Crucial Situations, Part 2

Editor’s Note: This is the second in a series of articles from Tom regarding Leverage and crucial situations. Last time, Tom revealed the secret recipes to Leverage Index. In this articles, he analyzes other versions of leverage.

Anthony from Long Island asks Keith Woolner:
After reading Baseball Between the Numbers, I have a question about the situational leverage chart in the chapter on closers. You wrote that teams should bring in their closer in the eighth inning or earlier when leverage exceeds 2.32, and in the ninth when leverage exceeds 1.66. The chart lists the leverage for ninth inning, no outs, tied game, bases loaded as 1.04. If that’s true, then that seemingly critical situation is of only average importance.

Drinen, Birnbaum, Woolner

Doug Drinen (Pressure), Phil Birnbaum (Importance), and Keith Woolner (Leverage) each introduced a measure along the same lines as Leverage Index. Let’s go through each one, and see how they work, and where they are different.

First up is Drinen’s Pressure. The viewpoint to Pressure is the Perfect Pitcher. How much does the win probability change, if the pitcher gets out of the inning without allowing a run? If we refer to Table 10 (Win Expectancy, By Game State) from The Book—Playing The Percentages In Baseball, we see that the win probability for the bottom of the ninth, with a three-run lead, men on first and second, and no outs is .841. Therefore, the Perfect Pitcher will not only get out of the inning, but will also win the game (win expectancy of 1.000), for a gain of +.159. That’s how many Perfect Pitcher wins are available.

Now, let’s go through some illustrations. What happens if it’s a tie game, with the bases loaded and no outs in the top of the ninth? In that case, the win expectancy for the home team is .168. If he’s perfect, he gets his team up in the bottom of the ninth with a .649 chance of winning. The Perfect Pitcher gain is +.481 wins (.649 minus .168). Bases empty, no outs and tie game in the ninth? By definition, the win expectancy is .500, giving us a Perfect Pitcher gain of +.149 wins.

How do those compare to the average inning? A tie game in the third inning is pretty average. Again, by definition, the win expectancy is .500 to start the top of the third. The Perfect Pitcher will get us to the bottom of the third with a win expectancy of .557, for a Perfect Pitcher gain of +.057 wins.

We have enough for our calculations. Though Drinen doesn’t scale things so that average equals 1.00, let’s do so. The ninth inning, bases empty situation would have an index level of .149 / .057 = 2.6. That certainly looks reasonable. The ninth inning, bases loaded situation would be .481 / .057 = 8.4. Is that reasonable? Perhaps from the manager’s perspective it is. The win expectancy at this point is .168. Even though it’s a tie game, the hitting team is way in control. And, the manager really has no reason to think he’ll escape without allowing even one run.

An average pitcher will allow a run in this case 87.5% of the time, while even a great pitcher will do so 83.6% of the time (see Tables 8, 9 of The Book). The Perfect Pitcher scenario will bring that 83.6% figure all the way down to zero. This is not reasonable. While a manager may see it from that perspective, rarely will this come to fruition.

Now, let’s see how Woolner’s Leverage stacks up. According to Table 2-2.4 of Baseball Between The Numbers, the Leverage value of the ninth inning, bases empty scenario is 3.44. As Anthony from Long Island told us, the bases loaded situation is 1.04. This is the polar opposite of Drinen. Woolner views things from the point-of-view of the One-Run Pitcher. What happens if the pitcher allows exactly one run?

Well, for the bases empty scenario, for the pitcher to allow that next run, a lot of bad things have to happen. We’re talking most likely a few baserunners, if not more. With the bases loaded situation, that next run will happen simply on a fly ball. So, to Woolner, the truly high-pressure situation is the bases empty, and not the bases loaded situation.

The problem we have with both Drinen and Woolner is that they start with a certain framework, win expectancy, and then abandon some of its principles. Win expectancy is based on the probability of certain things happening. How often will zero, one, two, etc runs score, and what impact does that have to the chances of winning the game. What they then proceed to do is take that distribution, and ask, “What happens if no runs score”, or “What happens if one run scores”. Why not also ask “What happens if two runs score”, or “What happens if three runs score”.

They only ask one question, when really they should be asking several questions. And, on top of that, they should be asking “How likely is this to happen”. Instead, they completely change the distribution of likely events into something that is unreasonable: “What happens if 100% of the time, no runs score”, etc. If they were instead to tweak the scenario, and alter the distribution of runs that score, that shift in run distribution will lead to a shift in win expectancy. And it is that shift that really concerns us.

With Leverage Index, the bases empty, ninth inning scenario has a leverage value of 2.4, while the bases loaded situation is 2.9. Those are the numbers that represent the true potential shift in winning and losing.


Birnbaum introduced Relative Importance, and it is a measure that is very close to Leverage Index. If we go back to Leverage Index, we look at the impact that each event has on the game, along with their frequency. So, for bases empty, tie game, top of ninth, the home team goes from .500 to .563 with an out, and to .174 with a home run. A walk or single puts the home team down to .418. So, the out is worth .063 wins, the single/walk is worth .082 wins, and the homer is valued at .326 wins.

According to Table 11 in The Book, the win value of the out is .026 wins, giving us a leverage in the ninth inning of 2.4 (.063 / .026). The walk has a random win value of .028, giving it a leverage of 2.9 (.082 / .028). The single has a win value of .042, giving it a leverage of 2.0 (.082 / .042). The home run, with a win value of .123, is leveraged at 2.7 (.326 / .123). As we can see, each event impacts the game about two to three times what it would normally impact otherwise. What Leverage Index does is the following:

sum(freq(i) * win impact here (i))
sum(freq(i) * win impact random (i))
Birnbaum’s overall Importance measure instead did the following:

sum(freq(i) * win impact here (i))
win impact random (i)
While this may seem a little nuanced, the impact is felt when you also consider things like a sacrifice bunt. Because this event has a low denominator, it will often end up with a very high Relative Importance figure. This is also true for a home run with bases loaded and two outs, among many other events and game situations. Birnbaum is mostly saved because the frequencies of all these events is always overshadowed by frequency of the out. In fact, the astute reader will have noticed that Leverage value of the out was 2.4, which is exactly what the overall Leverage Index value of all events came out to. Which leads us to …

Secret Recipe #5

Leverage Index is the Leverage value of the out.

The Incompleat Starting Pitcher
The end of the nine-inning start and how we got here.

This is very close in most cases, except with runners on third base and less than two outs, and a tight game. The reason here is that while most of the time, an out will leave the runner in the same spot, it will happen quite often that the out in this situation will lead to a run. Therefore, if you insist on using the value of the out (as a strikeout), you will be off in certain situations.

The most obvious situation is the bottom of the ninth, man on third base, tie game, one out. In this case, the win expectancy is .835. A strikeout will bring this down to .633, for a win change of .202 wins. Compared to the random out win value of .026, this gives us a Leverage Index value of 7.8. In fact, the Leverage Index value is 4.5. You see, the reason the win expectancy was .835 to begin with is because of the expectation that the out may lead to a run (and automatic win). If all outs were strikeouts, the starting win expectancy would not have been .835 to begin with, but rather something lower. Therefore, a strikeout in this situation would not have decreased the chances of winning by .202, but perhaps by only .100. And the Leverage Index value of the out would have been around 4.0.

Another try

Let’s look at the bases empty, two out, tie game, ninth inning scenario. Woolner gives us a leverage value of 4.21, and is in fact the highest leverage value of all possible game situations. Does this sound right at all? This is completely unreasonable. Drinen’s value would be .649 (win expectancy with no runs scored in the top of the ninth) minus .612 (win expectancy for this game situation) for a win change of .037.

Now, while the one-inning random change for the perfect pitcher is .057, it would be around .019 for the one-out random change. This gives us a leverage value for Drinen of 1.9. Leverage Index is 1.4. In this case, the Drinen value came out well because most of the time with two outs, no runs will score, and therefore the Perfect Pitcher viewpoint works well. The Woolner value did not come close at all because of the One-Run viewpoint. Expecting one run to score with 2 outs and bases empty will obviously give us a huge change in win expectancy.

We can go through all the different scenarios, and the holes would be exposed to both the Woolner and Drinen measures. Birnbaum’s process was right on the money, except for that final step, which is easily corrected.

Next time … One more Secret Recipe.

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