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March 15, 2010

Baseball Therapy

The Unintended Consequences of The Strike One Cult

by Russell A. Carleton

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"The most important pitch in baseball is strike one." – Attributed to many.

"Just be careful who hears you say that…" – RAC

There's no question that, for a pitcher, getting a first-pitch strike and a 0-1 count is much better than a first-pitch ball and a 1-0 count. It's not complicated to figure out why, and so as a result, coaches from Little League to the big leagues tell pitchers that they should get ahead of the hitter. Great. How?

Game Theory?

Well, that's a little more complicated, primarily because the confrontation between batter and pitcher is a game within a game. When I say it's a game, I mean that in the proper sense of the word. The batter/pitcher matchup is a game of strategy, bluffing, and sometimes, outright guessing. It's like poker, with projectiles. Like in poker, there's an element of the unknown. The batter doesn't know what the pitcher is going to throw. The pitcher doesn't know what the batter is looking for. We have entered the wonderful world of game theory. Game theory is an area of study that looks at these types of situations which involve strategic thinking. One person's actions affect the other person's. Game theory is the sort of thing to which one could devote , but here we'll stick to some basics.

It's not entirely true that the pitcher and batter have no idea what the other will do.  Both have doubtless studied the scouting reports and are perhaps drawing on their previous interactions with each other (and in other situations) to figure it out.  Anything to gain an advantage.  Before we continue, let me vastly oversimplify the possible things that might happen as the pitcher goes into the windup.  It'll make things a little easier to digest.  We'll assume that the pitcher has one pitch, a fastball, and that he can throw it either down the middle or out of the strike zone.  The batter, for his part, can either swing or not (at least that's real).  If it's one of those right-down-Broadway pitches, he will hit it.  If he swings at the ball out of the zone, he'll swing and miss.

Now, since we're playing with the boundaries of reality, let's make the pitcher psychic (or Greg Maddux; same thing.) He knows, even before he winds up, that this batter will take the first pitch, no matter what. Logically, he will throw a fastball down the middle. Even if we gave him back his full arsenal of pitches, he'd probably do the same thing. (What usually happens on a 3-0 pitch?) Why mess around with breaking stuff which might go flying all over the place when you have a gimme strike? If you know that poor predictable Bart will go with rock...

Both batter and pitcher are trying to gain what's called a second-move advantage. Imagine how easy rock-paper-scissors would be if the other player had to go before you. This is the reason that pitchers bother with breaking balls, off-speed pitches, and deception. A good curve breaks close to the plate, so that up to the last moment, when the batter is trying to decide whether or not to swing, he's not sure if it's a good idea or not. If he can pick up the spin on the ball out of the pitcher's hand (as Ted Williams was alleged to be able to do) and project its course from there, he has a second-move advantage in that the pitcher has already made his move and the batter knows what it is.

Back to our overly simplified universe where the pitcher has two pitches (down-the-middle fastball or fastball outside). Let's assume that the pitcher is not Maddux and that the batter is not Williams. Both batter and pitcher are essentially guessing. We're in a simultaneous move game. What should the pitcher do, since he’s the one who has to throw the ball?  (Note: the question is not "What will he do?" We'll take that up in a moment.) Game theory says that he should take a look at the rewards that go with each choice from the perspective of what the other person in the game might do, in this case, the batter. We can, with some ease, come up good estimates of what it's worth to a batter to hit the ball, or to have strike one or ball one. Those particular calculations are for another day. It's enough here to say that we could figure it out with a little elbow grease.

Warning! Mathematical Equations Ahead!

Instead, let's make up some numbers. For the pitcher, a batted ball is worth -12 (it doesn't matter 12 of what, I'm picking numbers for ease of calculation.) A strike is worth +4. A ball is worth -8. For the batter, you change the signs.

So, if the pitcher picks "down the middle," then the pitcher's rewards are:

Batter's swing % * -12 + (1 – Batter's swing %) * 4

If the pitcher picks "outside," then the pitcher's rewards are:

Batter's swing % * 4 + (1 – Batter's swing %) * -8

If our pitcher is playing the game correctly, he'll be looking to see which side gives him a better outcome (assuming that he has a decent idea of what the batter's tendency to swing is). So, if "down the middle:" will give him a better outcome, he'll go with that 100 percent of the time. The problem is that since the batter will see multiple pitches, it won't take him long to figure out that he should change his swing percentage to compensate.  As you might imagine, a cat-and-mouse game would ensue, and the batter would want to find a good balance where neither "down the middle" or "outside" gave better results, so that the pitcher was constantly guessing. So, we can set both of those sides equal to each other (using "p" to stand in for "batter’s swing %"):

-12 * p + 4 * (1 – p) = 4 * p + -8 * (1 – p)

p = 12/28 = 42.9%

If our batter swings 42.9 percent, he'll have the pitcher guessing. We could go through a similar set of calculations for the pitcher. Because the numbers are all the same—just in the opposite direction­—for the pitcher, it works out to the same number. At that point, the two sides have played each other to what's known as a Nash equilibrium, where neither one has an advantage. Again, the number is the product of made-up inputs, but the important thing to know is that there is an equilibrium, and that baseball is the sort of game where there is going to be strong natural pressure to reach that equilibrium.

That's how it plays out in theory. What about reality?

First off, pitchers have more than two selections for a pitch; this isn’t an Atari 2600, and the outcomes aren't so cut and dried (batters do hit home runs on balls out of the strike zone and swing and miss at balls down the middle). But we're looking at first-pitch strikes. A pitcher does have a bit of a tradeoff in locating his pitches. The closer to the heart of the plate he locates the pitch, the more likely it is to be called a strike if the batter takes it, but the more likely it is to be hit if the batter swings. If he tries to paint the corners, he runs a higher chance of the pitch being called a ball, but the batter isn't as likely to swing and probably won’t make as good of contact as he otherwise would.

And Then the Pitching Coach Messes Everything Up…

The calculus that goes into determining the equilibrium of how often and when a pitcher should throw a certain type of pitch is complex (not impossible), but still there is still an equilibrium.

Now, what happens when the pitching coach approaches this pitcher who is at equilibrium and reminds him that, "you need to be getting ahead of the hitters, because the most important pitch in baseball is strike one."

Uh oh.

The pitching coach means well. He thinks that he's just offering some words of encouragement and a truism about baseball. He may have completely messed everything up. Consider for a moment that the pitching coach is in a position of authority, and he is making a semi-direct request of the pitcher. The pitcher, honestly wanting to make his pitching coach happy (or perhaps not wanting to make him angry), considers how he will go about this.

In one little sentence, the pitching coach has introduced a bunch of different factors into the pitcher's mind. One is the fact that humans do not weight all information equally. The fact that the pitching coach said this very recently means that it's more likely that this will be more available in the pitcher's mind. The request also came from an authority figure, and there is always a good deal of pressure to obey authority, especially because he will be watching and he holds a certain amount of power over the pitcher's future.

The pitcher, if he is wise and he has reached an equilibrium in this delicate dance with the batters whom he faces, will ignore the pitching coach. But there's another force acting in his mind that might throw things off. It's natural that if I know that I'm going to be evaluated, I’m going to try to take as much control over the process as I can. Let's go back to our universe in which there were two types of pitches. There are two ways that I can get a strike: throw an outside pitch and hope that the batter swings or throw one down the middle and hope that the batter doesn’t swing. In throwing a ball down the middle, if the pitcher gets the strike, he can proudly claim that "I did that." If he throws a pitch outside, he is passively waiting for the batter to make a mistake. While this type of strike counts just as much, American (and especially male American) culture values being active rather than passive. All of these forces will likely lead our pitcher to begin to come into the zone more.

This may be exactly what he needs. If he's too far away from the equilibrium point in the direction of being too timid, this may fix him up. But if he's at the equilibrium point, it's going to knock him off that balance. He may indeed get more first-pitch strikes, but it may come at the cost of a bunch more balls in play, perhaps hard-hit balls. The tradeoff may not end up being all it was cracked up to be.

A Warning for Sabermetricians

There are always unintended consequences. Murphy's Law clearly states this. What may strike some people as counter-intuitive is that at no point did the pitching coach say anything that was incorrect. It really is a good idea to get a first-pitch strike and there are studies which prove it. However, putting this information into the wrong hands can have disastrous results.

There's a certain unspoken assumption in sabermetrics that if we simply spread the results of our studies far enough, someone with decision-making authority will read them and fix whatever problem we’ve identified. The problem is that information is not always perceived by everyone in the same manner, and even if it is, attempting to change human behavior is a task that must take into account a huge number of variables. It isn't that easy.

Special thanks to BP’s in-house professor of game theory (literally), Matt Swartz, for helping me out with calculating that Nash equilibrium.

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

Related Content:  Pitcher,  The Who,  Game Theory

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Is there a point?

Mar 15, 2010 08:55 AM
rating: -23
Andrew Kneeland

"It really is a good idea to get a first-pitch strike and there are studies which prove it. However, putting this information into the wrong hands can have disastrous results."

Mar 15, 2010 11:07 AM
rating: 6

Alexander Pope: A little learning is a dangerous thing.

Mar 15, 2010 09:28 AM
rating: 0
BP staff member Russell A. Carleton
BP staff

My father put it a little less elegantly when he said that he knew enough French to either get lucky or slapped, but that his face was red a lot.

Mar 15, 2010 09:30 AM

I recall an old baseball poem about a pitcher needing to be "inconsistently inconsistent" to keep the hitters guessing

Mar 15, 2010 10:00 AM
rating: 0

Or to quote Crash Davis, "Hit the mascot"

Mar 15, 2010 11:02 AM
rating: 4

It's also a maxim in (Nobel winner) Thomas Schelling's classic "The Strategy of Conflict."

Mar 15, 2010 18:52 PM
rating: 0

Not, of course, a book about baseball -- but instead about strategy in games.

Mar 15, 2010 18:53 PM
rating: 0

Awesome article. Basically, you're describing the battle of a hitter against a closer like Jonathan Papelbon, who throws tons of fastballs, including something like 14 in a row in his last game of last year's playoffs. I'd love for an intro on how complex this gets with more pitches and location in the mix, combined with hitter predilections.

Mar 15, 2010 10:08 AM
rating: 3
BP staff member Russell A. Carleton
BP staff

This "intro" of which you speak would probably end up being a book.

Mar 15, 2010 14:37 PM

Len Fisher talks about game theorists discovering that goalies and shooters in soccer penalty kick situations generally follow a strategy that optimizes (their own) success. PKs tend to be a pretty binary thing: although a shooter can choose to blast right down the middle, shots tend to be taken to one side or the other, and goalies tend to start diving one way before the shot is even struck.

I was thinking about how to apply this to baseball, and the case that I was thinking of was actually 3-2 counts. The "cost" of taking a strike or throwing a pitch well out of the zone is much higher, but knowing that a hitter is prone to swing at "anything close," is it more clever for the pitcher to throw a Glavine five inches off the plate, or a true strike? (The obvious confound here is that few pitchers really have that kind of control: Fausto Carmona will throw the ball "vaguely forward" and hope for the best, independent of hitter.)

It became obvious that I didn't know how to actually do this, but it seems like the 3-2 count would be more conducive to Actual Game Theory because it's a lot more likely to have a direct result than the first pitch.

Mar 15, 2010 10:12 AM
rating: 3

I think a clearer instance of game theory in pitching matchups is "how often should I throw a 3-2 curveball?" The same Nash equilibrium (with different fictional constants) applies, and it's a decision more fans are familiar with, and one more conducive to actually gathering data about real-life behavior, even if with respect to a different truism ("Only get beaten on your best pitch", or some paraphrasing thereof). We can't measure batter expectations, but we can measure pitch selection; I'd be very curious if 3-2 curveballs or 3-2 fastballs generate better outcomes on average, or if they're equal. (If they're not equal, that means somebody is failing to act rationally.)

Also worth noting is that if your opponent is known to be irrational, then the Nash equilibrium might not be the best strategy. If the batter always expects a first pitch strike, then you should throw fewer (or none) of them, even if you were throwing at the "correct" rate before. So, at least in our simplified game, where the hitter can't react to the pitch, the ubiquity of the "first-pitch-strike" theory means the pitcher should throw *fewer* of them, because the hitter assigns the event too high a probability.

Mar 15, 2010 13:18 PM
rating: 2

They might not be equal, because the value ascribed to an at bat is not constant. It would seem that you're not very likely to get the hitter to freeze and watch a third strike in a must-not-strike out situation (runner on third, less than two out, tie game late.)

Mar 15, 2010 14:37 PM
rating: 1

While he was with the Padres, I remember reading that Greg Maddux wasn't the biggest fan of strike 1, but felt that going 2-0 was the real problem. I was surprised to read it at that time, but this article probably fits into Maddux's thinking. If you don't mind going down 1-0, that really opens up your options in the Game Theory scenarios. Of course, that philosophy works a lot better if you have Maddux's control.

Mar 15, 2010 10:42 AM
rating: 5

I remember hearing this espoused by the pitching coach of one of the ML teams this year and thinking it was interesting. The idea was that trying to get strike 1 on the first pitch was too predictable, and that it was smarter to make sure you got strike 1 within the first 2 pitches. Of course, theoretically that would make the second pitch pretty predictable once you went to 1-0.

It may have been the Yankees, since I watch them the most. I just don't remember.

Mar 15, 2010 16:15 PM
rating: 0

I do remember Maddux in an interview once saying that he felt the most important pitch was actually 1-1. So I don't know what that's worth.

But I think he's the greatest pitcher of all time, so I kind of believe him...

Mar 15, 2010 17:20 PM
rating: 1

It may have happened more than once, but I remember this discussion in an interview on ESPN with Maddux and Tony Gwynn. They actually both agreed that the result of the 1-1 pitch was more important than the 0-0 one.

Mar 16, 2010 05:26 AM
rating: 1

I think the most important thing is to throw a good pitch for your first pitch... grooving a fastball to get strike one is not a good idea, however if you can mix up your pitches and location for that 1st strike, I think we can all agree that it is a good thing

Mar 15, 2010 12:12 PM
rating: 0

I know this doesn't occur in every PA, but isn't the third pitch in a 1-1 count the most significant? At least that's what I thought they said in Moneyball; a hitter's BA increases/decreases by 70 points after the third pitch of a 1-1 count.

Mar 15, 2010 12:31 PM
rating: 2
Fresh Hops

This is not run expectancy calculation, but looking at the AL Splits for 2009 (I always use AL stats for quick-and-dirty because there are are no pitchers batting except in a handful of inter-league games) I'd say that this is right: OPS in a after a 1-1 was .720; this drops to .527 in 1-2 and rises to .836 in 2-1.

However, it looks to me like it ain't over till the fat lady sings, so to speak: hitters have an OPS of .467 in 0-2 counts; this drops to .000 if the pitcher gets strike three and rise to .526 if the go to 1-2. But this OPS stuff is really crude and run expectancy would be better.

Mar 15, 2010 14:12 PM
rating: 1
Other readers have rated this comment below the viewing threshold. Click here to view anyway.

Did you really just post the OPS of batters who have seen three strikes? I mean, I get what you're trying to say, but do you even have to write that?

Mar 15, 2010 20:04 PM
rating: -4
Brock Dahlke

Another amazing article. BP is on fire the last few days.

Mar 15, 2010 12:16 PM
rating: 0

I have found two things about the pitchers I have examined in this regard:

First, while since batters never strike out on the first pitch, they have a high batting average when they put the first ball in play, they rarely do so. This means that most pitchers don't get hurt badly by their first-pitch strikes.

Second, most pitchers pitch FAR better after 0-1 than they pitch after 1-0.

I think the calculation naturally varies from pitcher to pitcher. But I think that most pitchers should take measures to improve their chances of throwing first-pitch strikes until their throwing "get it in" first pitches begins to backfire.

I also think an intelligent pitcher can determine which situations benefit from a "get it in" first pitch approach and which ones may involve too much risk.

And I further think that an intelligent pitcher can determine how MUCH he can afford take off a pitch in pursuit of a higher first-pitch strike rate.

I think the best part of this article is that it shows some of the dynamics that should go into a pitcher's decision on first pitches. And that it's not an all-or-nothing situation. There are various degrees between throwing the ball down the middle and trying to get the batter to swing at a first pitch outside the strike zone.

There are two firsts that make a big difference for a pitcher. One is getting that first strike. And even larger one is getting that first out. To the point where the first pitch to the leadoff batter in an inning may be the most important pitch of the inning moreso than any other particular pitch.

Mar 15, 2010 13:19 PM
rating: 0

I love, love, love that you are doing game theory. That said, I wish there was something closer to concrete results with the article. Can you introduce something using tAvg? For instance, what is the difference in tAvg between 0-0, 0-1 and 1-0 counts? Even better, could you add tAvg for swinging strikes, swinging balls, etc?

Mar 15, 2010 14:33 PM
rating: 0
BP staff member Russell A. Carleton
BP staff

When I originally conceived of this article, I thought about going there, but to be honest, I ran out of time this week... It's rather involved to run the numbers. I might at some point in the future though.

Mar 15, 2010 18:01 PM

This would be really interesting.

Mar 16, 2010 22:39 PM
rating: 0
Dr. Dave

As you sort of implied when invoking Maddux and Williams, the equilibrium point is different for every batter/pitcher matchup. That's because different pitchers have a different ability to miss bats in the strike zone (which is different for each pitch they throw), and different hitters have a different ability to avoid being fooled (which is different for the different pitches they might face).

In theory, you could look at a composite effectiveness of each pitch a pitcher throws, and a composite outcome matrix for the batter against each type of pitch, and figure out the pitcher's randomized optimal strategy.

(I'm not convinced, though, that the hitter really has a "strategy" here -- hitters who guess don't last. It might be more like the hitter has a different success matrix depending on what the last few pitches have been like -- a Markov decision model for the pitcher, rather than a Nash game.)

Mar 15, 2010 15:13 PM
rating: 0

The equilibrium point also changes because some hitters have a practice of nearly always taking the first pitch, whereas others frequently swing at the first pitch on the theory that the pitcher is trying to get strike 1.

Mar 15, 2010 16:20 PM
rating: 0

Brian, the equilibrium point does NOT change based on the pitcher's proclivity to throw a certain percentage of first pitch strikes and the batter's tendencies to swing at first pitch strikes and balls. The batter and pitcher have "choice" as to the equilibrium point. That is determined mathematically and is based on the pitcher's overall ability in terms of his stuff and his control and the batter's overall ability to hit, take, and swing at all the different pitches in all the different locations.

The trick is for the batter and pitcher to figure it all out. And of course of the batter is not acting optimally, then the pitcher's optimal strategy is not equal to the Nash equilibrium point and the same thing applies to the batters if the pitcher is not acting optimally.

Russell this is great stuff. The point about the pitching coaches, which is lost among these comments, is a great one. That is, the axiom, "Get the first pitch over" or, "The most important pitch is strike one," like all one-size fits all edicts, is dumb and is counterproductive toward guiding pitchers toward an optimal strategy.

Now, if a pitcher happens to be throwing too many pitches out of the zone, given his abilities, then obviously this advice will turn out to be good in general. If a pitcher, however, throws too many balls in the zone, given his stuff, it will turn out to be bad advice.

But, the worse part is that it ignores the different optimal strategies that different pitchers should have overall, and even worse, it ignores that fact that each pitcher should have vastly different first pitch strategies against different hitters.

On the first point, that different pitchers should have different first pitch strategies, for example, a pitcher with good in the zone stuff should be much more likely to throw first pitches (at any count actually) in the zone.

On the second point, to a hitter like Vlad, all pitchers should be much more likely to throw a first pitch out of the zone.

Anyway, this is a great example of where "scouting" (in this case tutelage from a pitching coach) PLUS analysis (game theory) equals the best strategy for teaching pitchers how to pitch optimally. BTW, the analysis applies much more to pitchers than to batters. For batters, the big "hitch" in using game theory to suggest optimal approaches is the batter's comfort zone. If you suggest to a batter that the correct strategy for Vlad is to take more first pitches, even though that might be correct on "paper" (given all of his various abilities), he simply might not be comfortable with any other strategy but the one he is using, and his performance might actually suffer even though on paper it should get better. Not so much with pitchers, and that is primarily because they "get to go first" and for batters their strategy is always ultimately reactive rather than pro-active, which it is for pitchers.

Mar 16, 2010 03:29 AM
rating: 1

"The batter and pitcher have "choice" as to the equilibrium point."

That should be "no" choice...

Mar 16, 2010 03:30 AM
rating: 0

"If a pitcher, however, throws too many balls in the zone, given his stuff, it will turn out to be bad advice."

Oy, that should be "too many STRIKES..."

Mar 16, 2010 03:31 AM
rating: 0
BP staff member Russell A. Carleton
BP staff

I'm not so sure that Markov would work so much as a repeated interaction Nash-type game. Surely, a batter's expectations are going to be influenced by the previous few pitches, because he intuitively knows that a pitcher is not truly randomizing his pitches and he might have an idea of the pattern that the pitcher is using.

Mar 15, 2010 18:07 PM
Dr. Dave

My point was that to talk about a hitter's 'strategy' you have to assume he's deciding courses of action before seeing the pitch, or at least looking for a particular pitch in a particular place. I'm not sure that's true for all hitters -- I think some of them really do react to the pitch.

In such a case, the most you could say is that the hitter's chance of successfully hitting a particular pitch might depend on the pitch sequence leading up to it. That makes it a Markov decision model for the pitcher -- but the state might be the entire pitch sequence, not just the most recent previous pitch. More likley, you could closely approximate the state space by looking only at (say) the last 3 pitches.

Mar 16, 2010 21:09 PM
rating: 0

Well, congrats on using the term game theory. Of course the real question is whether or not most pitchers would be better off throwing more first pitch strikes. I think you really overestimate even most professional pitcher's ability to be in a perfect equilabrium of game theory on throwing 0-0 strikes. Pitching coaches are aware of the cat and mouse game at play, they are simply saying their pitcher needs to throw more first pitch strikes to be effective (or in equilibrium). You provide no evidence that they are wrong.

Mar 15, 2010 21:03 PM
rating: 1

I love the article. I'm a big fan of game theory. Finding the Nash equilibrium is a really difficult subject. It really does come down to infinite degrees of variation between what part of the strike zone you were aiming for and what the results are.

I suppose at some point in the future, you might be able to graph exactly where the catcher put the target and the results on a per pitch basis to find the optimum placement and whether the actual placement was closer or farther from the center than optimum. This would be a very detailed project, but might eventually be worth it.

Mar 15, 2010 23:32 PM
rating: 0

Wow ... I can see it now ... the next generation of Pitch f/x data will include an x,y,z coordinate of the catcher's mitt at the time of pitcher's release! Yow!

Mar 16, 2010 12:54 PM
rating: 0

Every pitcher and every batter is different. But I think the "equilibrium point" for most pitchers would be a pretty high percentage of first-pitch strikes.

Do we know of many pitchers who throw a high percentage of first-pitch strikes and yet achieve poor results?

Mar 16, 2010 08:46 AM
rating: 0
BP staff member Russell A. Carleton
BP staff

Well, that assumes that most batters are up there taking on the first pitch. That's not a bad assumption, as it seems rather common. But, if you have a batter who swings a lot on the first pitch, it might be a good idea to throw a pitch outside and hope he chases it.

Here, we have to make a distinction between getting strike one (always a good thing) and how you get the strike. Should you throw something down the middle, try to paint the corner, or throw something that you hope the batter will chase?

Mar 16, 2010 13:21 PM

I just have to say . . . I'm all geeked out on this conversation.

Mar 16, 2010 09:13 AM
rating: 1
BP staff member Russell A. Carleton
BP staff

Geek pride!

Mar 16, 2010 13:14 PM
Richard Bergstrom

This discussion reminds me of David Sklansky's Theory of Poker. In a perfect world with perfect information, the player that does best is the one who plays most optimally. Thus, deception is added so that, while you may play less optimally, the move/tactic will increase the chances your opponent makes a drastic mistake.

In baseball terms, throwing a first pitch ball might make a batter overeager and prone to a mistake later in the at-bat.

Mar 16, 2010 09:47 AM
rating: 0

I would think that strategy's effectiveness would be tested by notorious bad-ball hitters like Vlad and Yogi.

Mar 16, 2010 09:55 AM
rating: 0
BP staff member Russell A. Carleton
BP staff

The permutations on this are maddening. Understanding the inner workings of this matchup is the next big task of Sabermetrics.

Mar 16, 2010 12:56 PM

Just wanted to say -- I thought this article was tremendous.

Mar 16, 2010 22:36 PM
rating: 0
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