Fuzzy RDD

In a fuzzy RDD, crossing the Cutoff changes the probability of treatment but does not deterministically assign it. The first-stage jump in treatment probability is less than 1: some units above the cutoff don't take the treatment, some below it do (e.g. scholarship offers that recipients can decline). The fuzzy design is estimated as IV with the cutoff dummy as the instrument: the Wald Estimator (jump in outcome / jump in treatment probability) is the LATE for compliers near the cutoff.

When to use

Use fuzzy RDD when the cutoff is "encouraging" rather than mandatory: eligibility for a program that some eligible people skip, age limits with exemptions, financial-aid offers that families can decline. The two extra requirements vs sharp RDD: a meaningful jump in treatment probability at the cutoff (analogue of instrument relevance) and the exclusion restriction (cutoff affects $Y$ only through the treatment).