithm will find a stable matching with high probability. In particular we allow the number of couples to grow at a near-linear rate. Furthermore, truth-telling is an approximated equilibrium in the game induced by the new matching algorithm. Our results are tight: for markets in which the number of couples grows at a linear rate, we show that with constant probability no stable matching exists.

Consider a matching problem on a graph where disjoint sets of vertices are privately owned by self-interested agents. An edge between a pair of vertices indicates compatibility and allows the vertices to match. We seek a mechanism to maximize the number of matches despite self-interest, with agents that each want to maximize the number of their own vertices that match. Each agent can choose to hide some of its vertices, and then privately match the hidden vertices with any of its own vertices that go unmatched by the mechanism. A prominent application of this model is to kidney exchange, where agents correspond to hospitals and vertices to donor-patient pairs. Here hospitals may game an exchange by holding back pairs and harm social welfare. In this paper we seek to design mechanisms that are strategyproof, in the sense that agents cannot benefit from hiding vertices, and approximately maximize efficiency, i.e., produce a matching that is close in cardinality to the maximum cardinality matching. Our main result is the design and analysis of the eponymous Mix-and-Match mechanism; we show that this randomized mechanism is strategyproof and provides a 2-approximation. Lower bounds establish that the mechanism is near optimal., Comment: 10 pages, 5 figures. Appeared in Proceedings of 11th ACM Conference on Electronic Commerce, Pages 305-314