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Register a new userCombinatorial Algorithms for Matching Markets via Nash Bargaining: One-Sided, Two-Sided and Non-Bipartite
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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.
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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.
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.
We design novel mechanisms for welfare-maximization in two-sided markets. That is, there are buyers willing to purchase items and sellers holding items initially, both acting rationally and strategically in order to maximize utility. Our mechanisms are designed based on a powerful correspondence between two-sided markets and prophet inequalities. They satisfy individual rationality, dominant-strategy incentive compatibility, budget-balance constraints and give constant-factor approximations to the optimal social welfare. We improve previous results in several settings: Our main focus is on matroid double auctions, where the set of buyers who obtain an item needs to be independent in a matroid. We construct two mechanisms, the first being a $1/3$-approximation of the optimal social welfare satisfying strong budget-balance and requiring the agents to trade in a customized order, the second being a $1/2$-approximation, weakly budget-balanced and able to deal with online arrival determined by an adversary. In addition, we construct constant-factor approximations in two-sided markets when buyers need to fulfill a knapsack constraint. Also, in combinatorial double auctions, where buyers have valuation functions over item bundles instead of being interested in only one item, using similar techniques, we design a mechanism which is a $1/2$-approximation of the optimal social welfare, strongly budget-balanced and can deal with online arrival of agents in an adversarial order.
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.
This paper addresses the paucity of models of matching markets, both one-sided and two-sided, when utility functions of agents are cardinal. The classical Hylland-Zeckhauser scheme cite{hylland}, which is the most prominent such model in economics, can be viewed as corresponding to the linear Fisher model, which is most elementary model in market equilibria. Although HZ is based on the attractive idea of using a pricing mechanism, from the viewpoint of use in applications, it has a serious drawback, namely lack of computational efficiency, due to which solving instances of size even 4 or 5 is difficult. We propose a variety of Nash-bargaining-based models, several of which draw from general equilibrium theory, which has defined a rich collection of market models that generalize the linear Fisher model in order to address more specialized and realistic situations. The Nash bargaining solution satisfies Pareto optimality and symmetry and the allocations it yields are remarkably fair. Furthermore, since the solution is captured via a convex program, it is polynomial time computable. In order to be used in industrial grade applications, we give implementations for these models that are extremely time efficient, solving large instances, with $n = 2000$, in one hour on a PC, even for a two-sided matching market. The idea underlying our work has its origins in Vazirani (2012), which viewed the linear case of the Arrow-Debreu market model as a Nash bargaining game and gave a combinatorial, polynomial time algorithm for finding allocations via this solution concept, rather than the usual approach of using a pricing mechanism.
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