THE BOOK--Playing The Percentages In Baseball

Just off the top of my head, I would go with normal tendencies -A- for both.  I am wanting to get outs in this case and with no one on, I would place the team in position to get outs.  Then if a player gets on, I might then change positioning.

Will have to think more about this, but that is my answer.

I’m going to go with A.  I’d play it straight up.

I think conventional wisdom says C tho......to guard against the extra base hit.

The question seems to be about when you would increase the chance of giving up an extra base hit to decrease the chance of the batter reaching base in the first place.  With a two run lead, clearly you play the infielders closer to second and the outfielders in since the cost of a double is effectively the same as the cost of a single-- outs are more valuable in this state.  With a one-run lead, there is no one on base so the question is which way decreases the odds of the team scoring a run in the first place.  I would guess you should always play that way, so I’d say the same.  For 1-run lead, A, and for a 2-run lead, B.

Without knowing the actual numbers, I would think that the results of these positions would be:

A: Minimizes opponents overall wOBA.
B: Reduces OBP but increases SLG of the batter.
C: Increases OBP but reduces SLG.

A should result in the lowest overall wOBA (or else it wouldnt be the normal configuration).

My choice is A.  We arent just playing where only 1 run matters here, one run ties it, but the second run is important as well. 

I love these questions of the day.  When I’m wrong, that means I’m learning something.

Same question but with a 2 run lead:

Probably B, because 1 run doesnt hurt you, its the 2nd and 3rd runs that are important.  So for the first guy, an extra base hit isnt much worse than a single, so it makes sense to give up more extra base hits if it would reduce OBP.

A for both.  B would be used if there was a guy on first and third with less than two outs in the bottom of the 9th or any extra ininng (although the infield would be in as well, probably.) C is the “No Doubles” defense.  That would be better used with a guy in first.

I am going to buck the trend and go with D) for both answers.  With nobody on base and a one run lead, I am going to play in on the corners to protect against the bunt.  With a two run lead I am probably going to put a shift on and load up the right side of the infield with three fielders, because the fans really love watching these gimmick plays and it’s all about fan experience afterall.

I think it’s A for both.  Better off taking an increased chance of outs by playing pitcher/hitter tendencies than guarding against anything.

A doesn’t necessarily maximize outs, as Alex mentioned.  A just minimizes opponent wOBA.  The real question is how teams fare when guarding against the lines or playing shallow in the outfield.  If playing shallow truly lowers OBP, is it enough to offset the likely gains in SLG?  Can the defensive alignment be gleaned from the play-by-play logs?

Thinking about this more, comparing the 1 run lead and 2 run lead scenarios:

For the two run lead, the first run doesnt matter.  Lets say we get a runner on.  We are then basically in the 1 run lead scenario (if they get a new runner on and score him, they will also score the first runner, so thats 2 runs), but with a small chance of getting that runner out in some way.

So 2 run lead, runner on is fairly equivalent to 1 run lead, no runners on.  Its a little bit better if the runner is on first so we can get a double play on a grounder, but only a little better.

Therefore with a 2 run lead and no one on, OBP is far more important than SLG. 

Definitely B in the 2 run lead scenario, minimizing OBP, its mostly irrelevant where the batter ends up if he gets on, what matters is if he gets on. 

* When 1 run is irrelevant, in the 9th inning, only OBP matters, not SLG.

The 1 run lead is far more complicated.  The first run gets us to 50% of a win chance, the second run gets the other 50%. 

What we really need to look at is the win expectancy of each of these score and base/out states.  Then we need to find the
g;e chance of each of those configurations leading to each outcome, and calculate their change to the win expectancy.

http://www.tangotiger.net/welist.html
(2002, so there might be better data by now, but this is what I am using right now)

Bottom of 9th, home team down 1, home team has WE .194

Result of an out: Home team’s WE goes to .108: -.086
(We gain +8.6%)
Result of a single/BB/HBP/RBOE: Home team’s WE goes to .208: +.100
(We lose 10.0%)
Result of a double: Home team’s WE goes to .282: +.174
(We lose 17.4%)
Result of a triple: Home team’s WE goes to .412: +.304
(We lose 30.4%)
Result of a homer: Home team’s WE goes to .634: +.526
(We lose 52.6%)

Lets assume a league average hitter in the AL (NL stats will include pitchers). 
For 2008, I found league average data here:
http://www.baseball-reference.com/leagues/AL/2008-standard-batting.shtml
Dividing various results by plate appearances:
HR: .026
3B: .005
2B: .049
1B: .159
BB: .086
RBOE:.016 ?
HBP: .010 ?
(Summing those 4 into results of runner on 1st: .271
Out: .649

Not 100% certain on a couple of those, but its close enough for this.

Now we need data for what the actual effect on the result of the hit is for different fielder configurations. 

I cant find a chart of the win expectancy for 2 run leads right now, but if it shows what I expect (that there is not much difference between 1B/2B/3B/HR in terms of win expectancy gain, and that most of the gain comes just from getting on base), then that would mean that we want to do scenario B, limiting OBP and increasing SLG.

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Result of an out: Home team’s WE goes to .108: -.086
(We gain +8.6%)
Result of a single/BB/HBP/RBOE: Home team’s WE goes to .208: +.100
(We lose 10.0%)
Result of a double: Home team’s WE goes to .282: +.174
(We lose 17.4%)
Result of a triple: Home team’s WE goes to .412: +.304
(We lose 30.4%)
Result of a homer: Home team’s WE goes to .634: +.526
(We lose 52.6%)

This is what you do with this data.  Compare it to the random win value of each event, to get the LI by event.

LI of out: .086/.027 = 3.2
LI of walk: .100 / .030 = 3.3
LI of single: .100 / .042 = 2.4
LI of double: .174 / .070 = 2.5
LI of triple: .304 / .1 = 3.0
LI of HR: .526 / .13 = 4.0

(I don’t have my win values handy from The Book, so I took a bit of an educated guess on the numbers.  They’ll be close anyway.)

As you can see, presuming Alex’s numbers are correct (I’d suggest going with the Markov numbers in The Book, which you can see for free from Amazon), then on a BIP, it’s the out that has the bigger impact than the single/double.  But the threat of the triple stops you from completely abandoning the line.

Anyway, play to get the out, at the expense of giving up more doubles.

Wait, was the “normal” defense supposed to be the normal defensive alignment in general or the normal defensive alignment with no one on base and zero outs?  My answer assumed the latter.  Otherwise, it makes perfect sense that OBP is relatively more important with bases empty and SLG is relatively more important with men on, while OBP is relatively more important with fewer outs and SLG is relatively more important with more outs.

I do A for both.  I would play the tendencies over the conventional wisdom, even in the 2 run lead example.  Is there any evidence playing deep or guarding the lines prevents enough extra bases to be worth while?

Alex - You made a mistake in reading Tango’s WE charts.  For the single, double, triple, HR you were reading off the 1 out chart instead of the 0 out chart.  HR’s, walks, HBP, K, obviously don’t apply to fielder positioning since nothing much a fielder can do with positioning can prevent them except the rare and hence trivial occurences of outfielders preventing inside the park home runs and reaching over the fence to pull back a home run ball.

1-run lead: C
2-run lead: B

Let x be the WE if the batter makes an out,
y be the WE if the batter gets a single,
z be the WE if the batter gets a double or triple.

The quantity to look at is (z-x)/(y-x). The higher it is, the farther back the outfielders should play.

Ordinarily with the bases empty and no outs, the ratio is about 1.4. In the bottom of the 9th, home team trailing by 1, it is about 1.5. Home team trailing by 2, it is close to 1.

So with a 1 run lead, you shade towards a no-doubles defense. With a 2 run lead you are maximizing the probability of an out, which means the outfielders play closer in.

Youre right, I used the wrong chart.  (I went from the 0 out to the 1 out chart to see the change from an out, and then I accidentally kept using that chart).

Correct values:
Out: +.086
1B/BB: -.137
2B: -.243
3B: -.357
HR: -.383

Using Tango’s method of comparing it to the random win value of the event to get the leverage index:

LI of out: .086/.027 = 3.2
LI of walk: .137 / .030 = 4.6
LI of single: .137 / .042 = 3.3
LI of double: .243 / .070 = 3.5
LI of triple: .357 / .1 = 3.6
LI of HR: .383 / .13 = 2.9

With the correct numbers it now looks a bit different. 

As Peter said, fielder positioning shouldnt effect walks/HBP or HRs, so we can probably ignore those.  (Perhaps the difference in position would effect the way the pitcher pitches or what the batter tries to do, resulting in a change to the chance of these results, but probably not). 

It looks like, in the 1 run scenario, with no one on, that we want to play to avoid doubles and triples, because those have the highest leverage values.  (We also REALLY want to avoid a walk).  So Scenario C, in the 1 run case.

I cant find the Win Expectancy table for 2 runs down, and I dont have The Book on me, but we’d just do the same process with that data to determine the correct strategy.

My guess is:

1 run game: C
2 run game: B

Great quiz. Thanks. I haven’t yet read the comments, where I’m assuming I will learn a lot (I always do here.) This is my first guess:

I think the right answer is that your defenders should do what they always do (given the tendencies of the pitcher and the weather and all of that), i.e. A. The reason for this is that what your defenders should always do is what minimizes run scoring. The only thing that is special about this case is that one run comes with a higher cost than it does in other innings, two with the highest possible cost (a loss), and in earlier innings this is never so. Nevertheless, my suspicion is that the strategy that minimizes run scoring in general is the strategy that minimizes the chance of scoring one or two runs, and so the strategy shouldn’t change. I’m even more confident this is the right strategy with a 2 run lead.

I think dcj #17 gets the utilities right but ignores the probabilities of the outcomes. What we want to do is arrange the defenders to minimize prob(extrabase hit)*win value(extra base hit) + prob(single)*win value (single).

**Note that using standard win expectancies for ninth inning outcomes incorporates what managers standardly do in this situation!** Consequently, if we use WE we are asking whether managers typically behave correctly, and A would be the right answer if they are. But if we want to ask how the defenders should play differently from usual, we need to use run expectancies of these events, not win expectancies.

I’m going to keep thinking about this.

*win value(x) is the value above an out of the event.

BTW, don’t presume that I know the answers to all of these questions.  For some I do, for others, I suspect one way or another, and for others, I have no idea, without doing some analysis.

Anyway, I find it curious that using Tango’s method, the walk and the single can have completely different leverage. I understand why of course.  In fact, the walk leverage will always be higher than that of the single with no one on base.

However, if we are using that leverage to come up with the correct answer in a question like this, what if the question were a certain strategy that changes the frequency of the walk and the single given the same scenario (last inning, down a run, etc.)?  Using Tango’s method, you would conclude that you want to minimize the walk.  That makes no sense of course, as the walk and single have the same impact with no one on base. Tango, how is that conundrum resolved?

Someone above mentioned the bunt.  Although I did not specifically ask about “bunt defense,” it might change of course in these two situations.  I also did not specifically say, “as compared to no outs and no one on base” or “an average position across all innings and scores (and everything else that can affect positioning).” I don’t think the answer will change, but I meant across all situations, but if you want to compare to no outs and no one on (with an average score and inning) that is fine too.  Just make sure you specify that you are comparing it to no outs and no one on or we will assume you mean a “normal position across all situations.”

Along those lines, interestingly, the position of the outfielders (and infielders) probably requires constant changing based on the score and inning.  That is one of the reasons I brought up this question in the first place.  You ONLY hear about outfielders and the first and third baseman playing in a “no doubles (and triples of course) defense” late in a game, when in fact, as I said, they should constantly be shifting back and forth, from inning one to inning last, depending on the relative win values of the single and extra base hit.  And you NEVER hear a commentator (ex-player or manager) talk about a “no singles” defense, which calls for the outfield to play up and the first and third baseman to play closer to second base.  Clearly this should be used at certain times during a game.  How many managers and coaches do you think are aware of this?  Again, I have never heard a commentator mention this.  If they know about it and teams are implementing it, you should hear about it on TV from ex-managers like Buck Martinez and Kevin Kennedy, and from ex-players who played the infield and outfield.

How many coaches and managers do you think realize this?

RE: MGL #21

I wonder whether there is a “no singles” defense. I think it’s pretty clear that no singles is the same as no hits (you can’t have a double worth less than a single.) “No doubles” defense is essentially decreasing the chance of a double at the expense of increased OBP. But I think it’s pretty reasonable to think that changes to the infield of the “standard postioning” of infielders is “no-singles” and the’re no significant ability of the OF to decrease OBP as a whole. Hence, you can’t really have a “no-singles” defense.

Often WE is much too generalized to be of any help to in providing a solution to real life problems, but in this bottom of the ninth situation with many of the extraneous factors filtered out it is definitely the way to go.  I figured my own WE empirically from 2005-2008 pbp data because I have a distrust of Markov generated WEs but the numbers don’t differ very much in this case from Tango’s.  As I said above the ones we are interested in are the ones that fielders can affect: Out, single and runner on error, double, and triple.  Here is the chart of the WE’s that I calculated.

-----------------1 run behind---2 runs behind
Starting WE--------21.1%------------9.9%
After Out----------11.95%-----------4.9%
After Single-------33.8%-----------18.55%
After Double-------45.3%-----------20.5%
After triple-------57.7%-----------24.15%

Which makes the change in WE for these events:

---------1_run_Behind--------2_runs_behind
out--------(-9.15%)--------------(-5.0%)
single------12.7%------------------8.65%
double------24.2%-----------------10.6%
triple------36.6%-----------------14.25%

The problem , as I see it, is that we want to position the fielders so that the distribution of hit balls on either side of the fielder in his area of responsibility have exactly the same average win value.  That should minimize the overall win value.  Here is where I don’t think I have enough information to answer the question accurately.  Since the Gameday hit location information only gives the location where the ball is picked up and not where it lands or passes an infielder it is not accurate enough for this kind of study.  Plus we can only guess where the fielder is normally positioned.  But I will give it my best guess anyway to illustrate how I think it should be done when we do get accurate information.

There are two ways to estimate the position of fielders.  For infielders taking the average position where they field line drive outs seems to work best.  But this method doesn’t work at all for outfielders because they have so much more time to react and they don’t go backward at the same speed that they can go forward.  So I use the rational positioning theory.  This assumes that outfielders play a position that equalizes the RE of the balls hit behind and in front and to either side of the fielder.  This is a pretty big assumption and I’m not sure that it is correct but it is the best I can do.
There are different hit ball distributions depending on the handedness of the batter.  I am going to demonstrate the methodology with one example for an infielder and one for an outfielder.

1B-L
-Event------------N------WE-------TOTAL--
-Outs-2B---------7333-(-0.092)---(-670.97)
-single-2B-------4896---0.127------621.79
-double-2B---------29---0.242--------7.02
-triple-2B----------3---0.366--------1.10
-TOTAL----------12261-------------(-41.06)
-AVG---------(-0.00335)

-Outs-1B--------11058-(-0.092)--(-1011.81)
-single-1B-------1196---0.127------151.89
-double-1B--------974---0.242------235.71
-triple-1B--------182---0.366-------66.61
-TOTALS---------13410------------(-557.60)
-AVG----------(-0.042)

Outfield 7-R
Location--Event---N-----WE-------total
Deep------Out----8786--(-0.092)--(-803.92)
----------Single-1473----0.127-----187.07
----------Double-6758----0.242----1635.43
----------triple--214----0.366------78.32
--------totals--17231-------------1096.91
--------avg-----0.063

Shallow---Out-----8009-(-0.092)--(-732.82)
----------Single-11251---0.127----1428.88
----------Double---966---0.242-----233.77
----------triple-----9---0.366-------3.29
----------totals-20235-------------933.12
----------avg----0.046

The first chart shows the hit distribution for the first base zone with a left handed batter and win values with the batting team behind by 1 run.  One thing that this distribution illustrates is that there are singles, doubles and triples hit on both sides of the 1st baseman, it is not an eitheror decision.  But the average value of a hit ball to the 2nd base side is worth slightly more than one hit nearer the foul line so the first baseman should move toward 2nd.

In the outfield example hit balls that are hit deeper than the left fielders usual position have greater win value than those hit shallower so the left fielder should move back.

My final answer based on the imperfect data that I have is:

1 run behind ALL batters - Move 1st toward 2nd and 3d toward the line. Move outfield back.

2 runs behind - Move 3d toward the line with a left handed batter otherwise move both 1st and third toward second.  Move outfielders in.

But as I said before I don’t have much confidence in these numbers even though I feel the methodology is correct.

Oh, and I forgot to add that moving the fielders around in this manner is contingent on believing MGLs assertion that batters don’t have enough control over where they hit the ball to be able to try and hit the ball on one side or the other of the 1st or 3d baseman.  I believe they do have at least some ability to do this when they are pulling the ball, hardly any ability to do this when they are hitting the other way, and almost no ability to control distance in the outfield.  So that might change the entire analysis on the pull side.

Along those lines, interestingly, the position of the outfielders (and infielders) probably requires constant changing based on the score and inning.

And the base out situation, and the type of pitch that is being thrown.  I was taught as an outfielder to play up with no outs and nobody on, back with two outs and nobody on, way up with men on 2nd and third and 2 outs, a couple of steps to the opposite field with a good fast ball pitcher in a fast ball count, a couple of steps to pull if I new it was going to be a change up, etc.  But that was in Babe Ruth league where I had a very smart and innovative coach.  I never got that coaching in high school, college, or semipro ball and I always wondered whether it was because the coaches just weren’t that smart or just assumed that I had already learned it earlier.

I’ll start with the easy one.  With a 2-run or more lead, of course a single and double affect the win expectancy by almost the same amount.  All we need is for the gap between a single and double to shrink (in terms of WE) in order to position the fielders differently (in favor of allowing fewer hits at the expense of more extra base hits), so it is a no-brainer with a 2 or more run lead.

You play your outfielders more shallow and your first and third basemen more toward second base.  I don’t know where or how Peter gets that you play your first baseman more towards second and your third baseman more towards third against a left-handed hitter, but that almost certainly is incorrect.  The ONLY thing you are trying to do is to decrease the chances of a hit at the expense of the extra base hit which is worth almost the same as the single or walk in this situation.  By definition playing your outfield in and your first and third baseman more towards second base HAS to do this.  We don’t need ANY data to prove this.  If a normal position balances the WE between a single and extra base hit, then a “no singles” defense must, by definition, decrease the chances of a hit and “no doubles” defense must increase the chances of a hit (while increasing the ratio of singles to extra base hits).  And I don’t know what is wrong with calling it a “no singles” defense.  We call the outfield playing back and the infielders guarding the lines a “no doubles defense” even though SOME doubles will occur.  Why can’t you call the OF playing in and the corner IF’ers bunching up a “no singles” defense?

Anyway, there is no controversy with this one.  The answer to part II of this question, with the 2 or more run lead, is B.  You don’t need ANY analysis to figure it out.

Part I, the one-run lead, is a little more complicated. The only question that needs to be answered, I think, it whether the impact of the extra base hit is greater or lesser than that of the single in that situation.  If it is greater than you play a “no doubles” defense in both the infield and outfield.  If it is lesser, you play the “no singles” defense and if it is around the same, you play “normally.”

The trick is to determine the relative values of the impact.  Tango seems to think that LI is the right way to go, and he may be right although I am not 100% convinced of that.  I think there might be some flaw in that methodology as evidenced by my comment on the walk and single.  However if this is the correct methodology then the answer is C because the LI of the double and triple are higher than that of the single, although not that much higher, which suggests that if playing deeper (the OF) is the right play, it is close.  But, as I said, it seems curious to me that the LI of the walk and the single suggests that the walk is much more valuable when we know that the single and the walk have exactly the same value in this situation.  (However, in defense of this methodology, and maybe the answer to my own question, is that IN THIS SITUATION you would play exactly the same way with respect to the walk and the single, if fielder positioning could influence a walk, but that in an “average” situation, you would play completely differently with respect to the walk and the single.)

Anyway, if we merely look at the relative differences between the WE of the single and the double, I think we can get an idea as to which way to play.

With a 2-run lead the difference in WE impact (before and after) between a single and extra base hit is 2.35% (taken from Peter’s numbers above).  I don’t know the difference between a single and extra base hit in an “average” situation, so I will use the bottom of the first inning of a tie game with no outs for comparison purposes.  In that situation, the difference between the impact of a single and extra base hit is 5.66%.  The ratio is 2-1.  (I don’t know if it is better to use the difference or the ratio.) The ratio for our situation - bottom of the 9th, 2 runs down, is only 1.27.  So clearly the difference and the ratio is much less than in the first inning and surely much less than for all situations combined, as you would easily expect.  That is why we play in a position to minimize ALL hits at the expense of the ratio of extra base hits to singles increasing.

Back to the one-run game.  With a one run lead in the 9th, remember I just said that the difference between the impact of the single and extra base hit is 12.7%.  The ratio is exactly 2-1.  Compared to the first inning numbers, the difference is a lot higher (12.7 to 5.66) and the ratio is the same, 2-1.  If using the difference is more correct, then the answer is C, play the no doubles defense. If the ratio is the one to use, then the answer is A, play the same, as the ratio is the same.  I think the difference is the important one, so I am going to say that the answer to Part I (one-run lead) is C.

Another way to approach this is to assume a certain change in the hit frequency and the extra base hit to hit ratio and see what happens.

Let’s start with these numbers from Peter.

1 run lead

WE impact of single: 12.7%
extra base hit: 25.2% (weighted average of the triple and double)

“Normal” situation (we’ll have to use the first inning numbers):

WE impact of single: 5.9%
extra base hit: 11.6%

We know that with a normal configuration, batters hit a single around 16% of the time, and an extra base hit around 5.5% of the time

So in a normal situation, we have:

5.9% * .16, or .944% for a single
11.6% * .055, or .638% for an extra hit.

.944 + .638 = 1.582

Let’s make up some numbers for playing a little back and make sure that the impact is greater than 1.582.  It has to be higher otherwise the “normal” position would be incorrect.

Let’s say that we increase singles by 1% and decrease extra base hits by .5%.  Now we have 17% for the single and and 5% for the extra base hit.

With that, the total WE impact in a “normal” (first inning) is:

5.9% * .17, or 1.003% for a single
11.6% * .05, or .580% for an extra hit.

1.003 + .580 = 1.583

Which is indeed a little higher (for the batting team).  So these are reasonable numbers (singles increase 1% and extra base hits decrease .5%).

Let’s apply those numbers to our bottom of the 9th situation:

Normal positioning:

12.7% * .16, or 2.032% for a single
11.6% * .055, or .638% for an extra hit.

2.032 + .638 = 2.67

Playing a little back:

12.7% * .17, or 2.159% for a single
11.6% * .05, or .580% for an extra hit.

2.159 + .580 = 2.739

So this seems to be a bad thing, playing the OF back, at least if it increase singles on the order of 1% and decreases extra base hits on the order of .5%.

You could plug in other numbers and see how it affects the WE impacts above, but I am going to say that the answer to the first part is also B, that you play a little closer, trying to decrease hits at the expense of extra base hits, but I think it is close.  It might be A and it might be C.

Of course, the answer is not that important.  The important thing is that teams recognize that position of the fielders must constantly change according to the score, inning and base runners, and not just according to the pitcher, batter, count, and type of pitch being thrown.  Not only that, but teams should have their “statistics” and “sabermetrics” departments working on these problems and telling the teams how to position their fielders “in general” as a function of innings, score, and base runners.  If all of the smart people on this blog can’t figure out what to do with a 1 run lead in the bottom of the 9th, which comes up A LOT, how is a manager supposed to figure it out?  He can’t!  But knowing that there is a right answer and that he doesn’t know what it is is the first step in making an optimal decision.

Remember what LI is: delta wins in THIS play divided by delta wins in RANDOM play

Presume the average win value of a walk is .03 and the single is .042.

If in a particular game state (like bases empty) the win value of the walk and single is the same (say +.06 wins), the numerator is the same for the LI, but the denominator will be different (.06/.03 or .06/.042).  This is why the LI is different.

So, as someone pointed out, when you ask “compared to what” in terms of positioning the fielders, are we presuming THIS base/out state, but compared to some RANDOM inning/score/base/out (game state)?  Or, are er presuming THIS game state state, but compared some RANDOM game state?

This way, if random win value for a walk and single, PRESUMING the base/out state of empty/0 outs might be say +.04 wins.  Then, if it’s +.06 wins in some inning/score combination, you will get the same LI of 1.5

Tango, I see my error now.  If in fact the position of fielders could affect the walk, then because the LI is so much higher for a walk as compared to a single in this situation (bottom of the 9th, etc.) we see this scenario:

The fielders would play in the same position if we were balancing the single versus the walk.  BUT, in a random situation, the fielders would play such that the frequency of the walk would be increased and that of the single would be decreased.

So, your method seems to work, I think.  And if Alex’ numbers are right, the leverage of the double and triple are a little higher, which would suggest the OF playing back a little.  But I think the numbers are unreliable enough that we cannot be sure what the correct strategy is (given that the LI of the single, double, and triple are all pretty close) OR that they would play back ONLY a little, as compared to a random situation.

Tango, what do you think of my method of assuming a certain change in the frequency of the single and extra base hit and then running the numbers on those, as I explained above?  I think that works even better than your method, or maybe the same.  Unless I made a mistake with the math, the fact that that method suggests that playing UP is the correct thing to do and the fact that your method (using LI) suggests that playing BACK is the correct thing to do, suggests that the real answer is very close.  I think it might be “flip a coin” in that situation!

As I always say when it comes to ANY strategy alternatives in baseball.  There are always three possible answers:  One, clearly one way.  Two, clearly, the other way (assuming a “two-sided” alternative), or three, either it is too close to call, or there are enough variables that we cannot account for AND it is too close to call now being able to account for all the variables. In this case, I think it might be too close to call although there are NO other variables that we cannot account for.  The reason there are no other variables to account for is that the question assumes that ALL of those variables are the same no matter where the fielders play…

In the third to last sentence above, “now” should be “not.”

Is this situation another way of looking at the calculations of base stealing break even points?

In basestealing a runner on first can either stay on first or try and steal second risking an out.

So you are trading a man on first for a mixture of men on second and outs and RE or WE tables give you the breakeven point needed for sucessful steals.

In this scenario a defence can play “normally” allowing the “ususal” number of singles or move up and get more outs at the risk of giving up more doubles (ignoring triples for the moment)

So you are trading singles for a mixture of doubles and outs and in the same way as base stealing you can work out the breakeven point where extra doubles balance the extra outs.

If the defence plays back you are doing the reverse converting a mixture of doubles and outs into singles. This will have the same breakeven point.

IF WE ASSUME that the normal defense is the optimal at reducing RE (or in other words if playing back or playing in were better at reducing RE than a normal defense everybody would do it normally as they don’t they must be worse so that playing in leads to too many doubles to offset the gain in outs while playing back leads to too many extra singles to ). Imagine a graph of alignment vs RE, it would be U shaped with normal defense in the middle with playing in and playing out being higher.

Therefore it is only whent the WE differs from RE such (as in this situation) that you would deviate from normal alignment.

USing an RE table the breakeven point is 69.3% i.e. 69 EBHs (merging doubles and triples into a single category of EBH (10:1 double to triple ratio)) and 31 outs are the same as 100 singles. Using The WEs from Peter Jensens post and the break even point for changing a single into EBhs and outs is 63.5% for a 1 run lead but 85.7% for a two run lead. 

So in a 1 run lead doubles are more costly so i would play back giving up more singles PROVIDED that i didn’t give up more than 1.57 (1/63.5%) singles for every EBH I prevented. Or to put it another way I would rather have 100 singles than 64 EBH and 36 outs

With a two run lead I’m not as worried about doubles so I would play in PROVIDED that I didn’t give allow more than one EBH for every 1.16 (1/85.7%) singles I prevented. or to put it another way I would rather have 85 EBhs and 15 outs than 100 singles.

Unfortunately I don’t think we will ever know how good different defenses are at preventing EBHs or singles given the amount of data needed and the amount of variation between different fielding strategies. 

I’m probaly just rephrasing what others have said but the similarity to basestealing is striking.

James, yes, that is how to look at it.

James, perfect!  While we don’t know exactly (and probably never will) how much to play in or out in the various situations, we do know that it HAS TO be correct to play either more in or more out (as compared to “normal") as the game situation changes.  That is not only true in the last inning but throughout the game as well.  That is the part that teams don’t get right.  They will adjust to the batter and pitcher (and stadium and weather) and occasionally with a runner on second and two outs they will play a little more shallow to have a chance to throw the runner out at home, but by and large they do not change their position as the relative value of the single and the extra base hit changes (which they should); that is largely because they don’t KNOW when that relative value changes and by how much (and in what direction).  In fairness to them, it is not always obvious.