A few years ago, I wrote an article called The One About Win Probability. It was meant to serve as a reference article, something that writers could easily link to when they wanted to explain WPA to their audience. It’s been pretty successful at that—I’m pretty sure it’s the most linked-to article in THT history. Several sites, such as Lookout Landing, link to it nearly every day.

But something has always bothered me about that article. I used Chris Shea’s Win Expectancy Finder as a hands-on way of demonstrating WPA in action. Chris’ application uses historical baseball games to show how win expectancy (another name for win probability) has actually played out during major league games, which is pretty cool.

However, there are some problems with that approach. Most importantly, there are “sample size” issues. For example, there are only 118 games in the dataset (all games from 1977 to 2006) that had a bottom-of-the-ninth situation, with the home team down by one, no outs and runners on the corners. The home team managed to win 76 times in the 118 games (64 percent of the time).

You can also find that the home team won 42 of 62 games (68 percent) in the same situation, but with runners on second and third instead of first and third. That might lead you to say that the WPA value of a stolen base is 4 percent. (68 percent minus 64 percent).

But that wouldn’t be quite right. As the following table shows, the “better” win probabilities are 72 percent and 66 percent, for a difference of 6 percent.

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

4.50 | Bottom of the ninth | Runners on first and third | 0 | -1 | 0.6610 |
0.3390 |
4.47 |

4.50 | Bottom of the ninth | Runners on second and third | 0 | -1 | 0.7237 |
0.2763 |
4.59 |

#### The Win Probability Added of the second situation is 0.0627

Where did I get this information? From the Hardball Times’ newest baseball tool, the WPA Inquirer.

The WPA Inquirer represents a step forward from Chris’ Win Expectancy Finder because it is based on the mathematical model behind the best implementations of WPA. When you use math, you don’t run into sample size issues. You can feel confident that each situation you investigate is the best interpretation of the WPA model.

You may think that the difference between 4 percent and 6 percent isn’t a lot, and you’d be right. Chris’ WE Finder is a mighty good tool. But if you were to track an entire game using the data in the WE Finder, you wouldn’t finish with a full win allocated to the winning team. That’s because there is no continuity between each historical situation—you’re pulling data from many different games. If you want something with more precision, use the WPA Inquirer.

Another reason the WPA Inquirer output is “better” is that history isn’t free of context. In the above examples, the historical win outcomes for the home team in the bottom of the ninth were actually lower than the theoretical win probability. That’s because opposing teams used their best closers in those situations, lowering the home teams’ ultimate success. Because the WPA Inquirer is built on straight math, it doesn’t assume anything about the opposing team’s strategy.

Plus, the WPA Inquirer has a few more bells and whistles. You might have noticed that we included the Leverage Index of the two above situations. Of course, the bottom of the ninth of a one-run game is a very critical situation, with an LI over 4.0 in these cases.

Another feature of the WPA Inquirer is that you can set the run environment. Chris’ WE Finder uses data from many years and many ballparks, each with their unique run environment. In the WPA Inquirer, you choose the run environment. As this table shows, it can make a real difference:

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

3.00 | Bottom of the ninth | Runner on first | 0 | 0 | 0.6779 |
0.3221 |
3.38 |

6.50 | Bottom of the ninth | Runner on first | 0 | 0 | 0.7480 |
0.2520 |
3.00 |

Same critical situation in the bottom of the ninth (no outs, runner on first, tie game), but the home team’s win probability is higher (seven points difference) in the higher run-scoring environment because they’re more likely to score.

Here’s another typical use of the WPA Inquirer, the sacrifice bunt. Most standard analytic tools will tell you that the sacrifice bunt isn’t usually a good move, and our new tool is no different. With a runner on first and no outs in the bottom of the ninth, a successful sacrifice actually lowers the home team’s chance of winning.

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

4.50 | Bottom of the ninth | Runner on first | 0 | 0 | 0.7112 |
0.2888 |
3.19 |

4.50 | Bottom of the ninth | Runner on second | 1 | 0 | 0.6981 |
0.3019 |
3.17 |

This is a superficial analysis of the sacrifice bunt, however. James Click did some very thorough analysis of bunting situations at Baseball Prospectus a couple of years ago, and MGL took the sacrifice bunt question even further in The Book. So you can use the WPA Inquirer for this sort of analysis, but don’t rely on it very much. The real answer depends on the quality of the batters at the plate and on deck.

Actually, there is (at least) one situation in which the WPA Inquirer does recommend a sacrifice bunt: tie game in the bottom of the ninth, no outs, runner on second:

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

4.50 | Bottom of the ninth | Runner on second | 0 | 0 | 0.8105 |
0.1895 |
2.58 |

4.50 | Bottom of the ninth | Runner on third | 1 | 0 | 0.8274 |
0.1726 |
4.72 |

#### The Win Probability Added of the second situation is 0.0169

You can also use the WPA Inquirer to confirm something you already suspected, that sacrifice bunts make more sense in a low-scoring environment. Here’s the same situation, but in a low-scoring environment:

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

3.00 | Bottom of the ninth | Runner on second | 0 | 0 | 0.7799 |
0.2201 |
2.85 |

3.00 | Bottom of the ninth | Runner on third | 1 | 0 | 0.8082 |
0.1918 |
5.43 |

#### The Win Probability Added of the second situation is 0.0283

As you can see, the WPA of the successful sacrifice bunt has a bigger impact when there are only three runs scored per game instead of 4.5.

Feel free to use the WPA Inquirer to explore the many nuances of WPA. According to WPA, for instance, a one-run home run that gives a visiting team a one-run lead is more important in the ninth inning than the first. Now you can calculate how much more. The first-inning homer is worth about ten WPA points:

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

4.50 | Top of the first | Bases empty | 0 | 0 | 0.5000 |
0.5000 |
0.87 |

4.50 | Top of the first | Bases empty | 0 | -1 | 0.4018 |
0.5982 |
0.79 |

#### The Win Probability Added of the second situation is -0.0982

While the same home run, with the same impact on the score, is worth more than three times as much in the ninth inning:

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

4.50 | Top of the ninth | Bases empty | 0 | 0 | 0.5000 |
0.5000 |
2.32 |

4.50 | Top of the ninth | Bases empty | 0 | -1 | 0.1622 |
0.8378 |
0.66 |

#### The Win Probability Added of the second situation is -0.3378

You can also use the Inquirer to calculate the impact of different situations on Leverage Index. I like this example: top of the ninth of a tie game at the beginning of the inning vs. having a runner on third and two out. The Win Probability of the two situations are virtually the same, but the Leverage Index of the second one is much higher:

The Situation | Win Probability | ||||||
---|---|---|---|---|---|---|---|

Runs/Game | Inning | Bases | Outs | Score Diff | Home Team | Visitors | LI |

4.50 | Top of the ninth | Bases empty | 0 | 0 | 0.5000 |
0.5000 |
2.32 |

4.50 | Top of the ninth | Runner on third | 2 | 0 | 0.5170 |
0.4830 |
4.40 |

#### The Win Probability Added of the second situation is 0.017

I’ll let you play with the tool yourself now. Here it is. Have fun.

**References & Resources**

Many thanks to Tangotiger, who donated the tables behind the Inquirer.

Pizza Cutter said...

Ooooooh… pretty.

Adam Guttridge said...

Good stuff.

WPA stats make for interesting MVP arguments and the like; anytime you’re looking for retrospective stats without caring about predictive value.

Also—and I don’t know how much of this has been done, so anyone with links, please share—I’ve always thought they’d provide a new perspective into clutch studies of any sort.

Dave Studeman said...

Hey Adam, yes a lot of work has been done with WPA and clutch hitting. The Book blog has a lot of work along those lines, as does Fangraphs. Plus, I’ve written about the subject in the last two THT Annuals.

Pizza Cutter: pretty, yes, but brainy too.

Pizza Cutter said...

Sounds like Mrs. Cutter.

Nick said...

Me likey.

Thomas Beck said...

Forgive my ignorance, but does the utility of the sac bunt change based on the identity of the batter? Say you’re in a situation where your light-hitting but great-fielding shortstop is due up and you can’t pinch hit for him (no spare infielders on the bench). Granted the sac bunt is usually a bad play, but in the situation I describe, would it be better to have your light-hitting shortstop bunt or hit away and possibly hit into a double-play? I don’t know the answer to this, which is why I’m asking.

Dave Studeman said...

Yes, absolutely. As I said, the utility of the sac bunt changes depending on who’s batting and who’s on deck. Well, I didn’t say exactly that. But pretty close.