We have studied the problem of finding a constrained welfare-maximizing allocation, that is, a Pareto-efficient allocation of highest welfare. This problem is NP-hard even under strong conditions, but we have also identified settings in which the problem is polynomial-time solvable. For the general problem, we formulated an integer program. We used this program to solve a real-world instance of kindergarten admissions in which edge weights represent travel distance, and it was quick to determine the solution. However, there are still many open questions left to study.
Implicit throughout has been that Pareto-efficiency takes precedence over welfare. (Under aligned interests or agent-based weights, the order is irrelevant: we can attain Pareto-efficiency and welfare-maximization simultaneously.) That is, the planner wants to select the Pareto-efficient allocation of highest welfare. A different approach is to define a measure of how Pareto-efficient an allocation is and then select a welfare-maximizing allocation of “highest Pareto-efficiency”. This is left for future research.
Also outside the scope of the current paper, an interesting extension is to equip objects with
“standard” priorities over agents and, say, find the stable allocation of highest welfare. Extending the model is unlikely to overturn the negative results but can be interesting for the positive results.
For instance, an extension of the condition imposed in Theorem 7, aligned interests, is to align the added priorities with the edge weights.
Another extension is to allow agents to receive multiple objects. In this case, it is already challenging to elicit the preferences of the agents over the bundles, as the number of bundles is exponential in the number of objects. This is often resolved by eliciting ordinal or cardinal prefer-ences over single objects, and then extending this to preferprefer-ences over bundles through, for instance, responsiveness or additivity. However, in the cardinal setting it can already be computationally hard to decide whether an allocation is Pareto-efficient (Aziz et al., 2016). But in line with our positive results in Section 3, finding constrained welfare-maximizing solutions may be still tractable under some conditions.
Appendix A. Solutions: Definitions and algorithms
In this section, we provide more formal definitions of the solutions referred to throughout the paper. For the reader interested in learning more, we refer to the excellent surveys by Manlove (2013) and Haeringer (2017).
We adopt the terminology of Section 5 and refer to agents as students and objects as seats at schools. We impose some restrictions compared to the model introduced in Section 2. First, every school is acceptable to every student. Second, students have strict preferences over schools (no indifferences). Third, each school ahasqa seats (its “quota”) and there are sufficiently many seats in total to assign every student. Finally, weights represent the distances between the students and the schools as in Section 5.
Most of the mechanisms are priority-based in the sense that students have different priorities at different schools. We examine two types of priorities. First, “distance-based priorities” give higher priority to students living closer to schools. Second, “distance-based priorities adjusted by allocationx” prioritize students assigned to the school underxto those not and otherwise prioritize on distance. As an example, suppose that studentsi,j, and klive100,200, and 1000meters from school a. Distance-based priorities yield the order i, j, k with student i given the highest priority.
Suppose further that the allocationxassigns both studentsjandk, but noti, to schoola. Adjusting the distance-based priorities to x then results in the order j, k, i. In what follows, priorities are adjusted for two reasons. First, by adjusting priorities toxand then executing Top Trading Cycles, we obtain a measure of how far from Pareto-efficient x is. Second, by adjusting priorities to the welfare-maximizing allocation, we can study solutions that “lie between” the original solution (say, Deferred Acceptance) and the welfare-maximizing solution. This will be used to define the new Optimal Priority Deferred Acceptance.
Top Trading Cycles. This mechanism is defined using a directed bipartite graph. Nodes are given by the students and schools. Each student has an outgoing arc to her most preferred school; each
school has an outgoing arc to the student with highest priority at the school. The graph contains at least one cycle. Each student in the cycle is assigned to the school she points to and then removed from the graph. Similarly, school a is removed from the graph once it has filled its quota qa. In succeeding rounds, students previously pointing to school a point to their most preferred school among those that remain in the graph. Schools redirect their arcs similarly to the remaining student with highest priority. If there are several cycles at some stage, the cycles not selected remain cycles in the subsequent round. Therefore, the outcome is independent of the order in which the cycles are selected.
In Section 5 and Appendix B, we report values on “swaps in post-TTC”. These are computed as follows. We adjust priorities (as described above) to whichever allocation that we are examining, and we then executeTop Trading Cycles. Whenever we process a cycle of at least two students, we keep track of how many students change schools. The number of “swaps in post-TTC” is the total number of changes across all cycles.
Immediate Acceptance. In this mechanism, students “propose” to their most preferred schools. If a school can seat all its proposers, then it does so. A school receiving more proposal than it has seats “immediately accepts” the proposals from the students with highest priority and rejects the others. All rejected students proceed to propose to their most preferred school that still has vacant seats (“Immediate Acceptance with Skips” in Harless, 2019).
Deferred Acceptance. This operates asImmediate Acceptancewith the exception that the allocation is not made final until at the end of the algorithm. Students again propose to their most preferred schools. Each school then tentatively accepts proposers up to its quota. Rejected students proceed to propose to their most preferred school that has not yet rejected them. Each school then, again, tentatively accepts proposers up to its quota–choosing both from those newly proposing and those tentatively accepted in the previous round. In this way, a school may tentatively accept a student at first only to reject the student at a later stage.
Optimal Priority Deferred Acceptance. 7
This mechanism is implemented in two steps. First, we compute the welfare-maximizing allo-cation and adjust the priorities on the basis of it. That is to say, students assigned school aunder the welfare-maximizing allocation are prioritized over those not; otherwise, students are prioritized on distance. From a practical perspective, this step can be undertaken as soon as the geograph-ical distribution of the students is known (for larger cities, the distribution likely only changes slowly over time). The allocation can then be announced to the students in the form of “catch-ment areas” (compare Figure B.7 in Appendix B). That is, we announce to the students that there has been an initial assignment according to the catchment areas. Students are guaranteed
7We are grateful to Philippe Jehiel for suggesting this mechanism to us.
seats at their respective schools, but if they prefer to relocate elsewhere, they are given the option to submit preferences over preferred schools. At that point, Deferred Acceptance is executed and identifies Pareto-improvements over the initial assignment. Put succinctly, Optimal Priority De-ferred Acceptance is Deferred Acceptance executed on the instance with priorities adjusted to the welfare-maximizing allocation.
The mechanism results in an allocation that is stable with respect to the adjusted priorities.
Recall that a stable allocation is one that refuses a student a preferred school only if all its seats are assigned to higher-priority students. In this case, if studentiprefers school aover her assignment, then a is filled with students who all either are from a’s catchment area or who live closer to a thanidoes. It is immediate thatOptimal Priority Deferred Acceptance, alike Deferred Acceptance, cannot be manipulated. Moreover, as both finding the welfare-maximizing allocation and executing Deferred Acceptance can be done in polynomial time, we quickly obtain the solution. Among its drawbacks, the solution is neither Pareto-efficient nor stable with respect to the original distance-based priorities.
Alternatively, the solution can be implemented by first having the students report their pref-erences, then adjusting the priorities for the welfare-maximizing allocation, and finally executing Deferred Acceptance. In this case, it is important that the preferences do not affect the welfare-maximizing allocation. Specifically, even if studentireports schoolaas unacceptable,ishould still be allowed to be assigned to awhen the welfare-maximizing allocation is determined. If not, if we look for the welfare-maximizing allocation that assigns students to acceptable schools only, then students will be able to manipulate the solution. This is illustrated in Example 6, which also shows that Optimal Priority Deferred Acceptance may unnecessarily leave students unassigned. To sum-marize, provided that the welfare-maximizing allocation is computed independently of the reported preferences, it does not matter for the solution and its strategic properties whether we collect pref-erences before or after we determine priorities. However, the benefit of first determining priorities is that it assigns each student a guaranteed school. The students then only need to report schools preferred to the guaranteed school. In the Estonian case, 80 out of 152 families need not report anything–they are assigned their preferred school under WM. Out of the remaining 72 families, 30 only have to report their top choice. In contrast, if we collect preferences first, then all students may have to report complete preferences.
Example 6. Consider students 1 and 2 together with schools aand b arranged along a line as in Figure A.6. The parents of1 work “to the right” and only accept dropping of1at school b. Hence, ais unacceptable. The parents of 2 work “to the left” and preferb slightly overa. Still,aremains acceptable. We assume further that 1lives closer tob than2 does.
a 1 b 2
Figure A.6: Students and schools in Example 6.
Consider first the welfare-maximizing solution that ignores the acceptability constraints. It selects the allocation {(1, a),(2, b)}. When we adjust priorities accordingly and execute Deferred Acceptance, student 2 prefers and has highest priority at b. Hence, we assign2 to b. As Deferred Acceptancedoes not assign students to unacceptable schools, student1remains unassigned. Hence, Optimal Priority Deferred Acceptance selects{(2, b)}.
Consider next the welfare-maximizing solution that only assigns students to acceptable schools.
Assuming this is set up to include a large cost to leaving a student unassigned, it selects the allocation{(1, b),(2, a)}, which also is the output of the following execution ofDeferred Acceptance.
Hence,Optimal Priority Deferred Acceptance selects{(1, b),(2, a)}.
Consider finally the case when student 1 prefers school b but is willing to accept both schools.
The welfare-maximizing allocation will then be as in the first case, {(1, a),(2, b)}, which also will be the final output ofOptimal Priority Deferred Acceptance. In this way, if the first step ofOptimal Priority Deferred Acceptance is set up to only assign students to acceptable schools, then student
1 can manipulate by reporting thatais unacceptable. ◦
Appendix B. Simulations
We run a simulation study to further compare the different solutions.8Each instance is composed of 1,000 students and 10 schools with 100 seats each. Every student is acceptable to every school, so all solutions output complete allocations in which no student is left unassigned. Geographical coordinates for the students and schools are drawn randomly within the unit circle through a distance (uniformly from [0,1]) and angle (uniformly from 0 to 360 degrees) from the center.
School priorities are distance-based, giving higher priority to student living closer (measured by the Euclidean distance). Weights are set up as in Section 5: the edge weight between studentiand school ais w(i, a) =D−d(i, a), where D is the maximum distance in the data andd(i, a) is the distance between iand a. Finally, each school is assigned a quality level uniformly from [0,1].
Student preferences are derived through a linear combination of school quality κ, distance d, and a random noise term ε. The utility that student i derives from being assigned to school a is positively affected by the quality of aand negatively by the distance:
u(i, a) =ακ·κ(a)−αd·d(i, a) +αn·ε(i, a).
The parametersακ,αd, andαncontrol how much weight is put on quality, distance, and the random noise, respectively. As an example, with αd = 1 and ακ = αn = 0, preferences are completely distance-based and we obtain the case of aligned interests. If instead ακ = 1 and αd = αn = 0, we obtain the case of common preferences. Finally, with ακ =αd= 0 and αn = 1, preferences are
8The simulations are run in Python 3 with the free open source software PuLP using its default CBC solver.
uncorrelated with distance and school quality.
In Table B.3, we report the simulation results for three different parameters settings, in each case averaged across 100 instances. The top rows labelled “distance” refer to primarily distance-based preferences with parametersαd= 3/5 andακ=αn= 1/5. The middle rows, “quality”, refer to primarily quality-based preferences with parametersαd=αn= 1/5and ακ= 3/5. For the final rows, “random”, we set αd=ακ=αn= 1/3. Among the results in Table B.3, we wish to highlight the following:
1. The average distance is very similar for CWM and OPDA under all three parameter settings and always considerably better than for DA, IA, and TTC. This is in some contrast to Section 5, in which OPDA and DA produced similar results while CWM significantly reduced the average distance.
2. The average preference ranks are considerably worse than in Section 5. That is, in the Estonian case study, more students are assigned their top schools. The fraction of blocking agents is considerably higher here than in Section 5. To explain for instance the high number for TTC under quality-based preferences, imagine that studentiliving close to popular schoolawishes to go elsewhere–possibly to the distant schoolb. In the cycle that assigns itob, a student j living close tobmay get assigned to the popular school a. Due to the limited number of seats at a, a lot of students living closer to a than i does will not be assigned toa and therefore block the allocation.
3. As in Section 5, the average preference rank is often lowest for IA. Among the other solutions, CWM typically assigns students to more preferred schools than OPDA does. Still, OPDA is an improvement over WM.
4. The average distance rank is almost identical for CWM and OPDA.
5. In terms of number of Pareto-improving swaps, OPDA is an improvement over WM but considerably worse than DA.
Figure B.7 graphically illustrates the relation between WM, OPDA, and the students’ preferred schools for a simulated instance. In the left figure, comparing WM and OPDA, we can note the Pareto-improving swaps between the blue schools as well as the yellow and dark schools. These students live closer to the school they are assigned under WM but prefer the more distant school that they are assigned under OPDA. In the right figure, we see that the pink school is of high quality.
Therefore, students who live close to it also are likely to prefer it. Confirming this intuition, WM and OPDA coincide for the pink school in the left figure.
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Stable Pareto-efficient
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Distance
Average preference rank 2.16 2.09 2.15 1.93 2.05 2.12 Average distance rank 2.27 2.31 2.34 1.76 1.74 1.7 Average distance 36,758 36,922 37,351 31,580 31,443 31,203
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Swaps in post-TTC 11 0 0 0 112 158
Quality Average preference rank 4.17 3.89 4.08 4.1 4.27 4.37 Average distance rank 2.5 2.82 2.83 1.84 1.81 1.74 Average distance 39,072 42,274 42,331 32,565 32,375 31,745
Blocking agents 0 422 606 642 639 649
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Average preference rank 2.76 2.56 2.69 2.68 2.96 3.38
Average distance rank 2.59 2.84 3.03 2 1.98 1.77
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