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June 5, 2003

Lies, Damned Lies

Solving a Ninth Inning Quandary

by Nate Silver

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NEW YORK (May 28) -- Ball four. With the Yankees batting in the bottom of the ninth and the score tied 5-5, Red Sox Manager Grady Little ordered the bases walked full, preferring that his nominal relief ace, Brandon Lyon, face Yankee catcher Jorge Posada instead of the strikeout-prone Alfonso Soriano or the slumping Jason Giambi. Perhaps Little should have consulted with nerd-in-residence Bill James first. The move backfired, as Posada walked following a borderline call on a 2-2 pitch, sending Hideki Matsui in a slow trot home and blowing the game for the Sox. Long live the curse.

Stop me if you've heard this one before. Any loss to the Yankees causes consternation from Barnstable to Bangor, and especially one with so many elements to pick on. Poor performance by the bullpen? Check. Questionable managerial decision? Check. Controversial call by the umpires? Yessiree, Bob. The reaction from the local scribes was fairly tame, in no small part because Bob Ryan is still serving his suspension for being a moron, but make no mistake: This was exactly the sort of loss that enables Red Sox Nation's impulses toward self-destruction (this column brought to you by Samuel Adams Lager).

But while the loss was painful for Boston fans, it was also the sort of game that Diamond Mind and Strat players love. The bottom of the ninth was a classic chess match, with classic endgame strategy. As D.H. writes:

"Statheads...often lament the intentional walk with an argument that usually goes like this: With a runner on third and one out, the expected runs scored for the inning are X. With the bases loaded and one out, that number is Y (higher than X). This argument normally makes sense, but in a situation where one run is all that matters, the manager should instead try to maximize the probability that no runs will score...Does walking the bases loaded with one out make sense on this basis?"

It goes without saying that the situation that Grady Little faced was unusual. As D.H. points out, the only thing that each manager need concern himself with is whether that one essential run scores. All the strategic elements of the game--hitting, baserunning, pitching, defense--are profoundly different under these conditions. For convenience sake, let's give the situation a name: THNT (Tie game, Home team at bat, Ninth inning or later, runner on Third).

When THNT arises, the visiting manager effectively has three ways to play his hand:

  • No intentional walk--runner on third.
  • Intentionally walk one hitter--runners on first and third.
  • Intentionally walk two hitters--bases loaded.

Is there a built-in, structural advantage to beginning the endgame in any of these states? Under THNT, and with one out, the potential outcomes can be summarized as follows:

  • Home team wins: Any hit, or sacrifice fly.
  • Visiting team "wins" (sends the game into extra innings): Double play.
  • Add an out, stir, and repeat: Strikeout, any other out.

The missing outcomes here are walks and hit-by-pitches; they produce an automatic loss if the bases are full, but the visiting team gets another chance if there's a base free.

Because the THNT situation is relatively unusual, we need to reach pretty far back in order to pull together a reasonable sample's worth of data. I looked at 15 seasons of play-by-play data, spanning 1982-1992, and 1999-2002--the intervening years are often referred to as the 'Dewan Gap.' Here's how THNT situations have played out over those seasons when there were fewer than two outs, and thus the possibility of a double play:


THNT, <2 Outs AB      H       BB      K       HBP     SF      DP      OUT
3rd             144     58      20      29      4       21      1       56
1st/3rd         257     96      22      36      7       28      20      105
Loaded          797     293     44      143     9       100     80      281

THNT, <2 Outs   H       BB      K       HBP     SF      DP      OUT
3rd             30.7%   10.6%   15.3%   2.1%    11.1%   0.5%    29.6%
1st/3rd         30.6%   7.0%    11.5%   2.2%    8.9%    6.4%    33.4%
Loaded          30.8%   4.6%    15.1%   0.9%    10.5%   8.4%    29.6%

A couple of points of clarification: 'Outs' refers to generic groundouts and flyouts which do not produce double plays or score any runners. Barring highly unusual plays, it's impossible to record an out under THNT while scoring the runner, unless the out is a sacrifice fly. This is why batting averages are so high under this scenario--the visiting team will always gun for the runner at home, and so the batter gets credit for a lot of hits on balls that would ordinarily have him thrown out at first.

'BB' refers to unintentional walks only; we're interested in what happens only after the visiting manager decides to play a situation straight up. I've also excluded sacrifice squeezes from the table, which were attempted only a handful of times in THNT situations.

As you'd expect, walks decrease as additional runners reach base, and especially with the bases loaded. Double plays move in the opposite direction, with the other figures remaining about the same. It's also helpful to describe each situation in terms of the potential outcomes it has on the result of the game:


THNT, Runner on 3rd, 1 out
Visiting win = 0.5% (DP)
Home win = 41.8% (H, SF)
Runners and 1st and 3rd, 1 out = 12.7% (BB, HBP) 
Runner on third, 2 outs = 44.9% (K, OUT)

THNT, Runners on 1st and 3rd, 1 out
Visiting win = 6.4% (DP)
Home win = 39.5% (H, SF)
Bases Loaded, 1 out = 9.2% (BB, HBP) 
Runners on 1st and 3rd, 2 out = 44.9% (K, OUT)

The second strategy--intentionally walking one batter to put a runner on first--looks like a winner here. It increases the chance of an "instant win" for the visiting team by means of a double play, and slightly reduces the chance of a game-winning hit or sacrifice fly by the homeboys (perhaps because of the additional force play at second--though this may also be a sample size fluke).

As for giving a free pass to a second hitter...


THNT, Bases Loaded, 1 out
Visiting win = 8.4% (DP)
Home win = 46.8% (H, SF, BB, HBP)
Bases loaded, 2 outs = 44.7% (K, OUT)

At first glance, this doesn't appear to be an advantageous strategy. Although the chance of a double play improves slightly, the chance of a home win increases more dramatically, as walks and HBPs are added to the list of game-ending events.

To be certain, though, we need to complete the chain and look at THNT outcomes when two outs are present.


THNT, 2 Outs  AB      H       BB      K       HBP     OUT
3rd             223     48      37      49      1       175
1st/3rd         331     82      23      53      2       249
Loaded          594     140     80      102     3       454

THNT, 2 Outs    H       BB      K       HBP     OUT
3rd             18.4%   14.2%   18.8%   0.4%    48.3%
1st/3rd         23.0%   6.5%    14.9%   0.6%    55.1%
Loaded          20.7%   11.8%   15.1%   0.4%    52.0%

It's clear that once the second out is recorded, the outcomes shift dramatically in favor of the visiting team. Sacrifice flies disappear as a run scoring strategy, and batting averages drop from close to .400 to the Mendoza Line as the defense needs only worry about recording any out on a ball in play, rather than gunning down the man heading home. Here's what the win probabilities look like with two outs:


THNT, Runner on 3rd, 2 outs
Visiting win = 67.1% (K, OUT)
Home win = 18.4% (H)
Runners on 1st and 3rd, 2 outs = 14.6% (BB, HBP)

THNT, Runners on 1st and 3rd, 2 outs
Visiting win = 70.0% (K, OUT)
Home win = 23.0% (H)
Bases Loaded, 2 outs = 7.1% (BB, HBP)

THNT, Bases Loaded, 2 outs
Visiting win = 67.1% (K, OUT)
Home win = 32.9% (H, BB, HBP)

Interestingly, with two outs and THNT, unintentional walk rates are significantly higher with the bases loaded than with men on first and third. It may be that, instead of trying to put the ball into play with forces at every base, the hitter realizes under these circumstances that his best chance to be a hero is to take a free pass. In any event, this is another reason to avoid loading the bases when you have a choice about it.

Since we're dealing with a finite number of outcomes, the probabilities can be chained to determine the probability of a home team win given each starting state. We'll start with the bases loaded case, which is the simplest.


Bases loaded, 1 out                      Home win %      Visit win %
Visiting win = 8.4%                             --              8.4%
Home win = 46.9%                                46.9%           --
Bases loaded, 2 outs = 44.7%                    --              --
        Visiting win = 44.7% x 67.1%            --              30.4%
        Home win = 44.7% x 32.9%                14.7%           --
Total                                           61.6%           38.4%

With one out and the bases full, the home team has a 61.6% chance of winning the game outright. The rest of the time, the visiting team will succeed in advancing the game by another inning.

By comparison, here's the same table when the visiting manager elects to walk one batter, and pitch to the opposition with men on first and third.


1st and 3rd, 1 out                          Home win %      Visit win %
Visiting win = 6.4%                                 --              6.4%
Home win = 39.5%                                    39.5%           --
Bases loaded, 1 out = 9.2%                          --              --
        Visiting win = 9.2% x 8.4%                  --              0.8%
        Home win = 9.2% x 46.9%                     4.3%            --
        Bases loaded, 2 outs = 9.2% x 44.7%         --              --
               Visiting win = 9.2% x 44.7% x 67.1%  --              2.8%
               Home win = 9.2% x 44.7% x 32.9%      1.4%            --
1st and 3rd, 2 outs = 44.9%                         --              --
        Visiting win = 44.9% x 69.9%                --              31.4%
        Home win = 44.9% x 23.0%                    10.3%           --
        Bases loaded, 2 outs = 9.2% x 44.7%         --              --
               Visiting win = 9.2% x 44.7% x 67.1%  --              2.1%
               Home win = 9.2% x 44.7% x 32.9%      1.0%            --
Total                                               56.6%           43.4%

In this situation, the visiting team has improved its chances of sending the game into extra innings from 38.4% to 43.4%, or about five percent. Is that a lot? It might not sound like it, but I'd venture to guess that there aren't too many situations in which a single managerial decision makes that much of a difference, and especially when a team has its back against the wall.

As for the third alternative, walking nobody on purpose:


Runner on 3rd Base, 1 out                   Home win%       Visit win%
Total                                              58.9%           41.1%

I won't print the full table here because it gets a little bit messy. (What's that nasty phrase that undermined my attempts to cheat on my econometrics homework? Left as an exercise for the reader?) But the win probability here is somewhere in between: Leaving two bases open gives the pitcher plenty of wiggle room, but effectively eliminates the possibility of a quick end to the inning via a double play.

So from the standpoint of the visiting team, the three THNT strategies rank as follows:

  1. Walk one batter: 43.4% chance of sending game into extra innings;
  2. Walk no batters: 41.1% chance of sending game into extra innings;
  3. Walk two batters (load bases): 38.4% chance of sending game into extra innings.

All else equal, intentionally walking one batter is the optimal approach.

Now, it might be objected that I've looked only at the win probabilities for 'generic' pitchers and hitters, and not for the particular players in the game in question. Certainly, the probabilities are close enough that there are circumstances in which the context of the game in question ought to take precedence--walking Barry Bonds to load the bases for Benito Santiago is gonna be a good move.

But I'm not sure that gets Grady Little off the hook. Yes, Jason Giambi has a much better track record than Jorge Posada; he's also slow, and having a ton of trouble making contact because of his vision problems this year, which makes him a good candidate to strike out or be doubled up. And Posada's greatest strength is his ability to draw walks; he's exactly the sort of guy you don't want to pitch to in a tie game with the bases loaded.

More to the point, the conventional wisdom that you ought to walk the bags full when the opportunity presents itself to you is the wrong one. The situation that Grady Little faced has come up literally thousands of times before. Each manager was trying to optimize his chance of winning given the particular players on the field. Those that intentionally walked one batter--but not two--came out a little bit ahead.

I think part of the reason that many managers favor a suboptimal strategy is the lingering perception that a walk represents an unforced error on the part of the pitcher, rather than a skill on the part of the hitter. Tie game, bottom of the ninth: You've presumably got one of your best pitchers on the mound, and well, he ought to be smart enough not to walk in the winning run. But walks happen under these circumstances, and they happen quite a lot--in a tie game, with the bases full, and two outs in the bottom of the ninth, the opposing hitter manages to draw a walk around 12% of the time, a rate considerably higher than average. Unless there's a compelling reason to think otherwise, that's not a risk that a manager should volunteer to bring upon himself.

Nate Silver is an author of Baseball Prospectus. 
Click here to see Nate's other articles. You can contact Nate by clicking here

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