The Efficiency Adjusted Deferred Acceptance Rule (EDA) is a promising candidate mechanism for public school assignment. A potential drawback of EDA is that it could encourage students to game the system since it is not strategy-proof. However, to successfully strategize, students typically need information that is unlikely to be available to them in practice. We model school choice under incomplete information and show that EDA is regret-free truth-telling, which is a weaker incentive property than strategy-proofness and was introduced by Fernandez 2020. We also show that there is no efficient matching rule that weakly Pareto dominates a stable matching rule and is regret-free truth-telling. Note that the original version of EDA by Kesten 2010 weakly Pareto dominates a stable matching rule, but it is not efficient.