Baseball folklore claims Indians manager Lou Boudreau devised the defensive shift to combat a Ted Williams hot-streak. Boudreau admitted the strategy was meant as more of a psychological ploy rather than a legitimate baseball tactic. In some sense, Boudreau’s unconventional stunt worked since Teddy Ballgame stubbornly refused to take advantage of all the open real estate he was presented with.

Even if its origin has been debunked, the defensive shift is widely accepted as a conventional baseball strategy today. Broadcasters and mainstream writers only seem to point out the gaping hole on the left side of the infield when the hitter who is subject to the shift is going through a slump. If you ask the player himself why he doesn’t lay down a bunt or attempt to hit a ground ball where the third baseman (and sometimes even shortstop) should have been standing, he’ll probably tell you he’s paid to drive the ball and hit homeruns, not bunt. The reality of the situation is runs are runs, regardless of how they’re generated. How good does a player need to be to justify continuously attempting to over-hit the shift?

Since quality offensive performance comes in many forms, this becomes a complicated question to answer. Conversely, another question that yields a similar answer is: how successful does a hitter need to be at dropping down a bunt or slapping a ball to the left side to match his offensive output with his current approach? To answer this, we have to first make a couple assumptions.

The first is that we’ll only consider situations with the bases empty. While the shift is used with runners on base, it’s less likely and these situations increase complexity. Secondly, we’ll only consider singles as a successful outcomes for a hitter’s attempt to hit left of the shift. Depending on the severity of the shift, a double is possible, but less likely. A couple weeks ago, I used the 2009 to 2011 run expectancy charts to show break-even points for stolen bases – I’m going to use these same charts for the backbone of this analysis.

If we individually divide the number of singles, doubles, triples, homeruns, walks plus HBP and outs by plate appearances over a full season, we’ll have the rates for each of those outcomes. If we treat these as probabilities (assumption number three), we can then multiple these rates by the difference in run expectancy between the situational outcome of the event and a bases empty situation.

For singles, we take the run expectancy with a runner on first and a given number of outs and subtract it from the run expectancy with the bases empty and the same number of outs – then multiply this difference by the hitter’s 1B%. This gets repeated for 2B%, 3B%, HR%, BB% and OUT% and summed to give the expected runs added per plate appearance. Once we have this number we need to duplicate the procedure for just singles and outs.

The percentage at which this curbed run expectancy added per at bat equals the total from the original calculation is the success rate of hitting left of the shift that a hitter needs to have to equal the expected value of his normal approach. Essentially, this number is a batting average. The fourth assumption I didn’t mention above is that a hitter wouldn’t accept a walk when attempting to hit around the shift. This would definitely not hold up in practice, so instead of taking the break-even rate as a batting average, we can also interpret it as an OBP instead – since a BB, HBP and single all cause the same effect with the bases empty.

Obviously this framework is unique to the player. Since I watched managers deploy the shift on Mark Teixeira all season, I ran through the numbers for his 2011 season. The table below shows Teixeira’s frequencies for each type of outcome.

The graph below shows the summed difference in run expectancy per at bat on the vertical axis and the success rate of hitting left of the shift on the horizontal axis. The colored lines are plots of how the run expectancy changes with success rate for the limited scenario of only singles and outs. The horizontal lines mark Teixeira’s run expectancy difference based on his overall season rates for each number of outs. The vertical lines pointing down to the x-axis give us the number we want. When we average these numbers, it tells us that if the Yankees’ first baseman thought he could have successfully shot a ball through the open hole on the left side of the infield more than 55% 45% of the time, he should have done just that – given the frequencies of the types of his hits.

(Update: Many thanks to BronxBaumer for pointing out I originally used the wrong set of splits. The table and chart have been corrected.)

The disclaimer here is that this analysis isn’t an evaluation of what his season was on an event-by-event basis and does not definitively say he would have provided more value to the Yankees with this approach. Of course, over a full season, once managers notice Teixeira’s unwillingness to play into their game, the defense would return to normal alignment – which of course is advantageous to Teixeira with his regular approach. This cat and mouse game might continue, but Teixeira will always benefit from whatever the opposing manager shows him because he can stay one step ahead.

What does this number look like when we push offensive performance into near unfathomable levels? Here’s the same chart, except using the offensive rates from Barry Bonds’ record-breaking 2001 season.

When we average these values, we see that Bonds would have had to bat .788 to justify attempting to hit left of the shift. The word “stubborn” need not apply here.

Obviously this number varies by hitter, but for hitters from earth, it suggests that they are in fact being less productive by attempting to hit over a shift – especially with fewer outs in the inning – than trying to take advantage of the gigantic hole on the left side of the infield. To take it one step further, the worse the hitter is, the less forgiving fans should be if Dustin Pedroia snatches a line drive in short-right field when a routine ground ball to the other side of the field would have done the job.