Weighted Matching Markets with Budget Constraints

Abstract

We investigate markets with a set of students on one side and a set of colleges on the other. A student and college can be linked by a weighted contract that defines the student's wage, while a college's budget for hiring students is limited. Stability is a crucial requirement for matching mechanisms to be applied in the real world. A standard stability requirement is coalitional stability, i.e., no pair of a college and group of students has any incentive to deviate. We find that a coalitionally stable matching is not guaranteed to exist, verifying the coalitional stability for a given matching is coNP-complete, and the problem of finding whether a coalitionally stable matching exists in a given market, is SigmaP2-complete: NPNP-complete. Other negative results also hold when blocking coalitions contain at most two students and one college. Given these computational hardness results, we pursue a weaker stability requirement called pairwise stability, where no pair of a college and single student has an incentive to deviate. Unfortunately, a pairwise stable matching is not guaranteed to exist either. Thus, we consider a restricted market called a typed weighted market, in which students are partitioned into types that induce their possible wages. We then design a strategy-proof and Pareto efficient mechanism that works in polynomial-time for computing a pairwise stable matching in typed weighted markets.

Matching Market Design with Constraints

“…Next, we consider related models in which budgetary constraints are involved that are essentially single-dimensional constraints. Ismaili et al (2019) considered the case where hospitals hire doctors. To hire a doctor, a hospital need to pay a certain wage, which can vary according to the expertise of the doctor.…”

Matching Algorithms: Fundamentals, Applications and Challenges

“…Many-to-one Matching - [7], [11], [28], [29], [30] job matching [31], [32], [33] pay matching Many-to-many Matching Optimization algorithms [6], [34], [35], [36] cognitive radio networks; D2D communications…”

“…Gale and Shapley [1] first proposed the concept of matching theory in 1962, which is a mathematical framework based on the game theory, and applied it to the marriage market and college admission. Since then, this theory has attracted the interests of a panoply of economists as it can solve many matching problems where the money is not involved [10], [11]. When only one side of the agents has a preference list, it is regarded as one-sided matching, while matching with preference lists on both sides is a two-sided matching.…”

Cutoff stability under distributional constraints with an application to summer internship matching

“…There is also work on matching with budget constraints (see, e.g., Iwasaki [2017, 2018], Ismaili et al [2019]). The models considered in these papers are different in several respects.…”