# Stability and the immediate acceptance rule when school priorities are weak

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## Abstract

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.

### Keywords

School choice Stability Immediate acceptance### JEL Classification

C78 D71 D78### References

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