# Leverage of Win Probabilities

#### Introduction

As defined by the MLB site, the Leverage Index measures the importance of a particular play event by quantifying how much the win probability can change as a result of the event. High leverage situations correspond to exciting moments for the baseball fan. As described by Tom Tango in this 2006 post, the leverage of a particular game state (defined by inning, runners on base, and score) is found by

- considering all of the possible plays during that situation
- computing an average change or swing of win probabilities where the changes are weighted by the frequencies of their occurrence.
- since this value of this average swing is easy to understand at face value, one computes a leverage index where one divides the average swing value by the average swing value in a neutral situation. So a leverage index larger than 1 can be interpreted as a “high” leverage situation, and a value less than 1 can be thought as a low leverage situation

Actually, it is easy to compute these leverage values from Retrosheet play-by-play data. For each play, one has associated run values and these can be converted to WPA values that indicate how the home team’s win probability has changed based on the play. (I talk about computing win probabilities in a recent post.) What is the leverage of a particular situation, say runners on 1st and 2nd with one out in the bottom of the eight inning? An average change in WPA values is just a standard deviation calculation. One collects the WPA values for all plays in that particular situation in the 2018 season, and one computes

Leverage = sd(abs(WPA)),

the standard deviation of the sizes of the WPA values. Here we’ll explore how these leverage values change as a function of the inning, bases/outs situation, and score. Also we’ll list the situations that have the highest leverages, and the hitters who have the top and bottom leverage values. Since we are focusing on patterns of these values, it is not necessary to change these to leverage index values.

#### Leverages Across Innings

First we explore how leverage (as defined by sd(abs(WPA))) changes across innings — in the below graph I distinguish the top of the inning (BAT_HOME_ID = 0) from the bottom of the inning (BAT_HOME_ID = 1). As one might expect, the leverage increases as the game progresses. One interesting thing is that the bottom of the late innings have unusually high leverages.

#### Leverages Across Runner/Outs States

Next we see how leverages change as a function of the runners on base and number of outs. As one might expect, there are high leverages with runners on base — when you think of high leverage, think of the despair when a team leaves runners in scoring position on the bases and the joy when the runners are brought home. The least exciting situation is when there are no runners on base — there is little variation in the size of the WPA.

#### Leverages for Different Scores

Also, leverage values are dependent on the current score. The following graph shows that the leverage values are highest for tied games or games where the margin is 1 or 2 runs. Situations where one team has a commanding lead will have little leverage.

#### The Most Exciting Situations

Putting this all together, the high leverage situations correspond to late in the game, runners on base, and close games where the lead is small. These are the situations when a fan wants to stay in his/her seat. To confirm this, I have computed the leverages for all situations defined by these three factors. The table shows the top 15 leverage situations. The top one, interestingly, is bases loaded with 2 outs and the leading margin is 3 runs. Looking down the list, they are primarily 8 or 9th innings, most with 2 outs, and 1-3 winning margins.

#### Leaders in Volatility and Stability

Here’s a new leaderboard (maybe not, but I doubt you see it in Baseball Reference). I have looked at all 2018 players with at least 500 PA and I have ranked the top 10 with respect to leverage. Denard Span is the leader — he was the most volatile hitter in that he had the greatest spread in absolute WPA. (It would be interesting to explore Span’s hitting data in detail to better understand this large leverage value.)

A the other extreme, here are the 2018 leaders with respect to the smallest standard deviation of the absolute WPA. Lorenzo Cain is the leader and Mike Trout is number 7. These players are the stable ones in the sense that their WPA values have low spread. I think of myself as a consistent tennis player that keeps the ball in play, so I appreciate these players that show consistency in their PA results. Of course, consistency is not necessarily a good thing — consistently bad hitters will have short stints in MLB

#### Why Do We Care?

I think baseball teams are interested in these leverage measures. One reason is that leverage provides guidance when one should play their relief pitchers. The best relief pitcher should be put in a innings/bases/score situation where the leverage is high. On the other hand, if your team has a large lead in the bottom of the 9th, the leverage value is small and you don’t need to put in your best reliever. Also, I think managers would like consistent hitters in the top of their lineup — a team consisting solely of volatile hitters may have more scoring droughts than a team with consistent hitters. In any event, I believe leverage plays an important role in understanding the hills and valleys in a baseball game.

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