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Register a new userOne-Sided Matching Markets with Endowments: Equilibria and Algorithms
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The Arrow-Debreu extension of the classic Hylland-Zeckhauser scheme for a one-sided matching market -- called ADHZ in this paper -- has natural applications but has instances which do not admit equilibria. By introducing approximation, we define the $epsilon$-approximate ADHZ model, and we give the following results. * Existence of equilibrium under linear utility functions. We prove that the equilibrium satisfies Pareto optimality, approximate envy-freeness, and approximate weak core stability. * A combinatorial polynomial-time algorithm for an $epsilon$-approximate ADHZ equilibrium for the case of dichotomous, and more generally bi-valued, utilities. * An instance of ADHZ, with dichotomous utilities and a strongly connected demand graph, which does not admit an equilibrium. Since computing an equilibrium for HZ is likely to be highly intractable and because of the difficulty of extending HZ to more general utility functions, Hosseini and Vazirani proposed (a rich collection of) Nash-bargaining-based matching market models. For the dichotomous-utilities case of their model linear Arrow-Debreu Nash bargaining one-sided matching market (1LAD), we give a combinatorial, strongly polynomial-time algorithm and show that it admits a rational convex program.
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This paper is an attempt to deal with the recent realization (Vazirani, Yannakakis 2021) that the Hylland-Zeckhauser mechanism, which has remained a classic in economics for one-sided matching markets, is likely to be highly intractable. HZ uses the power of a pricing mechanism, which has endowed it with nice game-theoretic properties. Hosseini and Vazirani (2021) define a rich collection of Nash-bargaining-based models for one-sided and two-sided matching markets, in both Fisher and Arrow-Debreu settings, together with implementations using available solvers, and very encouraging experimental results. This naturally raises the question of finding efficient combinatorial algorithms for these models. In this paper, we give efficient combinatorial algorithms based on the techniques of multiplicative weights update (MWU) and conditional gradient descent (CGD) for several one-sided and two-sided models defined in HV 2021. Additionally, we define for the first time a Nash-bargaining-based model for non-bipartite matching markets and solve it using CGD. Furthermore, in every case, we study not only the Fisher but also the Arrow-Debreu version; the latter is also called the exchange version. We give natural applications for each model studied. These models inherit the game-theoretic and computational properties of Nash bargaining. We also establish a deep connection between HZ and the Nash-bargaining-based models, thereby confirming that the alternative to HZ proposed in HV 2021 is a principled one.
In 1979, Hylland and Zeckhauser cite{hylland} gave a simple and general scheme for implementing a one-sided matching market using the power of a pricing mechanism. Their method has nice properties -- it is incentive compatible in the large and produces an allocation that is Pareto optimal -- and hence it provides an attractive, off-the-shelf method for running an application involving such a market. With matching markets becoming ever more prevalant and impactful, it is imperative to finally settle the computational complexity of this scheme. We present the following partial resolution: 1. A combinatorial, strongly polynomial time algorithm for the special case of $0/1$ utilities. 2. An example that has only irrational equilibria, hence proving that this problem is not in PPAD. Furthermore, its equilibria are disconnected, hence showing that the problem does not admit a convex programming formulation. 3. A proof of membership of the problem in the class FIXP. We leave open the (difficult) question of determining if the problem is FIXP-hard. Settling the status of the special case when utilities are in the set ${0, {frac 1 2}, 1 }$ appears to be even more difficult.
We consider the problem of welfare maximization in two-sided markets using simple mechanisms that are prior-independent. The Myerson-Satterthwaite impossibility theorem shows that even for bilateral trade, there is no feasible (IR, truthful, budget balanced) mechanism that has welfare as high as the optimal-yet-infeasible VCG mechanism, which attains maximal welfare but runs a deficit. On the other hand, the optimal feasible mechanism needs to be carefully tailored to the Bayesian prior, and is extremely complex, eluding a precise description. We present Bulow-Klemperer-style results to circumvent these hurdles in double-auction markets. We suggest using the Buyer Trade Reduction (BTR) mechanism, a variant of McAfees mechanism, which is feasible and simple (in particular, deterministic, truthful, prior-independent, anonymous). First, in the setting where buyers and sellers values are sampled i.i.d. from the same distribution, we show that for any such market of any size, BTR with one additional buyer whose value is sampled from the same distribution has expected welfare at least as high as the optimal in the original market. We then move to a more general setting where buyers values are sampled from one distribution and sellers from another, focusing on the case where the buyers distribution first-order stochastically dominates the sellers. We present bounds on the number of buyers that, when added, guarantees that BTR in the augmented market have welfare at least as high as the optimal in the original market. Our lower bounds extend to a large class of mechanisms, and all of our results extend to adding sellers instead of buyers. In addition, we present positive results about the usefulness of pricing at a sample for welfare maximization in two-sided markets under the above two settings, which to the best of our knowledge are the first sampling results in this context.
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 platforms 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.
Motivated by the emergence of popular service-based two-sided markets where sellers can serve multiple buyers at the same time, we formulate and study the {em two-sided cost sharing} problem. In two-sided cost sharing, sellers incur different costs for serving different subsets of buyers and buyers have different values for being served by different sellers. Both buyers and sellers are self-interested agents whose values and costs are private information. We study the problem from the perspective of an intermediary platform that matches buyers to sellers and assigns prices and wages in an effort to maximize welfare (i.e., buyer values minus seller costs) subject to budget-balance in an incentive compatible manner. In our markets of interest, agents trade the (often same) services multiple times. Moreover, the value and cost for the same service differs based on the context (e.g., location, urgency, weather conditions, etc). In this framework, we design mechanisms that are efficient, ex-ante budget-balanced, ex-ante individually rational, dominant strategy incentive compatible, and ex-ante in the core (a natural generalization of the core that we define here).
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