Approximately Stable Matchings with Budget Constraints

This paper considers two-sided matching with budget constraints where one side (firm or hospital) can make monetary transfers (offer wages) to the other (worker or doctor). In a standard model, while multiple doctors can be matched to a single hospital, a hospital has a maximum quota: the number of doctors assigned to a hospital cannot exceed a certain limit. In our model, a hospital instead has a fixed budget: the total amount of wages allocated by each hospital to doctors is constrained. With budget constraints, stable matchings may fail to exist and checking for the existence is hard. To deal with the nonexistence of stable matchings, we extend the “matching with contracts” model of Hatfield and Milgrom, so that it handles approximately stable matchings where each of the hospitals’ utilities after deviation can increase by factor up to a certain amount. We then propose two novel mechanisms that efficiently return such a stable matching that exactly satisfies the budget constraints. In particular, by sacrificing strategy-proofness, our first mechanism achieves the best possible bound. Furthermore, we find a special case such that a simple mechanism is strategy-proof for doctors, keeping the best possible bound of the general case. Introduction This paper studies a two-sided, one-to-many matching model when there are budget constraints on one side (firm or hospital), i.e., the total amount of wages that it can pay to the other side (worker or doctor) is limited. The theory of two-sided matching has been extensively developed. See the book by Roth and Sotomayor (1990) or Manlove (2013) for a comprehensive survey. In this literature, rather than fixed budgets, maximum quotas are typically used, i.e., the total number of doctors that each hospital can hire is limited. Some real-world examples are subject to matching with budget constraints: a college can offer stipends to students to recruit better students while the budget for admission is limited, a firm can offer wages to workers under the condition that employment costs depend on earnings in the previous accounting period, a public hospital can offer salaries to doctors in the case where the total amount relies on funds ∗A full version can be found at http://arxiv.org/abs/????.?????. Supported in part by JST ACT-I and KAKENHI 26280081, 16K16005, and 17H01787. Copyright c © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. from the government, and so on. To establish our model and concepts, we use doctor-hospital matching as a running example. Most papers on matching with monetary transfers assume that budgets are unrestricted, e.g., (Kelso and Crawford 1982). When they are restricted, stable matchings may fail to exist (Mongell and Roth 1986; Abizada 2016). There are several other possibilities to circumvent the nonexistence problem. Abizada (2016) modifies the notion of stability in a different way and proposes a variant of the Deferred Acceptance (DA) mechanism that produces a pairwise, instead of coalitional, stable matching, and is strategy-proof for doctors. Dean, Goemans, and Immorlica (2006) assume that hospitals’ priorities are lexicographic to ensure the existence of a stable matching. We instead allow each hospital to have an additive utility, but the existence is not guaranteed yet. Kawase and Iwasaki (2017) focus on near-feasible matchings that exceed each budget of the hospitals by a certain amount. Their mechanisms find a “nearby” instance with a stable matching for each instance of a matching problem. This paper focuses on approximately stable matchings where the participants are willing to change the assignments only for a multiplicative improvement of a certain amount (Arkin et al. 2009). This idea can be interpreted as one in which, a hospital in a blocking pair changes his match as soon as his utility after the change increases by any (arbitrarily small) amount. Arkin et al. (2009) examine a stable roommate problem, which is a non-bipartite, one-to-one matching problem, while we examine a bipartite, one-tomanymatching problem. It would be enough reasonable that a hospital would change his assignment only in favor of a significant improvement (the grass may be greener on the other side, but it takes effort to cross the fence). Furthermore, it must be emphasized that those studies except the works by Abizada (2016) and Kawase and Iwasaki (2017) discuss no strategic issue, i.e., misreporting a doctor’s preference may be profitable. The literature on matching has found strategy-proofness for doctors, i.e., each doctor has no incentive to misreport her preference, to be a key property in a wide variety of settings (Abdulkadiroğlu and Sönmez 2003). The contribution of this paper is twofold. First, we modTable 1: Summary of the Results: UB and LB stand for upper and lower bounds, respectively. Let s be the maximum fraction of a wage to budget among the given contracts and |D| be the number of doctors. (a) Non-Strategy-Proof mechanisms: Thm. 2 holds if s > 1/2. Hosp.’s Utils Approximation Ratio Additive UB: 1 1−s (Alg. 2 and Thm. 4) (incl. Proportional) LB: 1 1−s (Thm. 2) Proportional UB: 1.62 (Alg. 4 and Thm. 7) LB: 1.62 (Thm. 6) (b) Strategy-Proof mechanisms. Hosp.’s Utils Approximation Ratio Additive UB: ⌈ 1+ln(|D|−1) 1−s ⌉ (Alg. 3 and Thm. 5) (incl. Proportional) LB: open Proportional UB: 1 1−s (Alg. 5 and Thm. 8) LB: open ify the generalized DA algorithm and devise a new property, which we call α-approximation, on the matching with contract framework (Hatfield and Milgrom 2005). This is because the existing class of mechanisms and properties are not sufficient to characterize a choice function that produces such an approximately stable matching. Second, we propose two novel mechanisms that efficiently return such a stable matching that exactly satisfies the budget constraints. In particular, by sacrificing strategy-proofness, the best possible bound is achieved. We further examine a special case where each hospital has a utility that is proportional to its size of each contract. We find two mechanisms achieving better approximation ratios and, in particular, by sacrificing strategyproofness, the ratio is bounded by a constant. Table 1 summarizes the results on approximation ratios.