Battal Dogan

Responsive affirmative action in school choice

School choice programs aim to give students the option to choose their school. At the same time, underrepresented minority students should be favored to close the opportunity gap. A common way to achieve this is to have a majority quota at each school, and to require that no school be assigned more majority students than its majority quota. An alternative way is to reserve some seats at each school for the minority students, and to require that a reserve seat at a school be assigned to a majority student only if no minority student prefers that school to her assignment. However, fair rules based on either type of affirmative action suffer from a common problem: a stronger affirmative action may not benefit any minority student and hurt some minority students. First, we show that this problem is pervasive. Then, we uncover the root of this problem: for some minority students, treating them as minority students does not benefit them, but possibly hurts other minority students. We propose a new assignment rule that treats such minority students as majority students, achieves affirmative action, and never hurts a minority student without benefiting another minority student.

Efficiency and stability of probabilistic assignments in marriage problems

We study marriage problems where two groups of agents match to each other and probabilistic assignments are possible. When only ordinal preferences are observable, an extensively studied efficiency notion is stochastic dominance efficiency (sd-efficiency). First, we provide a characterization of sd-efficient allocations in terms of a property of an order relation defined on the set of man-woman pairs. Then, using this characterization, we constructively prove that for each probabilistic assignment that is sd-efficient for some ordinal preferences, there is a von Neumann-Morgenstern utility profile consistent with the ordinal preferences for which the assignment is Pareto efficient. Next, we analyze stability of probabilistic assignments. We show that, when the preferences are strict, for each probabilistic assignment that is ex-post stable for some ordinal preferences, there is a von Neumann-Morgenstern utility profile consistent with the ordinal preferences such that the assignment belongs to the core: no coalition can deviate to another probabilistic assignment among themselves and achieve a higher total expected utility.

Responsive Affirmative Action in School Choice

School choice programs aim to give students the option to choose their school. At the same time, underrepresented minority students should be favored to close the opportunity gap. A common way to achieve this is to have a majority quota at each school, and to require that no school be assigned more majority students than its majority quota. An alternative way is to reserve some seats at each school for the minority students, and to require that a reserve seat at a school be assigned to a majority student only if no minority student prefers that school to her assignment. However, fair rules based on either type of affirmative action suffer from a common problem: a stronger affirmative action may not benefit any minority student and hurt some minority students. First, we show that this problem is pervasive. Then, we uncover the root of this problem: for some minority students, treating them as minority students does not benefit them, but possibly hurts other minority students. We propose a new assignment rule that treats such minority students as majority students, achieves affirmative action, and never hurts a minority student without benefiting another minority student.

A New Ex-Ante Efficiency Criterion and Implications for the Probabilistic Serial Mechanism

We introduce and analyze an efficiency criterion for probabilistic assignment of objects, when only ordinal preference information is available. This efficiency criterion is based on the following domination relation: a probabilistic assignment dominates another assignment if it is ex-ante efficient for a strictly larger set of utility profiles consistent with the ordinal preferences. We provide a simple characterization of this domination relation. We revisit an extensively studied assignment mechanism, the Probabilistic Serial mechanism (Bogomolnaia and Moulin, 2001), which always chooses a “fair” assignment. We show that the Probabilistic Serial assignment may be dominated by another fair assignment. We provide conditions under which the serial assignment is undominated among fair assignments.

Eliciting the Socially Optimal Allocation from Responsible Agents

We consider designing a mechanism to allocate objects among agents without monetary transfers. There is a socially optimal allocation, which is commonly known by the agents but not observable by the designer. The designer possibly has information about the existence of responsible agents. A responsible agent, when indifferent between his objects at two different allocations, prefers the first allocation to the second if the first allocation is closer to the optimal allocation than the second, in the sense that all the agents who are allocated their optimal objects in the second allocation are allocated their optimal objects also in the first allocation, and there is at least one more agent in the first allocation receiving his optimal object. We show that, if the designer knows that there are at least three responsible agents, even if the identities of the responsible agents are not known, the optimal allocation can be elicited.

On Acceptant and Substitutable Choice Rules

Each capacity-filling and substitutable choice rule is known to have a maximizer-collecting representation: there exists a list of priority orderings such that from each choice set that includes more alternatives than the capacity, the choice is the union of the priority orderings’ maximizers (Aizerman and Malishevski, 1981). We introduce the notion of a critical set and constructively prove that the number of critical sets for a choice rule determines its smallest size maximizer-collecting representation. We show that responsive choice rules require the maximal number of priority orderings in their smallest size maximizer-collecting representations among all capacity-filling and substitutable choice rules. We also analyze maximizer-collecting choice rules in which the number of priority orderings equals the capacity. We show that if the capacity is greater than three and the number of alternatives exceeds the capacity by at least two, then no capacity-filling and substitutable choice rule has a maximizer-collecting representation of the size equal to the capacity.

Stability and the Immediate Acceptance Rule When School Priorities are Weak

In a model of school choice, we allow school priorities to be weak and study the preference revelation game induced by the immediate acceptance (IA) rule (also known as the Boston rule), or the IA game. When school priorities can be weak and matches probabilistic, three stability notions—ex post stability, ex ante stability, and strong ex ante stability—and two ordinal equilibrium notions—sd equilibrium and strong sd equilibrium—become available (“sd” stands for stochastic dominance). We show that for no combination of stability and equilibrium notions does the set of stable matches coincide with the set of equilibrium matches of the IA game. This stands in contrast with the existing result that the two sets are equal when priorities are strict. We also show that in the presence of weak priorities, the transition from the IA rule to the deferred acceptance rule may, in fact, harm some students.

Maskin-monotonic scoring rules

We characterize which scoring rules are Maskin-monotonic for each social choice problem as a function of the number of agents and the number of alternatives. We show that a scoring rule is Maskin-monotonic if and only if it satisfies a certain unanimity condition. Since scoring rules are neutral, Maskin-monotonicity turns out to be equivalent to Nash-implementability within the class of scoring rules. We propose a class of mechanisms such that each Nash-implementable scoring rule can be implemented via a mechanism in that class. Moreover, we investigate the class of generalized scoring rules and show that with a restriction on score vectors, our results for the standard case are still valid.