THE BOOK--Playing The Percentages In Baseball

For an excellent study on the sacrifice bunt, I highly recommend Tom Tippett’s resent presentation at the New England Symposium on Statistics in Sports:

http://video.yahoo.com/watch/6197243/16088891

Not that much less than 50 pages!

I would have to say that I don’t think MGL could have made it any shorter.  He basically had to lay it out like that…

I’m curious, has anyone actually applied Game Theory to a certain bunt situation?  MGL laid it out beautifully, but it would be nice to see it be applied using real numbers.

I kid, Tango, I kid.

Guy: that’s the way I took it.

***

I was just using it as a jumping off point to state that this is a tough thing to evaluate in isolation. 

This is no different that saying that a pitcher should not have thrown a fastball in that spot, or Vlad should not have swung at the low pitch in the dirt.

Funny though how they never say he should not have swung at a pitch at his ankles that goes for a double.

That’s what you call a cowardly analyst.

Tango, I have a long comment with no hyperlinks which doesn’t seem to want to post.  When I click preview, it just sends me back to this page.

Happens for some reason… me too.

Copy/paste to a text editor, then open a brand new browser, and paste it back.

See if that helps.  Otherwise, email it to me at tom~tangotiger~net and I’ll take care of it.

Tango, thanks for the link. I thought it was a really good article but the comments on Fangraphs are pretty weak to say the least…

I thought it was a great article.  My 2 comments:

1. In Game 6 of the ALCS, I want my manager to maximize the WE of that particular game.  I do not care about repeated iterations or other managers learning from what I do in this one particular game.  There will be no coin-flipping in the playoffs (unless the defense really is optimally positioned).  I am going to assume that Charlie Manuel is not going to learn much from this particular bunt decision--and if he does learn, I’m more than willing to trade the increase in WE in Game 6 for any potential decreases in my chances in Game 7/World Series. 

2. Any advantage to be gained by telling my defense to wait until the 3B coach is finished with his signs before moving into position?  If I understand the article correctly, it shouldn’t matter if both teams are playing optimally.  But if a real offensive manager has to guess at what the defense is going to do instead of knowing for certain--that has to help the defense.  Even if I end up playing the optimal defense as described in the article every time, I might be able to fool the non-optimal playing opposing manager.  (I assume the MLB offensive manager would not give most batters the freedom to make their own bunt decision based on where the defense lines up).  Also, I know that if I want to create real doubt about what I’ll do, I’ll have to change the defense after every pitch in random (as that term is used in MGL’s article) ways, and I don’t think that would be worth the trouble.

Again, great article.

Tom, did you get my email?

Should the randomness of choices carry over during the at bat as well, maybe even during change of state (throw to first, pitch time, batter time, etc)?
i.e. if you are going to bunt 20% of the time does it make sense to ‘roll you 5 sided die’ every chance you get taking into account defensive position.
Is showing bunt, calling time, and hitting on a move up infield too obvious?

blackadder: I’m having trouble posting too.  I’ll look into it tonight.

Ok, I’ll just post it in two comments, this seems to work.

Part 1

This is very good, although I am not sure I agree with some of the details. I’ve been grading a lot of freshman calculus exams lately, so apologies in advance for this little model, which is obviously heavily inspired by MGL’s discussion.

We suppose both managers behave optimally. Let t ranging from 0 to 1 be the possible positions the infielders can play, so t=0 means that the infielders are playing as far in as possible, and t=1 means that they are playing as deep as possible (or as deep as realistically possible, since I guess they could play as deep as the fences!) We assume for simplicity that there is only one free parameter in terms of how “deep” the defense can play, while of course there are many in reality.

For a given hitter, in a given game state, with a given defense, we have the functions b(t) and h(t), where b(t) is the expected value of the bunt with the defense playing position t and h(t) the expected value of hitting away. We assume that b(t) is increasing, h(t) is decreasing, h(0) > b(0) (so hitting is preferred to bunting with the defense playing in) and b(1) > h(1). At the Nash equilibrium, the defense will play some position t_0 between zero and 1 and the offense will bunt some fraction 0 < c < 1.

t_0 is easy to find, as MGL indicates. Namely, our assumptions about h(t) and b(t) guarantee that there is a unique point x between 0 and 1 where h(x) = b(x). MGL’s argument in the link shows that if the defense is playing any position t’ not equal to x, it cannot be in a Nash equilibrium (the offense would have to bunt or hit 100% of the time, and the defense could then improve by changing their position.) Of course, if the hitter is a good bunter then b(t) is bigger, so x is closer to 0, while if he is a poor bunter or a good hitter x is closer to 1, all of which obviously makes intuitive sense.

Part 2 next

Part 2

Now, what should the batter do? He bunts some fraction c of the time, and no matter what of c he picks the total value of his PA is c*b(x)+(1-c)*h(x)=(c+1-c)*b(x)=b(x) since b(x)=h(x). However, there is still a constraint: for the optimal value c, the function f(t) = c*b(t) + (1-c)*h(t) (which is the value of the PA as a function of t if the batter bunts with probability c) must have a local maximum at t=x. Otherwise, by either moving in or out, the defense can decrease the value of the PA, which contradicts the Nash equilibrium assumption.

So we must have c*b(t) + (1-c)*h(t) with a local max at t=x. From freshman calculus, we know that a necessary condition for a local max is that the derivative vanishes at that point. So

0 = f’(x) = c*b’(x)+(1-c)*h’(x)

Which gives

c = - h’(x)/(b’(x)-h’(x))

Our assumptions about b(t) and h(t) guarantee that this is always strictly between 0 and 1, which is obviously good.

Note, though, and this is the key point, that the optimal ratio is independent of how good a bunter the batter is! All that the matter is the LOCAL behavior of the two functions at the indifference point.

For instance, suppose ARod is a bad bunter and a good hitter, so the defense plays him very far back, say at .9. But at .9, it may well be the case that b’(.9) >> h’(.9), or in other words, that the marginal improvement in ARod’s bunting value when the defense steps back is much bigger than the corresponding marginal decrease in his hitting ability. In this case, ARod should be bunting most of the time. Similarly, you can be a very good bunter and poor hitter, but still be advised to swing away when the defense is playing you optimally. So I don’t think I agree with MGL’s recommendations about how much different players should bunt

This explains why the offensive manager’s job is much harder the the defensive manager’s. The defense only has to know when the two functions are roughly equal, which is something that can be discovered implicitly through trial and error. It is very hard, however, to get a sense of the derivatives of these functions through trial and error. So it makes a lot of sense that offensive manager’s make more bunting mistakes than defensive ones.

Does this make sense? Is there some other parameter that people think should be included in this ridiculously simple model?

I think Blackadder may be right that the optimal percentages for each batter (given the pitcher, game situation, etc.) may not have anything to do with how good a hitter or bunter the batter is. It may only have to do with the h(t) and b(t) function and how they gain and lose as t changes.  Is that what you are saying, Blackadder?  The ability of the batter to hit and bunt may play into that.  For example, if the batter never hit ground balls or pop flies to the third base side, then h(t) would never change as the third baseman moves.

It may be this, in fact, if for all batters, h(t) and b(t) changes by the same amount for any value t:

It may be that batters are either supposed to bunt 100% of the time (if b(0) > h(0)), hit away 100% of the time (if b(1) < h(1)), or bunt/hit away 50% of the time (or some other percentage - depending on the b(t) and h(t) functions) in all other cases, regardless of the batter (more or less).

So I may have been very wrong in stating that for batters who should probably bunt and the defense plays very far in, the percentage should be like 80 or 90%, and for batters who should probably not bunt, and the defense plays almost all the way back, that they still have to bunt 10% of the time.  It may be that for batters against whom the defense plays almost all the way back or almost all the way in, and everything in between, they should bunt 50% of the time (or some other fixed percentage).

Yeah, sorry, I should have made the “punchline” clearer.  It is precisely what you are saying here, that the shape of the functions h(t) and b(t) are what are important for determining whether to bunt, not how good the bunter is.  The quality of the bunter matters a lot, of course, but it matters in terms of setting the defenders position (i.e. x), not in terms of how much he should bunt conditional on the defense playing optimally.

I actually don’t have a good sense for how the functions would roughly look in practice.  My roommate tonight made the argument that when t is small, b(t) increases at a much faster rate than h(t) decreases.  In other words, when you are already playing well in for the bunt, moving in or out has a big impact on the bunt, but a smaller impact on your chances of getting a hit.  I don’t know whether that is true, but it is plausible.  If it is true, it would imply that when the defense is playing in, and doing so rationally (i.e. the hitter is a good bunter), the player should bunt over 50% of the time. 

It should be emphasized, of course, that all of this assumes optimal decision making on both managers.  It also, perhaps more worryingly, assumes away the asymmetry in the situation, i.e. that hitters can see the defensive alignment before deciding to bunt.  Perhaps considerations of iterated games would fix that.

In any case, the analysis at the end of MGL’s piece, of the decision making of the Girardi and Sciosca, was clearly not in the realm where both managers were acting optimally, and so I don’t think gets touched by any of this.

Yup, I agree with all.  If the offense does not get to see the defense before making its decision, it would still have to bunt/non-bunt in the same optimal proportion, if it assumes that the defense is going to play optimally.  The only difference in the dynamic when the offense gets to see the defense, is that if the defense makes a mistake (by not positioning itself such that b(t) = h(t)), the offense can alter its original strategy by either bunting 100% of the time or not-bunting 100% of the time, if it is not afraid that it will teach the defense the position themselves better next time.  If it is afraid that the defense will correct itself next time, then the offense can alter their strategy a little, but not to the point that it now bunts (if the defense is playing too far back) or hits away (if the defense is playing too close) 100% of the time.

Of course the mistake that defenses would be likely to make against the rare optimal offense that it might encounter would be playing too far in when most teams tend to bunt (late in a close game and a weak or mediocre batter at the plate) or playing too far back when a hitter like A-Rod is at the plate.

The key is figuring out that optimal percentage for the offense to bunt or hit away when it is not 100% (when the correct defensive position is somewhere between back and in) for each batter.  I am trying to do that, with no luck so far.

I’ve been playing around with the numbers and I think it is going to depend on how much the defense moving affects hitting away versus how much it effects bunting.  I suspect that moving affects bunting more than hitting away, which means that all batters would have to hit away more than bunt.  The correct percentages might be 70/30 or something like that.

That relationship between the effect of moving on hitting away and bunting might be different depending on where the defense is supposed to be playing, as your friend suspects, so that means that the proper percentages might indeed be different for different kinds of batters, depending on where the optimal defensive position is against them.

For example, say we have a batter against whom the proper defensive position is half way.  Again at this position, t, b(t) = h(t).

And let’s say that moving back reduces h(t) as much it increases b(t), and the same in reverse for moving up.  Then clearly if the batter bunts 50% of the time, there is nothing the defense can to do take advantage of that.  If b(t) = h(t) = .500 wins with the defense playing optimally and the batter bunts 50% of the time, what happens if the defense moves in?  If b(t) is now .480 then h(t) will be .520 (we are assuming that the value of one goes up or down the same amount as the value of the other as the defense moves).  So 50/50 yields the same WE as if the defense doesn’t move.

So there is nothing the defense can do to change the WE if the b(t) and h(t) values change at the same rate and the batter bunts 50% of the time.

But what if the bunt value, b(t) changed at twice the rate of the hit value, h(t) as the defense moves?  Now the batter cannot bunt 50% of the time.  If he did, it would be correct for the defense to move in from their original optimal position where b(t) = h(t).  Let’s see how that works.

If the defense moves in such that b(t) is now .48, but h(t) is only .51 (the bunt value changes by .02 wins but the hit value only changes by .01 wins), then if the batter bunts 50% of the time then with this new defensive configuration, the average WE would now be .5 * .48 + .5 * .51, or .495, which is a gain for the defense.

So if the bunt value moves more quickly than the hit value, then the batter has to hit more than he bunts. How much more?  Depends on how much more quickly the bunt value moves than the hit value as the defense changes its position.  In the above example, the batter would have to bunt around 35% of the time and hit away 65% of the time for it to not matter if the defense moved.

Now, we know that with the good bunters/weak hitters, the defense will be playing up quite a bit and with the lesser bunters/stronger hitters, the defense will be playing back quite a bit.  Whether that changes how much b(t) and h(t) move as the defense moves, I don’t know.

If h(t) and b(t) both moved around the same amount when the defense was playing pretty far back, then the lesser bunters should bunt around 50% of the time.  If the bunt value changes a lot when the defense is playing pretty far in but the hit value doesn’t change all that much, then the good bunters should actually hit away most of the time, like in the 65/35 example above. I don’t know if this is the case.

Off the top of my head, I would say that it is more likely that the bunt value changes faster than the hit value as the defense moves, if only because the hit value is only affected by the infield defense on ground balls hit down the third base and to some extent the first base line.  If that is the case, that b(t) changes faster than h(t) for all starting positions, then all batters who are not supposed to bunt or hit away 100% of the time should hit away more often than they bunt.  I would probably recommend 65/35 off the top of my head, assuming that the defensive was playing optimally.

However, I think that defenses will tend to play too far in against batters and game situations where the bunt tends to be common, so I would recommend that in these situations, offensive teams hit away 70-80% of the time.  With players at the plate who typically don’t bunt, like an A-Rod, but who could, the defense probably plays too far back (although I am not as sure of this as I am of the other), which means that batters should bunt maybe 40%-50% of the time rather than 35%.

Again, these are all educated guesses.

Interesting.  You can actually read of the exact ratio from the model.  If the rate of chance of bunting value is a and hitting value is b, then the correct ratio is b/(a+b).  I actually made a stupid mistake and misread this, so some of the statements I made in discussing the implications are exactly wrong.  Note, though, that this formula is totally consistent with your numerical examples.

Those educated guesses seem very plausible to me.

Good comments Blackadder!  Even though I have talked about bunting and game theory many times on this blog, no one has ever addressed the issue of how to figure out the correct ratio of bunt/non-bunt and what it might be.

Thanks!  And thank you for the article.  I only came up with the little model by trying to understand where you were getting your ratios from; I would not have come up with it without your discussion.

I would love for someone to be able to take the empirical data, make some assumptions about where the defense was playing, and come up with an estimate of what the b(t) and h(t) functions might be. If we could do that, we could then come up with the correct ratios (or close enough for government work) for the offense.

It looks like the BP staff could stand to read your primer.

http://www.baseballprospectus.com/unfiltered/?p=1421

I can’t believe this actually came from BPro.

When you have one of your best hitters at the plate, one who has a knack through coming through with runners on base—Jeter has a career .308 average when runners are in scoring position—you’d surely give him a chance to knock in some runs, especially one with decent base running skills, like Melky Cabrera, is on second.

BA w/ RISP?  Really?

There’s always a chance that Jeter hits into a double play—Jeter ranked 20th in the league in DP% with 17 percent—and he is a ground-ball hitter, but when you’re in the World Series and have one of your best hitters at the plate, it’s better to take the risk.

That’s one of those classic BS things you here from announcers.  Even ignoring the part where they use a generic RE table to apply to a very specific situation, the last line is unbelievable.  Because it’s the world series, you should take more of a risk?  Where is the basis for that?

"It looks like the BP staff could stand to read your primer.”

You know what, Mike.  People actually do read it, or claim to read it, and even acknowledge it (as being correct) sometimes, and then when their fingers hit the keyboard on the computer, they write whatever crap comes to mind.  That happens all the time.

Nick, yes, that is mind-boggling B.S from B.P.  Maybe they should just change the P to an S.

i am having trouble posting a comment here.

ok lets try it again.
based on the defensive alignment, the batter should bunt with frequency either 100%, 0%, or it doesnt matter.
i’ll continue the assumption that the batter gets to see the defense before he decides. so the value of the at-bat (using Blackadder#14 notation) is:
V = c*b + (1-c)h
but once the defense is set, he has no control over b and h, they are constant. so V is a function of c:
V(c) = (b-h)c + h
if the defensive alignment is optimal, b=h, and plugging that in gives V=h (which equals b). the choice of c (the frequency for bunting) doesnt matter.
if the alignment is not optimal, and b>h, then you maximize with the biggest possible value for c, so c=1 or 100%, always bunt. vice versa, if h>b, the batter chooses c=0, always hit.

so if the defensive alignment is optimal, it doesnt matter what the batter chooses. if the alignment isnt optimal, you exploit that every time for all its worth.

the big trick, i think, is to figure out the actual optimal defensive alignment - no easy way to do that. also, i bet that for some hitters, like Pujols, h>b always - its better to hit than to bunt, no matter where the defense stands.

"i bet that for some hitters, like Pujols, h>b always - its better to hit than to bunt, no matter where the defense stands. “

Yes of course.  And for some pitchers at some time b is always greater than h even with the defense playing all the way up.

“the big trick, i think, is to figure out the actual optimal defensive alignment - no easy way to do that.”

That is true too.  So if you think that they are playing anywhere near optimally, you bunt and hit some proportion of the time.  If you think they are playing too far in, you hit more than that.  If you think they are playing to far back, you bunt more than that.

“based on the defensive alignment, the batter should bunt with frequency either 100%, 0%, or it doesnt matter.”

It doesn’t matter for that PA, but unless you bunt and hit at an optimal proportion, you will quickly “teach” the defense to correct their mistake.

I actually think jsolid is right, IF both the hitter and pitcher are perfectly rational.  It really doesn’t matter, since the offense can see the defensive alignment.  If the defense tries to move in, say, to take advantage of a non-optimal ratio, the hitting team just immediately starts hitting 100% of the time until they move back.

MGL’s view is more of an adaptive expectations sort of position.  So the offense has developed the optimal ratio, then the defense moves in, but the offense does not adjust their strategy until they start to get burned.  The adaptive expectation part is also evident when he argues that teams should not hit away 100% of the time even when it is advantageous to do so, in order not to “teach” the defense about its error.

@MGL#30
‘If you think they are playing too far in, you hit more than that.’
if they are playing too far in you hit *always*.
‘unless you bunt and hit at an optimal proportion, you will quickly “teach” the defense to correct their mistake.’
the defense is playing there because they think that is the proper alignment. so, for example, if they are at a certain depth and you think it is optimal to bunt all the time (at that depth, for that batter), i dont think there is anything the defense will learn. first of all, even at the optimal alignment, a 100% bunt strategy is still possible. second, perhaps the defense will think, boy i cant believe they are pursuing that non-optimal strategy, they are playing into our hands.
the really big obstacle here is that it is hard to determine the optimal defensive alignment, so both teams may arrive at different conclusions. further, each can stick to those conclusions even after repeated interactions. however, to the extent it is possible for the defense to learn, they will learn by judging the outcomes from multiple hit and bunt events (for a given scenario). all they need is the average value of each, the ratio is irrelevant.

Blackadder is actually right, since the defense has to play first.  If the defense is aligned optimally, then it truly doesn’t matter what the offense does. (If the offense had to act without seeing the defense, then they would have to mix up their bunts and hits.)

In fact, the offense probably wants to do something like bunt a lot (or hit a lot) in order to try and fool the defense into changing their alignment in the future.  And when they do - bam - they do the opposite.

Now of course if the defense starts out in an optimal position (b(t)=h(t)) and the offense, say, bunts too much, and the defense moves in, and the offense still bunts too much, then the defense will keep moving in until the offense starts to bunt a little less.  As I said, that is essentially the situation now in “obvious” bunt situations.  Offenses bunt too much and defenses play way in.  It is actually the offense not playing optimally and the defense taking advantage of that - not the other way around. 

If the defense is not aligned properly (which, as I said, is often the case - they play too far in - then again that is because offenses bunt too much when it is an “obvious” bunt situation), then the offense cannot just bunt or hit away all the time. If they do that, they will teach the defense to play more optimally.

If, for example, the defense is playing too far in, such that it is correct for you to bunt hit away 100% of the time - you can’t do that.  The defense will see that you never bunt so they will move back, probably to a point which is now close to optimal. 

So when the defense is out of alignment, you have to punish them but not so much that you force them to play too optimally. What the proper ratio is for the offense to bunt/hit away when the defense is not playing optimally, I don’t know.  It depends on how astute the defense is (how willing they are to adjust).

@MGL#33
ok, i see what the point is now.
i think i settled the question about what happens when both sides pursue an optimal strategy.

however, you are talking about what happens when the strategy is non-optimal for one or both - e.g. the offense will definitely bunt here, no matter what. so yes, that will lead to some kind of cat-and-mouse, because one player has to make decisions based on how stupid they think the non-optimal player will be, and how slow to learn (e.g. how far in can i play but the offense will still bunt, thus “stealing” some value).

i think the learning by a non-optimal player will be very slow, glacial even, since the sample size for any type of situation - specific hitter with specific runner alignment against one team - will be very low. so i’m less concerned than you are with one side learning based on repeated events. also, since b(t) & h(t) are continuous functions even in this simplified state, its hard to make a nice data set and learn from that.

jsolid, you are exactly right. Keep in mind that what you do against a non-optimal opponent no only affects that opponent in future matchups but all other opponents. Plus, you always have your entire career ahead of you to worry about as compared to just this one time.  In poker, every time my opponent bluffs, when I know he is bluffing too much, I simply HAVE to lay my hand down sometimes (even though each time he does it, I am supposed to call) otherwise I know that he and every player at the table will eventually stop bluffing against me for ever, or at least bluff less often.

The instinct is to respond as if this is the last time you are going to play the game, in poker and in baseball, but you simply cannot do that. It is hard to explain, but you can’t.  Even the “Well this is a WS, isn’t a good excuse.” Teams scout the other teams. What if every time the defense played too far in last year in the WS, the Phillies hit away?  The Yankees would know that from their scouting and they would never play too far in.  By always hitting away last year in the WS, the Phillies would have “taught” every other team forever, who might otherwise play too far in, not to do that against the Phillies.  If the Phillies always hit away only 50% of he time (or whatever was a little more than optimal if the defense played optimally) then no one would probably know that they were actually taking advantage of sub-optimal defense.

If you think that a batter is looking for a fastball a little too much, do you think it is ever correct to throw him an off-speed pitch 100% of the time?  No!  How long would that last?  You have thousands of times in the future to face batters.  You cannot teach them to play more optimally.

In poker, an expert player does two things:  One, he encourages players to play less optimally than they are.  For example, against an amateur who bluffs too often, when he folds a hand that is a good hand, he may show it to that opponent to try and get him to bluff more.  If an opponent rarely bluffs, he occasionally calls when he can only beat a bluff and shows that to the opponent to encourage him to bluff even less.

The second thing he does is that in responding to a player who bluffs too little or too much, or an opponent who calls to little or too much, he plays in such a way as to be careful that he does not force an opponent too much towards optimal play.  He can’t help but force him a little towards optimal play, but not too much.  How much is tricky to figure out.

Baseball is really not unlike poker with regard to bunting and pitch selection.  You would be surprised at how much players actually respond and adjust to what you do on a continual basis.  You cannot play every “hand” in poker or in baseball as if it is your last, whether it be the WS or the regular season.

I agree with everything you are saying in post #35.  If I understand it correctly then it seems that you are advocating that the correct long term strategy should be to play non-optimally in situations where the payoff for paying optimally would be relatively low and then play optimally where the payoff would be relative high.  Not exclusively, of course, because that could be a discernible pattern for the opponent as well.

But if you do that won’t some nerdy little analyst go on the web and criticize you for the occasions when you are playing non-optimally?