THE BOOK--Playing The Percentages In Baseball

Flip a coin.  Choose the bunt with heads, swing away with tails.

Pinch hit.

I would prefer more facts.  Has he left a tiring starter in that the #2 guy has faced three times earlier?  What’s the platoon advantage?  Does his 1B throw lefty or righty?  What about for the next two hitters?  Who is up in his bullpen?  I smy lineup constructed with a high average hitter batting third (.330+)?  What’s my opposing manager’s tendency to IBB?  Is my #4 guy a GDP monster?

The first key is which pitcher is in.  If it’s his starter, and I have the platoon advantage, I swing away.  If it is a reliever, and I have the platoon advantage on either of my next two hitters, I bunt.  Heck, if it is a reliever, I bunt to the first base side.

Well, I’d be an awful manager anyway, but I say swing away.  You’ve got the middle of your lineup coming up, even bunting for a hit isn’t a sure thing, and even if you move the runner up, I’d rather take a chance at a big inning.

Presumably, the opposing manager employed a standard hitter-specific defense since the no-doubles defense would not increase the Win Expectancy since it would be best to avoid allowing a hit here.

I believe, then, the answer would be to have the batter swing away. For one, he’s a decent hitter, and this means he’s above average, though not necessarily exceptional. Nonetheless, the batter being decent increases the likelihood of a hit. Most importantly, Win Expectancy would decrease by using a sacrifice to move the runner to second because the WE is higher with a runner on first and no outs than it is with a runner on second and one out.

(What I could be missing here is if the WE would increase with a runner on second and one out and the third and fourth hitters coming up since they are presumably the team’s best hitters.)

I would tell my hitter to bunt if he gets the optimal pitch to do so (basically, telling him not to force it).

I would prefer for him to walk to put men on first and second with my 3 hitter up.

Play for the 3-run homer.

[1] Swing away.

[2] Consider the hit-and-run if the runner on first is speedy and the catcher/pitcher combination is poor at throwing out would-be base-stealers.

If the defense is playing for the bunt, swing away

If the defense is playing back, bunt

No one is even on the right track so far…

Steal second.

Please clarify this: “He positions his fielders accordingly (optimally).”

What is the optimal defensive positioning in this situation?

If he’s playing the optimal fielding arrangement, then by definition either option should be of equal utility to you.

To clarify that, an optimal defense would be one that could not be exploited.  If the bench coach (or whoever was setting the defensive arrangement) used his game-theory skills and saw that I (the manager) could gain a slight advantage by bunting, he’d bring in his corner infielders a little bit.  If he saw that there was an advantage to swinging away, he’d push them back.  You can’t gain an advantage either way because he gets to choose the defensive positioning before you choose to bunt or swing away.

What do you do? 

Absolutely nothing.

Stop the action!  Jake is the winner!  He is correct. If the defense is playing optimally, which I said they were, then it does not matter what you do - bunt or hit away. Your WE must be exactly the same either way - by definition.  If that were not the case - if one alternative was better than the other, then the defense would NOT be playing optimally.

Jake, send me your address and I’ll send you an autographed copy of The Book.  If you already have it, let me know what book you want.  Nice job!  This was a tricky one.

BTW, the lesson here (for actual teams) is that the defense is supposed to play optimally such that for each and every potential bunt situation, it does not matter what the offense does.

However, in reality, no one can know exactly what that position is.  So you play up more for batters who are better bunters and faster runners and weaker hitters, and back more for the opposite.  And of course you tend to play back more early in the game and up more late in the game, all other things being equal.

Now, this defensive position is ONLY correct if you know or suspect that the offense is going to play optimally as well.  In baseball that is rarely the case.  Most managers either bunt too little or too much and are quite predictable given the batter and the overall game situation.  In that case, if you KNOW that the batting team is going to bunt too much (that means that their chance of attempting a bunt is higher than game theory would dictate assuming that YOU, the defense, would play optimally), it is correct for you to play IN more than would be optimal if your opponent were a perfect strategist.  But, if you play too far in, you might force the batting team to correct their mistake and bunt less often.  You might force them to play optimally, in which case YOU would now be playing sub-optimally.  If the batting team bunts too infrequently, then it is correct for you, the defense, to play further back.  Same thing though.  You don’t want to force your opponent into playing optimally.

In practice, I think most teams “back into” playing more or less optimally on defense, however, I think that offensive teams make many, many mistakes in terms of their strategy.  It is almost NEVER correct for a batter to bunt even close to 100% of the time. If it were, then the defense would be able to play so close as to negate the advantage of the bunt.  And although it is often correct for certain batters to never bunt (because they bunt so poorly, are too slow, and/or hit too well), better batters who have some speed and can bunt a little generally bunt WAY too infrequently.  For example, if the defense is playing back and you are a very good hitter and a decent bunter with decent speed, you MUST bunt some percentage of the time, if nothing else, to force the defense to play in a little against you so that when you swing away, you get some extra benefit.  In fact, even for a very good hitter, if the defense is playing all the way back, and the very good hitter is also a decent or beter bunter with decent or better speed, his WE by bunting will actually be higher than hitting away.

Anyway, most of this is discussed thoroughly in the sac bunt chapter of The Book.

So the correct answer isn’t to show bunt in hopes the defense gets out of the optimal position? Would the cost of a (virtually) guaranteed strike cost more than the benefit from getting the fielders to shift slightly?

Why doesn’t Wayward win with post #1???

I was thinking the same thing...Wayward should be the winner.

No he does not win.  The answer is that it does not matter what you do. I also expected an explanation, as Jake provided.  I specifically said when I started this game that “work must be shown.” Flip a coin is not an acceptable answer without some explanation as to why that would work. Plus, given that it does not matter what you do, why is 50/50 the correct answer?  Why not “bunt all the time?” That works just as well doesn’t it? I have no idea what he was trying to say or whether he was just joking as some of the posters were.  If he was trying to tell us what the optimal play for the offense was, that is not necessarily right anyway.  I did NOT want the optimal play for the offense which is why I specifically said that this is your last game as a manager and that there were no future repercussions or issues.  Flipping a 50/50 coin is NOT the same as, “It doesn’t matter what you do.” That is the correct answer.  If there were future issues and this were not your last game, then there is NOT answer to the question because it would depend on a host of variables. It could be 50/50, it could be 70/30, or it could be 40/60 (or anything else).

The answer is that it does not matter what you do.

That’s pretty much the defnition of “flip a coin”.

I also expected an explanation,

FFR, say “Show your work”, or “Explain Why”.

"And although it is often correct for certain batters to never bunt (because they bunt so poorly, are too slow, and/or hit too well)”

Ok, I have to ask this somewhere, and since it’s sort of brought up here I’ll ask here and hope it’s not overlooked.

I’m starting to wonder if left-handed sluggers need to bunt more. Recently, Jack Cust has dropped down a couple nice bunt singles. Were they “good” bunts? Well, when the third baseman is playing shortstop, a “good” bunt is basically anything fair and not directly at the pitcher. So what happens here?

1) Cust gets on base, and since these bunts are coming with nobody on, it’s equal to a walk (something he does a lot).
2) Cust does not make an out (Obviously)
3) Cust does not get an extra base hit (though there’s the error possibility, which for sake of this question matters). He’s a home run threat, so he turns x% chance of a home run, and 1 run, into just a single.
4) He “beats” the shift this time, and if he drops down a successful bunt a few more times, teams might not shift. Or they might alter the shift to something that he can more easily beat by swinging away. The game theory aspect is the one I’m interested in most.

So somewhere in there is an equation that weighs the straightforward likelihood of bunt single vs. all the other possibilities. I don’t know the math, but I suspect it would not be hard to figure this out for Cust, Thome, Giambi, Ortiz, etc.

But what about game theory and forcing teams to not shift? How many times should a guy like this drop down a bunt (even if it’s not always successful, or even if the bunt is foul or just a fake attempt)?

I think all the questions that come from this can be inferred here. I suspect the answer is “guys like this should bunt for ‘easy’ singles more often,” but I’d like to know if that’s true.

Sal, since the defensive manager is the perfect strategist and assumes that you are as well (those were two of the conditions), he is going to play his defense at a position such that the WE is the same regardless of what you do.  You can try and “bait him” all you want, but he is not going to fall for it, according to my conditions.

In a “real” game, whether faking a bunt and then taking a pitch to force the defense to play up (and then swinging away) some more is worth the extra called strikes, I don’t know. You can try and figure that out if you want.  OK, I’ll try. If you take the first pitch, you are going to be 1-0 around 45% of the time and 0-1 around 55% of the time.  With an 0-1 count, the RE is around .878 (runner on first and 0 outs).  With 1-0, it is around 1.037.  So 55% 0-1 and 45% 1-0 is .950. At 0-0, it is .961. So you lose around .01 runs or so. If the pitcher throws 60% strikes in anticipation of a bunt, you lose .02 runs. You normally have around an 8% chance of a GB or line drive hit.  You would need to increase that by 2-4% (to 10-12%) to justify taking that first pitch in order to draw the IF even closer.  OF course once you don’t bunt on the next pitch and you don’t put the ball in play, the ruse is up.  And if you’ve done that before…

From the initial post:

You will only receive credit for a correct answer if you give a (correct) reason why.

And “flipping a coin” is NOT the same as “it doesn’t matter what you do.” At least not in this context.  As I said, if it doesn’t matter what you do, then everyone’s answer would be correct according to your logic.  “Flipping a coin” implies that there is some game theory involved and that the optimal ratio of bunts to non-bunts is 1-1.  That is simply not correct. Not only is there NO game theory on your part (because this is your last game and you already know EXACTLY how the defense is playing before you make your decision), but if there were, there is not reason why the correct ratio would be 50/50, as I said.  That would depend on many variables.  Presumably the defense knows all of these variables and how to process them (otherwise they would be able to play optimally and force you to have no decision).

JD, yes playing against the shift or potential shift is a similar situation and involves game theory.  It is likely that most if not all of the players against whom the shift is employed should drop a bunt down the third base line some percentage of the time (actually all the time until the shift changes).  As soon as the batters do that a certain percentage of the time, the defense would be forced to abandon the shift or to lessen it to some degree.  If the batters were bunting/not bunting optimally, then the correct position for a shift or non-shift would be such that it would not matter whether the batter bunted or not, given his bunting skill and speed, just like our sac bunt question.  Again, I suspect that against a shift that most of these guys can drop a bunt down enough of the time to “beat” the shift and force the defense to reconfigure itself.  But they don’t bother for various reasons.  Mostly because it’s not “cool” for a power hitter to do that.

Well I feel like a moron.

I also had it correct, in a way. : )

Although I posted after Jake and didn’t show my work. 

Thanks, mgl! These have been fun.

This was a game theory question, not a baseball question.  If one approaches it from a baseball standpoint, they’re going to ask the types of questions that Dial did in post 3.

Now, the way MGL stated the question, it was clear that those considerations didn’t matter.  Cool, I get what MGL was going for.  But it’s not realistic.  We’ve been told to assume that the opposing manager is playing the optimal strategy.  That’s not a realistic assumption; no matter whether you hold your opposing manager with ultimate esteem, you’re going to consider the handedness of the pitcher and first baseman, whether the pitcher is throwing from the windup or stretch, the defensive prowess of the 3B, etc.

If none of that stuff matters, if we’re asked to assume that the opposing manager was perfect on not only all that stuff, but also anything else that any of could come up with?  Well, then it’s not a useful discussion from a baseball standpoint.  Feel free to ask game theory questions, MGL, but don’t pretend they’re baseball questions.

Or, as I put it in the Primer Lounge tonight:

Ah, it would make managing so easy, if you could assume that your opponent was playing everything perfectly optimally; it would free one from actually looking at details like the abilities of your opponents, your lineup, etc.

Given that the opposing manager is game-theory perfect neither move can be better than average, however if you can create a situation of asymetrical information you can do better than average, for instance having your batter try to bunt on the first pitch makes the opposing manager play you to bunt more as a rational move. However if you know that you are planning to swing away on the next pitch (or the other way around) then you can gain an advantage.

I can see flipping a coin is not correct, because of the implied 50/50 ratio.  Maybe the correct thing would be to bunt 1/3 of the time, or 1/5.

But “It doesn’t matter what you do” doesn’t answer the question.  You still have a baserunner, a batter, and a tie game.  You have to do SOMETHING.  Even if it doesn’t matter based on WE%, you still have to make the call one way or another.

MGL - If there are more than 2 offensive choices that require more than 2 different optimal defensive positions, I am not sure that it would be able to find a single defensive position that would equalize the win value of all the offensive choices.  For example, the question states that the batter is a good bunter.  If the defense takes that into consideration and optimizes for hit away and the possibility of a sacrifice bunt, what happens if the batter attempts to bunt for a hit?  Bunting for a hit is a completely different strategy than a sacrifice.  The batter doesn’t show bunt early and usually bunts to a completely different location on the field.  The problems for the defense are even worse if the offensive manager sends the runner on a run and bunt.  Either the shortstop or 2nd baseman must abandon his optimized position to cover second for the possibility of a straight steal attempt or a hit and run.  This increases the possibility of a bunt for a hit toward the moved fielder’s initial position either resulting in a hit or a sacrifice.  I think that the offense, because it chooses its strategy last, can usually make an advantageous strategic choice if there are 3 or more strategies to choose from, each requiring a different optimal defense strategy.  But its been 40 years since I took my course in game theory so my memory might be completely faulty on this.

MGL - In case you missed it, I added some information to the 06-02 question thread on the rate at which pitchers are pinch hit for in the 4th, 5th, and 6th innings in BaseOut situations with a leverage index greater than 2 and meeting all the other conditions you set in your earlier post there.

If we accept that both sides had all the known information, both sides knew that they all had the known information, and both sides act with that knowledge, so that the defense positions itself optimally, you are still left with the specific choice of what the pitcher will throw and where he will throw it.

If optimally the fielders should position themselves such that the 3B is going to play at 85 feet, so that the bunt and swing away is a breakeven, this is still based on the fact that the 3B will expect the pitcher to throw what the manager wants him to throw (say a fastball) and where the manager wants him to throw it (say inside).  If the pitch actually ends up going outside, then this means that the 3B is out-of-position (in the optimal sense).

The question therefore is how to best take advantage if a specific chain in the link is broken.

I would presume therefore that it would be better for the batter to do as the other reader said:
“I would tell my hitter to bunt if he gets the optimal pitch to do so (basically, telling him not to force it). “

Or, tell the batter to swing only if the pitcher doesn’t hit his intended target (which in an optimal sense you would know what it is, if you knew it was a fastball coming).

The correct answer is to bunt the runner to 2nd base. Then have the next hitter bang a triple and listen to Steve Phillips remark how you scored the run by playing small-ball.

I also agree with the other reader who says if you “show” bunt, and not bunt (unless it’s the “perfect pitch” to bunt on), this will give the opponent an extra piece of information to consider.

By “showing” bunt, he doesn’t know if you did intend to bunt or not.  So, he will have to rearrange his alignment to be optimal.  But, since you have given him false information (and he doesn’t know it’s false information, or he only suspects, but is not certain it is false), you gain an advantage.

Basically, in order to beat the other guy’s alignment, you need to leverage information that the other guy doesn’t have.  And the only information he doesn’t have is:
- where the pitcher is actually throwing the ball (as opposed to intending to throw)
- what the batter is showing he is doing (as opposed to what he intends to actually do)

You as the offense actually have that information after the fielding alignment has been set.

Hit away.  But I am going to fake a bunt and then swing away drawing the infielders out of position raising my chances of getting a hit.  My batter is fast so I am also not too worried about hitting into a double play.

Tango - As long as the only choices are to bunt or to hit away I don’t think it makes any difference where the pitcher pitches the ball or what false information the batter has displayed.  As long as the defense stays in that alignment they positioned so that there is equal win value for the batter to bunt or not to bunt and each fielder is positioned individually so that if the batter hits away there is equal win value for him to attempt to hit to the left or right of that fielder or behind or in front of him given his likely hit distribution.  Remember, MGL has already told us that he is 100% certain that the batter cannot control his hit distribution.  I think you have to introduce a third option for the batter as I suggested in Post# 30.

Dang, that (game theory discussion - the other manager is playing optimally so there is no way for you to improve your WE) was what I was going to answer, if I had seen this last night.  But thats just because I have already read The Book, and its discussions on game theory and sac bunts.

I was thinking that we would need to calculate what the most optimal percentage of the time to bunt in this situation was, so as not to allow the opposing manager to know that we either bunt too much or dont bunt enough.  Then use a random method of determining whether to bunt.  (Like the flip a coin suggestion, but having calculated what the correct percentage is, which is probably lower than 50%).

But then there is the part in the question where it says this is your last game.  That means that you dont care about the effect of your decision on the future, this manager wont be able to apply anything he learns to future at bats, and therefore the answer is that it doesnt matter what you do.

Yes, it was somewhat of a trick question, with limited practical significance, as I was running out of difficult ones. However, it is useful for teams to understand how to set a defense so that the opposing team cannot get any advantage from a certain bunt strategy.

Tango, since when can a manager decide whether to bunt or not based on the pitch? Not only that, but even if the batter decides on his own what to do after he sees the pitch coming (which is almost never done, by the way), the opposing manager knows what the pitcher is going to throw and he adjusts his defense accordingly! Anyway, as far as I know, the manager gives the bunt or no bunt sign before each and every pitch. Other possibilities are fake bunt and swing, or fake bunt and take, and of course bunt and run.  And whether he fakes on one pitch and swings on another makes no difference.  If the defense is playing optimally, none of that affects the WE.  And as I already said in my last post, since the opposing manager is a perfect strategist, and he assumes that you are too, he is NOT going to fall for any trick strategy.  So what is the point of faking a bunt on one pitch, or something like that?  In fact, the opposing manager hopes you do that!  Now you have taken the first pitch, incorrectly so, and the opposing manager doesn’t move. You just screwed yourself!

Was mgl posting this from Europe?  I saw this question last evening.

Focus on the location of the pitch.  If the defense is playing based on the pitcher throwing a fastball inside, and the pitcher instead throws a fastball outside, suddenly, the fielding alignment is NOT optimal.

And there is nothing they can do about it, since the batter will have that information (pitch location) before any of the fielders (other than the catcher).

You are talking about it like a card game, where the knowledge is learned at the same time, before the next event occurs.  However, in this case, once the pitch is in flight, it is the batter (and catcher and pitcher) who have the extra information (by 0.20 seconds or whatever) to react to it over the rest of the fielders.  Well, the pitcher will know right away, but he won’t be able to impact the play.  The catcher is limited in his impact.  The batter has the biggest impact.

The question is whether the batter has the best chance to be able to counteract the suboptimal defense once the pitch is in flight, by swinging away or bunting.  IF he can make the decision to go one way or the other based on how he perceives the pitch location that is different from how the fielders are aligned, then he should do that.

IF he has to make the decision prior to the pitch being in-flight, then you are right, it won’t matter.

Herzog - The battle of wits has begun!  The battle ends when my pitcher delivers and your batter swings.  Then we discover who is right… and who is out.

LaRussa - But it’s so simple… based on what I know about you, I can easily deduce the optimal strategy.

Herzog - And that is?

LaRussa - You were an NL manager in a low-scoring environment, which places a premium on single runs over large innings, so I can expect you to defend against the sacrifice, so I can clearly choose to swing away.  BUT, you know that I have also managed in the AL, and have had some of the game’s most feared sluggers on my rosters, and anticipating this, you would expect me to play for the big inning; so I can clearly choose to bunt.

Herzog - So you’ve chosen?

LaRussa - HARDLY! Your infield is playing slightly back with the middle infielders ready to turn two, so I can clearly bunt - BUT it also very fast and strong-armed, so you can do this while still covering a bunt attempt, so I can clearly choose to swing away.

Herzog - Truly you have a dizzing intellect.

LaRussa - Wait ‘til I bat my pitcher seventh!  You’ve beaten my ace, which means your lineup is very strong and can score more runs, so I can clearly choose to swing away… BUT you’ve never beaten my closer, and if I can get even a one-run lead to him, my win expectancy is something like 92%, so I can clearly choose to bunt.

Herzog - You’re only stalling now.  It won’t work.

LaRussa - It already has!  You’ve given everything away!

Herzog - Then make your choice!

LaRussa - What in the world can that be?

Herzog - Where? [while he looks, LaRussa sends his runner on a delayed steal] I don’t see anything!

LaRussa - Well, no matter… heheheheheh…

Very clever nightfly

mgl: I have a question. What pop culture item is nightfly referencing?

(Sorry no presents for the correct answer)

Pop reference in post 5:

http://www.insidethebook.com/ee/index.php/site/comments/mlb_pitcher_agrees_with_dips_we_can_finally_come_out_of_our_basements/#5

Re #42:

Clearly its a reference to the Princess Bride.

I’ll go with what Alex said.  Is that truly a pop culture reference or just a parody of a movie?

If in fact it’s optimal to bunt half the time and to swing away half the time, then the optimal strategy is to pick one...in advance (you are, after all, an expert strategist and an expert in game theory)...not to toss a coin.  Picking one (in advance) means you are right half the time.  Flipping a coin means you are right 1/4 of the time.

I admit that I did not “show my work”.

I used to play a game (War and Peace) that involved a “tactical matrix” during battle.  You could choose to Charge, for example, but if your opponent chose “Square” you got to re-enact Waterloo.  My opponent knew me well so I would often make quite the show of randomly choosing my tactical move, since the matrix was essentially zero-sum.  Used to irritate him.

There were times when I needed to Withdraw, so I would then also make a nice show of a random pick and then make sure I did some sleight of hand to play Withdraw.  This worked about twice.

If the opposing manager played the optimal strategy then I would flip a coin, in full view of the opposing manager.  If I thought I really did have an advantage, such as knowing some piece of information that might lead to my advantage then I would still flip the coin (in full view) and then do what I had planned anyway.  I might have only two opportunities to do this against any particular opponent in one year.

I endorse Peter/30 and Tango/40. For both of those reasons, it’s not as simple as Jake’s answer.

If I were the manager I might call for a drag bunt past the pitcher. The opposing manager might be able to position the rest of his infield optimally, but he can’t change the positioning of the pitcher!

Tango/40 is right, but I think it is more subtle.  The opposing manager, playing the optimal strategy will base his positioning on the possibility that the pitch location will go awry with some probability.  He will have to optimize over the ensemble of potential pitches his pitcher will throw.  The batter still has an advantage, because this ensemble collapses before he makes his decision (this is clearly true, batters decide to take or swing at pitches only after acquiring information on their flight). 

But, of course, the opposing manager knows this!  So, he will position his fielders assuming the batter will be able to take best advantage of the pitch thrown.  Thus, I think the opposing manager/pitcher/catcher must also be indifferent to the pitch type and location, adjusting his fielders so that the batter does not gain advantage by producing the optimal outcome given the pitch flightpath.

(No this isn’t parody, I’m actually trying to think this through.)

Well, I was aiming for “homage.” The back and forth over optimal strategies and such reminded me of that scene with Westley and Vezzini.  Since I’m a relative duffer at sabermetrics it was the only contribution I could offer…

Sal, since the defensive manager is the perfect strategist and assumes that you are as well (those were two of the conditions)

I was considering this:

Either that or he has no idea what you are going to do and doesn’t suspect that you are either going to bunt or not bunt.

But it’s all good.