Stability and incentives for college admissions with budget constraints

Abstract

We study two‐sided matching where one side (colleges) can make monetary transfers (offer stipends) to the other (students). Colleges have fixed budgets and strict preferences over sets of students. One different feature of our model is that colleges value money only to the extent that it allows them to enroll better or additional students. A student can attend at most one college and receive a stipend from it. Each student has preferences over college–stipend bundles. Conditions that are essential for most of the results in the literature fail in the presence of budget constraints. We define pairwise stability and show that a pairwise stable allocation always exists. We construct an algorithm that always selects a pairwise stable allocation. The rule defined through this algorithm is incentive compatible for students: no student should benefit from misrepresenting his preferences. Finally, we show that no incentive compatible rule selects a Pareto‐undominated pairwise stable allocation.

Weighted Matching Markets with Budget Constraints

“…. This result is stark contrast to the model presented in(Abizada, 2016), in which a pairwise stable matching is guaranteed to exist.…”

“…Since every possible matching admits a blocking pair, there is no pairwise stable matching in Example 3. This result highlights the difference between our model and the model used in (Abizada, 2016), in which a pairwise-stable matching is guaranteed to exist. In our model, we allow different types of students to be in a market.…”

“…Abizada (2016) presented a strategy-proof mechanism that works for a different model as described in Section 1, where each college c has its budget limit b c and maximum wage m c , and all students are the same type θ. The mechanism proposed by Abizada (2016) can be considered as a special case of our component DA mechanism, where possible wages W c,θ is restricted to {w c 1 , w c 2 , 0} where w c 1 = m c , w c 2 = b c mod m c .…”

“…Although our model is a natural extension of standard maximum quotas, very little literature has addressed this issue, possibly due to its intractability; two conditions from the literature, substitutability and the law of aggregate demand, make analysis tractable (Hatfield & Milgrom, 2005), but neither is satisfied in our model. The most relevant work (Abizada, 2016) studies college admissions with budget constraints, in which a student receives a pair of a college and a stipend, and develops a strategy-proof mechanism that satisfies a weaker notion of stability. The major differences between this model and ours are that we deal with general ordinal (instead of quasi-linear) preferences of students, which becomes possible since we focus on discrete sets of wages (instead of a continuum) and allow different types of students to exist in a market (instead of assuming all students are of the same type).…”

Near-Feasible Stable Matchings with Budget Constraints

“…To date, 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.…”