Optimizing the tie-breaker regression discontinuity design

Motivated by customer loyalty plans and scholarship programs, we study tie-breaker designs which are hybrids of randomized controlled trials (RCTs) and regression discontinuity designs (RDDs). We quantify the statistical efficiency of a tie-breaker design in which a proportion $\Delta$ of observed subjects are in the RCT. In a two line regression, statistical efficiency increases monotonically with $\Delta$, so efficiency is maximized by an RCT. We point to additional advantages of tie-breakers versus RDD: for a nonparametric regression the boundary bias is much less severe and for quadratic regression, the variance is greatly reduced. For a two line model we can quantify the short term value of the treatment allocation and this comparison favors smaller $\Delta$ with the RDD being best. We solve for the optimal tradeoff between these exploration and exploitation goals. The usual tie-breaker design applies an RCT on the middle $\Delta$ subjects as ranked by the assignment variable. We quantify the efficiency of other designs such as experimenting only in the second decile from the top. We also show that in some general parametric models a Monte Carlo evaluation can be replaced by matrix algebra.