There’s an article up on Fangraphs by Eno Saris that talks about whether the pitch to Justin Turner in the bottom of the 9^{th} inning in Game 2 of the 2017 NLCS was the “wrong” pitch to throw in that count (1-0) and situation (tie game, runners on 1 and 2, 2 outs) given Turner’s proclivities at that count. I won’t go into the details of the article – you can read it yourself – but I do want to talk about what it means or doesn’t mean to criticize a pitcher’s pitch selection – on one particular pitch, and how pitch selection even works, in general.

Let’s start with this – the basic tenet of pitching and pitch selection: Every single situation calls for a pitch frequency matrix. One pitch is chosen randomly from that matrix according to the “correct” frequencies. The “correct” frequencies are those which result in the exact same “result” (where result is measured by the win expectancy impact of all the possible outcomes combined).

Now, obviously, most pitchers “think” they’re choosing one specific pitch for some specific reason, but in reality since the batter doesn’t know the pitcher’s reasoning, it is essentially a random selection as far as he is concerned. For example, a pitcher throws an inside fastball to go 0-1 on the batter. He might think to himself, “OK, I just threw the inside fastball so I’ll throw a low and away off-speed to give him a ‘different look.’ But wait, he might be expecting that. I’ll double up with the fastball! Nah, he’s a pretty good fastball hitter. I’ll throw the off-speed! But I really don’t want to hang one on an 0-1 count. I’m not feeling that confident in my curve ball yet. OK, I’ll throw the fastball, but I’ll throw it low and away. He’ll probably think it’s an off-speed and lay off of it and I’ll get a called strike, or he’ll be late if he swings.”

As you can imagine, there are an infinite number of permutations of ‘reasoning’ that a pitcher can use to make his selection. The backdrop to his thinking is that he knows what tends to be effective at 0-1 counts in that situation (score, inning, runners, outs, etc.) given his repertoire, and he knows the batter’s strengths and weaknesses. The result is a roughly game theory optimal (GTO) approach which cannot be exploited by the batter and is maximally effective against a batter who is thinking roughly GTO too.

The optimal pitch selection frequency matrix is dependent on the pitcher, the batter, the count and the game situation. In that situation with Lackey on the mound and Turner at the plate, it might be something like 50% 4-seam, 20% sinker, 20% slider, and 10% cutter. The numbers are irrelevant. Then a random pitch is selected according to those frequencies, where, for example, the 4-seamer is chosen twice as often as the sinker and slider, the sinker and slider twice as often as the cutter, etc.

Obviously doing that even close to accurately is impossible, but that’s essentially what happens and what is *supposed* to happen. Miraculously, pitchers and catchers do a pretty good job (really you just have to have a pretty good idea as to what pitches to throw, adjusted a little for the batter). At least I presume they do. It is likely that some pitchers and batters are better than others at employing these GTO strategies as well as exploiting opponents who don’t.

The more a batter likes (or dislikes) a certain pitch (in that count or overall), the less that pitch will be thrown. In order to understand why, you must understand that the result of a pitch is directly proportional to the frequency at which it is thrown in a particular situation. For example, if Taylor is particularly good against a sinker in that situation or in general, it might be thrown 10% rather than 20% of the time. The same is true for locations of course, which makes everything quite complex.

Remember that you *cannot* tell what types and locations of pitches a batter likes or dislikes in a certain count and game situation from his results! This is a very important concept to understand. The results of every pitch type and location in each count, game situation, and versus each pitcher (you would have to do a “delta method” to figure this) are and should be exactly the same! Any differences you see are noise – random differences (or the result of wholesale exploitative play or externalities as I explain below). We can easily prove this with an example.

Imagine that in all 1-0 counts, early in a game with no runners on base and 0 outs (we’re just choosing a ‘particular situation’ – which situation doesn’t matter), we see that Turner gets a FB 80% of the time and a slider 20% of the time (again, the actual numbers are irrelevant). And we see that Turner’s results (we have to add up the run or win value of all the results – strike, ball, batted ball out, single, double, etc.) are much better against those 80% FB than the 20% SL. Can we conclude that Turner is better against the FB in that situation?

No! Why is that? Because if we did, we would HAVE TO also conclude that the pitchers were throwing him too many FB, right? They would then reduce the frequency of the fastball. Why throw a certain pitch 80% of the time (or at all, for that matter) when you know that another pitch is better?

You would obviously throw it less often than 80% of the time. How much less? Well, say you throw it 79% and the slider 21%. You *must* be better off with that ratio (rather than 80/20) since the slider is the better pitch, as we just said for this thought exercise. Now what if the FB still yields better results for Turner (and it’s not just noise – he’s still better versus the FB when he knows it’s coming 79% of the time)? Well, again obviously, you should throw the FB even less often and the slider more often.

Where does this end? Every time we decrease the frequency of the FB, the batter gets worse at it since it’s more of a surprise. Remember the relationship between the frequency of a pitch and its effectiveness. At the same time, he gets better and better at the slider since we throw it more and more frequently. It ends at the point in which the results of both pitches are exactly equal. It HAS to. If it “ends” anywhere else, the pitcher will continue to make adjustments until an equilibrium point is reached. This is called a *Nash equilibrium *in game theory parlance, at which point the batter can look for either pitch (or any pitch if the GTO *mixed strategy* includes more than two pitches) and it won’t make any difference in terms of the results. (If the batter doesn’t employ his own GTO strategy, then the pitcher can exploit him by throwing one particular pitch – in which case he then becomes exploitable, which is why it behooves both players to always employ a GTO strategy or risk being exploited.) As neutral observers, unless we see evidence otherwise, we must assume that all actors (batters and pitchers) are indeed using a roughly GTO strategy and that we are always in equilibrium. Whether they are or they aren’t, to whatever degree and in whichever situations, it certainly is instructive for us and for them to understand these concepts.

Assuming an equilibrium, this is what you MUST understand: Any differences you see in either a batter’s results across different pitches, or as a pitcher’s, MUST be noise – an artifact of random chance. Keep in mind that it’s only true for each subset of identical circumstances – the same opponent, count, and game situation (even umpire, weather, park, etc.). If you look at the results across all situations you will see legitimate differences across pitch types. That’s because they are thrown with different frequencies in different situations. For example, you will likely see better results for a pitcher with his secondary pitches overall simply because he throws them more frequently in pitcher’s counts (although this is somewhat offset by the fact that he throws them more often against better batters).

Is it possible that there are some externalities that throws this Nash equilibrium out of whack? Sure. Perhaps a pitcher must throw more FB than off-speed in order to prevent injury. That might cause his numbers for the FB to be slightly worse than for other pitches. Or the slider may be particularly risky, injury-wise, such that pitchers throw it less than GTO (game theory optimally) which results in a result better (from the pitcher’s standpoint) than the other pitches.

Any other deviations you see among pitch types and locations, by definition, must be random noise, or, perhaps exploitative strategies by either batters or pitchers (one is making a mistake and the other is capitalizing on it). It would be difficult to distinguish the two without some statistical analysis of large samples of pitches (and then we would still only have limited certainty with respect to our conclusions).

So, given all that is true, which it is (more or less), how can we criticize a particular pitch that a pitcher throws in one particular situation? We can’t. We can’t say that one pitch is “wrong” and one pitch is “right” in ANY particular situation. That’s impossible to do. We cannot evaluate the “correctness” of a single pitch. Maybe the pitch that we observe is the one that is only supposed to be thrown 5 or 10% of the time, and the pitcher knew that (and the batter was presumably surprised by it whether he hit it well or not)! The only way to evaluate a pitcher’s pitch selection strategy is by knowing the frequency at which he throws his various pitches against the various batters in the various counts and game situations. And that requires an enormous sample size of course.

There is an exception.

The one time we can say that a particular pitch is “wrong” is when that pitch is not part of the correct frequency matrix at all – i.e., the GTO solution says that it should never be thrown. That rarely occurs. About the only time that occurs is on 3-0 counts where a fastball might be the only pitch thrown (for example, 3-0 count with a 5 run lead, or even a 3-1 or 2-0 count with any big lead, late in the game – or a 3-0 count on an opposing pitcher who is taking 100% of the time).

Now that being said, let’s say that Lackey is supposed to throw his cutter away only 5% of the time against Turner. If we observe only that one pitch and it is a cutter, Bayes tells is that there is an inference that Lackey was intending to throw that pitch MORE than 5% of the time and we can indeed say with some small level of certainty that he “threw the wrong pitch.” We don’t really mean he “threw the wrong pitch.” We mean that we think (with some low degree of certainty) he had the wrong frequency matrix in his head to some significant degree (maybe he intended to throw that pitch 10% or 20% rather than 5%).*

So, the next time you hear anyone say what a pitcher *should* be throwing on any particular pitch or that the pitch he threw was “right” or “wrong,” it’s a good bet that he doesn’t really know what he’s talking about, even if they are or were a successful major league pitcher.

* Technically, we can only say something like, “We are 10% sure he was thinking 5%, 12% sure he was thinking 7%, 13% sure he was thinking 8%, etc.” – numbers for illustration purposes only.