Weighted Matching Markets with Budget Constraints

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 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 to find whether a coalitionally stable matching exists in a given market, is $\text{NP}^\text{NP}$-complete (that is $\Sigma_2^P$-complete). Given these computational hardness results, we pursue a weaker stability requirement called pairwise stability, i.e., no pair of a college and single student has incentive to deviate. We then design a strategy-proof mechanism that works in polynomial-time for computing a pairwise stable matching in typed markets in which students are partitioned into types that induce their possible wages.