  As a baseball game progresses, batters appear to perform better the more
times they face a particular pitcher. The apparent drop-off in pitcher
performance from one time through the order to the next, known as the Time
Through the Order Penalty (TTOP), is often attributed to within-game batter
learning. Although the TTOP has largely been accepted within baseball and
influences many managers' in-game decision making, we argue that existing
approaches of estimating the size of the TTOP cannot disentangle continuous
evolution in pitcher performance over the course of the game from
discontinuities between successive times through the order. Using a Bayesian
multinomial regression model, we find that, after adjusting for confounders
like batter and pitcher quality, handedness, and home field advantage, there is
little evidence of strong discontinuity in pitcher performance between times
through the order. Our analysis suggests that the start of the third time
through the order should not be viewed as a special cutoff point in deciding
whether to pull a starting pitcher.
