A general characterization of optimal tie-breaker designs

In a regression discontinuity design, subjects with a running variable $x$ exceeding a threshold $t$ receive a binary treatment while those with $x\le t$ do not. When the investigator can randomize the treatment, a tie-breaker design allows for greater statistical efficiency. Our setting has random $x\sim F$, a working model where the response satisfies a two line regression model, and two economic constraints. One constraint is on the expected proportion of treated subjects and the other is on how treatment correlates with $x$, to express the strength of a preference for treating subjects with higher $x$. Under these conditions we show that there always exists an optimal design with treatment probabilities piecewise constant in $x$. It is natural to require these treatment probabilities to be non-decreasing in $x$; under this constraint, we find an optimal design requires just two probability levels, when $F$ is continuous. By contrast, a typical tie-breaker design as in Owen and Varian (2020) uses a three level design with fixed treatment probabilities $0$, $0.5$ and $1$. We find large efficiency gains for our optimal designs compared to using those three levels when fewer than half of the subjects are to be treated, or $F$ is not symmetric. Our methods easily extend to the fixed $x$ design problem and can optimize for any efficiency metric that is a continuous functional of the information matrix in the two-line regression. We illustrate the optimal designs with a data example based on Head Start, a U.S. government early-childhood intervention program.