Assignment Mechanisms under Distributional Constraints

Abstract

We generalize the serial dictatorship (SD) and probabilistic serial (PS) mechanism for assigning indivisible objects (seats in a school) to agents (students) to accommodate distributional constraints. Such constraints are motivated by equity considerations. Our generalization of SD maintains several of its desirable properties, including strategyproofness, Pareto optimality, and computational tractability while satisfying the distributional constraints with a small error.Our generalization of the PS mechanism finds an ordinally efficient and envy-free assignment while satisfying the distributional constraint with a small error. We show, however, that no ordinally efficient and envy-free mechanism is also weakly strategyproof. Both of our algorithms assign at least the same number of students as the optimum fractional assignment. s 1 is used to resolve this partial assignment. After the (i 4 , s 1 )-updates the constraints are:In the next assignment step, we have f (t 5 , s 1 ) = 0.5 and i 5 is partially assigned to s 1 . In the resolution step, s 2 is used to resolve this assignment. After the (i 5 , s 2 ) updates, the constraints are:Next, we have f (t 1 , s 1 ) = 0, and f (t 1 , s 2 ) = 0 and therefore i 6 is assigned to the outside option.Finally, when i 7 is about to get assigned, we have f (t 2 , s 1 ) = 0.5 and therefore i 7 is partially assigned to s 1 . In the following resolution step, we use the outside option, i.e. φ to resolve this partial assignment. After the (i 7 , φ)-updates, the constraints are:2 ≤ x t 1 ,

The Vigilant Eating Rule: A General Approach for Probabilistic Economic Design with Constraints

“…This approach works if there are upper quotas on agent-object groups that form a bihierarchical constraint structure; it does however not extend to lower quotas on agent-object groups or constraint structures that do not form a bihierarchy. Ashlagi et al (2019b) consider probabilistic allocation with lower and upper bounds on the types of students for each school without priorities and give a suitable generalization of the probabilistic serial. Our model captures the constraints considered by Budish et al (2013), Fujishige et al (2018), andAshlagi et al (2019b), which are all specified by lower and upper quotas or submodular constraints.…”

“…Ex ante constraints can include imposing ex ante stability requirements that a system designer may want to impose on the outcome. In that case, a probabilistic allocation is feasible if it satisfies the ex ante constraints and can be decomposed into deterministic allocations that meet the ex post constraints (see, for example, Ashlagi, Saberi, and Shameli, 2019b;Akbarpour and Nikzad, 2020).…”

Assignment Maximization

“…Irving and Manlove (2010) consider the problem of finding stable matchings with maximum cardinality when priorities have ties, which is known to be an NP-hard problem, and present heuristics for finding them. Ashlagi et al (2020) also consider object assignment problems under distributional constraints. The authors show that variants of serial dictatorship and Probabilistic Serial (Bogomolnaia and Moulin, 2001) mechanisms assign at least as many agents as one can match under the constraints, while the violations of the constraints are relatively small.…”

The Probabilistic Serial and Random Priority Mechanisms with Minimum Quotas

“…Katta and Sethuraman (2006) showed that when agents are allowed to report indifferences in their preferences, there is no mechanism which is ordinally efficient, envy-free, and weakly strategy-proof. In a recent work, Ashlagi et al (2020) consider the assignment of students to schools under distributional constraints, where each school imposes quotas on subsets of students according to their type. They generalise the PS mechanism based on the same underlying principle of appropriately restricting the menu offered to students, and show that there is no ordinally efficient, within-type envy-free and weakly strategy-proof mechanism in their setting.…”

“…They generalise the PS mechanism based on the same underlying principle of appropriately restricting the menu offered to students, and show that there is no ordinally efficient, within-type envy-free and weakly strategy-proof mechanism in their setting. With respect to Ashlagi et al (2020), we may treat the problem with upper and lower quota on objects as a middle ground between the classical random assignment problem and the random assignment problem with distributional constraints. It is therefore interesting to see that albeit we introduce significant constraints, the positive result of the existence of an ordinally efficient, envy-free, and weakly strategy-proof mechanism is retained.…”