May 4, 2005

Lies, Damned Lies

Introducing ORVY

by Nate Silver

Many basic baseball strategy decisions involve trade-offs between scoring or preventing one run, and scoring or preventing multiple runs. An intentional walk to a good singles hitter with one out and a runner on second, for example, will decrease the chance that the offense scores one run by reducing the likelihood of a base hit, and by creating a double-play opportunity. But it will increase the chance of a big inning by putting another runner on base, when he'd most likely have made an out if he'd been pitched to. Similarly, the sacrifice bunt and the stolen base are ploys to score one additional run that reduce the chance of a multi-run inning by potentially giving up an out.

There are some situations in which the decision to play for one run or multiple runs is straightforward. With the score tied in the bottom of the ninth inning, for example, one run will make all the difference, and teams are correct to employ strategies like the stolen base, sacrifice bunt and intentional walk very liberally. Most of the time, though, the situation is much more ambiguous. To take a couple of examples:

• The home team trails by one run in the bottom of the fifth, and has a runner on first base. Is this a good time to send the runner or drop down a bunt at the risk of sacrificing an out? Or is it still early enough in the game that the homers should seek to play for a big inning?

• The home team leads by two runs in the bottom of the seventh, with runners on second and third and one out, and a dangerous hitter at the plate. Should the visitors attempt to contain the damage by walking the hitter and hoping to complete a double play, or is putting an additional runner on and loading the bases compounding the problem?

In order to evaluate these questions, we need to consider the probability of a team winning the game given a certain scoring margin and in a certain inning. For example, assuming two sides of average strength, a home team that leads by one run heading into the bottom of the sixth inning will win the game about three-quarters of the time. This estimate, as well as those that will be used throughout the balance of this article, are derived from a model that I created here, which establishes win probabilities based on a historical distribution of MLB run scoring by inning (e.g. a team will score exactly three runs in an inning about 3.5% of the time). We will assume for purposes of this article that both the home and the road teams have average run-scoring and run-prevention abilities.

Let's take a look, for example, at the probability that the home team wins the game given a particular scoring margin following the bottom of the seventh inning. The model estimates the home team's winning percentage as follows:

Home Team Win Probabilities following 7th inning

Score         Home Win %
+5 runs         98.2%
+4 runs         96.3%
+3 runs         92.6%
+2 runs         86.0%
+1 runs         74.1%
Tied            50.0%
-1 runs         25.9%
-2 runs         14.0%
-3 runs          7.3%
-4 runs          3.7%
-5 runs          1.8%

In other words, a team trailing by two runs after seven innings will come back to win the game about 14% of the time. Here, derived from the table above, is the marginal benefit to the home team's win probability from scoring exactly one additional run during the bottom of the seventh given a particular scoring margin:

Marginal Benefit of One Additional Run for Home Team, after Top of 7th inning

Score         Home Win %  Marginal Increase
+5 runs         98.2%          --
+4 runs         96.3%          +1.9%
+3 runs         92.6%          +3.6%
+2 runs         86.0%          +6.6%
+1 runs         74.1%          +11.9%
Tied            50.0%          +24.1%
-1 runs         25.9%          +24.1%
-2 runs         14.0%          +11.9%
-3 runs          7.3%          +6.6%
-4 runs          3.7%          +3.6%
-5 runs          1.8%          +1.9%

As we would anticipate at this stage of the game, there is a profound inflection point around tied scores and one-run margins. A team that trails by one run and ties the score, for example, will go on to win the game about an additional 24% of the time. On the other hand, a team that trails by five runs and scores one run to cut the deficit to four will only win about an extra 2% of the time by making their job somewhat easier in the eighth and ninth innings. All of this should be fairly intuitive.

The rest of this article is restricted to Baseball Prospectus Subscribers. Not a subscriber? Click here for more information on Baseball Prospectus subscriptions or use the buttons to the right to subscribe and get access to the best baseball content on the web.

Cancel anytime. That's a 33% savings over the monthly price! That's a 33% savings over the monthly price! Already a subscriber? Click here and use the blue login bar to log in.