The Hardball Times -- Matt Swartz http://www.hardballtimes.com/main Baseball. Insight. Daily. en studes@hardballtimes.com Copyright 2013 2013-05-24T08:09:15+00:00 Game theory and baseball, Part 5: Generalizing the pitch selection model http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-5-generalizing-the-pitch-selection-model/ http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-5-generalizing-the-pitch-selection-model/#When:05:48:15 the past four articles, I have discussed various applications of game theory to baseball analysis. Last time, I developed a new framework for pitch selection wherein a pitcher named Chuck decides whether to throw a fastball or a curveball, after which a batter named Willie Bayes may get a noisy signal of whether a fastball has been thrown. Upon receiving the signal, Willie decides whether to swing.

The set-up is shown below as an extensive form game:

Figure 3A

image

Remember that “Nature” is not an agent with preferences, but a mechanism that acts after Chuck throws. It gives a signal of “fastball” with probability “x” when a fastball is thrown, but also gives a “fastball” signal when a curveball is thrown with a smaller probability “y.”

We assumed that the normal form of the game had the following structure:

Table 1

















Pitcher\Batter Swing Take
Strike (Fastball) -1,1 1,-1
Ball (Curveball) 1,-1 -1,1


In the equilibrium to this game, the pitcher randomizes—he throws both fastballs and curveballs with positive probabilities. The batter swings if and only if he gets a signal of fastball. In fact, this is always the equilibrium structure if we assume that the batter is relatively better off when he swings at fastballs than taking them, and relatively better off when he takes curveballs than swinging at them.

Last time, we made two sets of assumptions about “x” and “y” and solved for the equilibria:

1) We assumed that x = 0.9 and y = 0.4. This led to an equilibrium where the pitcher threw a fastball 31 percent of the time, and the batter swung only if he received a signal.
2) We assumed that x = 0.9 and y = 0.5. This led to an equilibrium where the pitcher threw a fastball 36 percent of the time, and the batter swung only if he received a signal.

The surprising discovery here was that the pitcher with the better (harder to detect) curveball threw it less often an equilibrium, so as to entice the batter to swing more often.

More generally, the batter preferred to swing when receiving a signal (conditional on pitchers throwing fastballs with probability “p”) as long as:

px / [px + (1-p)y] > 0.5

He preferred to take when he did not receive a signal as long as:

p(1-x) / [p(1-x) + (1-p)(1-y)] < 0.5

Chuck, the average pitcher


In the above pair of examples, I constructed a reasonable scenario where pitchers with better curveballs used them less frequently as out-pitches than pitchers with inferior curveballs. Now, let’s tweak the assumptions such that an average pitcher, “Chuck,” has a fastball that “looks like a fastball” just 60 percent of the time, while a mediocre curveball “looks like a fastball” just 10 percent of the time. Let’s solve for the equilibrium, borrowing from the formulas above.

To entice Willie to swing when he gets a signal (i.e. when the pitch “looks like a fastball”), we need 0.6p/(0.6p + 0.1(1-p)) > 0.5, which is when p > .14.

To entice Willie to take when he gets no signal, we need 0.4p/(0.4p + 0.9(1-p)) < 0.5, which is when p < .69.

Therefore, any equilibrium will require Chuck to throw a fastball between 14 percent and 69 percent of the time. What will Chuck want to do within these limits? His strategy consists of solving a “p” such that:

MAX [ p*(-1*0.6+1*0.4) + (1-p)*(1*0.1+-1*0.9) ]

which simplifies to

MAX [0.6p – 0.8].
Subject to .14 < p < .69

Therefore, Chuck wants to throw the highest percentage of fastballs that the equilibrium will allow—which is 69 percent.

Charlie, the good pitcher



Now suppose that Charlie has a dominant curveball that actually looks like a fastball 20 percent of the time, while having the same fastball as Chuck that looks like a fastball 60 percent of the time.

To entice Willie to swing when he gets a signal, we need “p” such that 0.6p/(0.6p + 0.2(1-p)) > 0.5, which is when p > 0.25.

To prevent Willie from swinging when he gets no signal, we need “p” such that 0.4p/(0.4p + 0.8(1-p)) < 0.5, which is when p < 0.67.

Therefore, Charlie will need to throw between 25 percent and 67 percent fastballs to entice Willie to swing only when he gets a signal. He will solve:

MAX [ p*(-1*0.6+1*0.4) + (1-p)*(1*0.2+-1*0.8) ]

which simplifies to

MAX [ 0.4p – 0.6 ].
Subject to .25 < p < .67

Charlie also wants to throw as many fastballs as possible: so he throws 67 percent fastballs and 33 percent curveballs. Chuck, who had an inferior curveball, actually threw 31 percent curveballs. So now we have the opposite outcome from yesterday: The pitcher with the superior curveball throws it more often as an out-pitch.

More generally


What was different about the second pair of pitchers? Why was Charlie less likely to throw curveballs as an out-pitch than Chuck under the first set of assumptions, and more likely to throw curveballs as an out-pitch under the second set of assumptions? This can be derived simply by looking at the variables more generally. Once we do that, we can figure out what type of assumptions makes more sense as a realistic description of the major leagues.

As long as we have the set-up of payoffs above, we know that any equilibrium requires:

y/(x+y) < p < (1-y)/(2-y-x)

And then we need the pitcher to maximize:

MAX [ p*(-1*x + 1*(1-x) + (1-p)*(1*y – 1*(1-y)) ]

which simplifies to

MAX [ 2p*(1-x-y) + (2y-1) ].

This structure makes it clearer—the maximand is always linear with respect to “p,” which means that whenever x+y > 1, the pitcher responds by setting “p” as low as possible, which is y/(x+y). This term increases with respect to “y,” the variable that describes how often a curveball looks like a fastball (i.e. how good it is). So the better the pitcher’s curveball is, the less he throws it.

When x+y < 1, the pitcher responds by setting “p” is high as possible, which is (1-y)/(2-y-x). This term gets smaller as “y” gets bigger (i.e. when the curveball gets better), so the pitcher throws more curveballs when his curveball is better.

When the fastball is as mistakable as the curveball


The borderline case is when x+y=1. As an illustration, let’s consider when x = 0.7 and y = 0.3. In this case, we know that Willie will swing when:

y/(x+y) < p < (1-y)/(2-y-x)
0.3 < p < 0.7

So anything between 30% and 70% will work, but what will the pitcher prefer? He will find the maximizer of:

MAX [ p*(-1*x + 1*(1-x) + (1-p)*(1*y – 1*(1-y)) ]

which cancels out all of the “p”s when you simplify and get:

MAX [ -0.4 ].

In other words, any “p” that the pitcher chooses will give him the same expected value, as long as “p” is between 30% and 70%. Therefore, we have multiple equilibria (in fact, infinitely many equilibria), and no general rule could be established.

The key assumption


So, does my surprise equilibrium accurately describe the world of major league baseball? Does x+y exceed 1? If so, then the surprising result that pitchers with great out-pitches should be showing them less often in two-strike counts than pitchers with mediocre out-pitches will hold. To me, it seems like a logical statement that x+y exceeds 1, which means that hitters are more likely to mistake a curveball for a fastball than mistake a fastball for a curveball.

One last wrinkle


Another assumption that I made in the equilibrium above was to suppose that the difference between the gap in payoffs between strike/swing and strike/take is equal to the gap between ball/take and ball/swing. Let’s relax this assumption. This will allow us to compare the value of pitches in different counts relative to each other.

Here is a generalization of this payoff table:

Table 2

















Pitcher\Batter Swing Take
Strike (Fastball) -0.5,0.5 0.5z,-0.5z
Ball (Curveball) 0.5,-0.5 -0.5z,0.5z


In this case, Willie’s value of swinging, conditional on Chuck’s strategy of throwing “p” fastballs is:

V(s) = 0.5*(xp/(xp+y(1-p))

His value of taking when getting a signal is:

V(t) = 0.5*z*(y(1-p)/(xp+y(1-p))

Therefore, to entice Willie to swing when he gets a signal, we need V(s) > V(t), which is when:

p > zy/(x+zy).

A batter’s value of taking when he gets no signal is:

V(s) = 0.5*((1-x)p/((1-x)p+(1-y)(1-p)))

And his value of taking when he gets no signal is

V(t) = 0.5*z*((1-y)(1-p)/((1-x)p+(1-y)(1-p)))

A batter won’t swing with no signal when V(s) < V(t), which is when:

p < z(1-y) /(z(1-y)+(1-x)).

Knowing this, Chuck selects “p” to maximize:

MAX(p*(-0.5x+0.5z(1-x)) + (1-p)*((0.5y – 0.5z(1-y)))

which simplifies to:

MAX(0.5*(y-z(1-y) + p*(z(2-x-y)-(x+y)))).

Therefore, the more general version of the hypothesis is that the pitcher should throw a greater out-pitch less often when:

x+y > z(2-x-y)

He throws a greater out-pitch more often when:

x+y < z(2-x-y)

Therefore, we get the following general solution:

{exp:list_maker}When x>0.5 and y>0.5, we know that x+y > z(2-x-y) definitely holds, which means that the pitcher minimizes fastballs. However, the higher "z" gets, the more fastballs the pitcher has to throw to entice the batter to swing.
When x<0.5 and y<0.5, we know that x+y < z(2-x-y) definitely holds, which means that the pitcher maximizes fastballs. However, the lower "z" gets, the more curveballs the pitcher has to throw and still entice the batter to take.
When x>0.5 and y<0.5, we know that x+y > z(2-x-y), only holds when "z" is small enough. When "z" is sufficiently small, the pitcher minimizes fastballs, but the smaller that "z" gets, the more fastball the pitcher has to throw fastballs to entice the batter to swing. On the other hand, if "z" is large enough, the pitcher maximizes fastballs, but the higher that "z" gets, the more the pitcher has to throw curveballs to still entice the batter to take. {/exp:list_maker}

Verbally, this says that:

{exp:list_maker}When most fastballs look like fastballs and most curveballs look like fastballs, then the bigger the benefit the pitcher gets from throwing a strike when the batter is taking (relatively to when he throws a ball and the batter is taking), the pitcher minimizes the amount of fastballs he throws. However, for the largest gaps, he must throw fastballs more frequently to entice the batter to swing.
When most fastballs look like curveballs and most curveballs look like curveballs, then the smaller the benefit the pitcher gets from throwing a strike when the batter is taking (relatively to when he throws a ball and the batter is taking), the pitcher maximizes the amount of fastballs he will throws. However, for the smallest gaps, he must throw curveballs more frequently to entice the batter to take.
When most fastballs look like fastballs and most curveballs look like curveballs, then the smaller the benefit the pitcher gets from throwing a strike when the batter is taking (relatively to when he throws a ball and the pitcher is taking), the pitcher minimizes the amount of fastballs he throws. However, for the smallest gaps, the more he has to throw fastballs to entice the batter to swing. When the gap is large enough, however, the pitcher wants to throw as many fastballs as he can, but the larger that gap, the more he must throw curveballs to entice the batter to take. {/exp:list_maker}

Of course, this is only the tip of the iceberg in how we can vary these payoffs, and more general rules could be found with sufficient variation.

Going forward


These articles have only served to develop a framework. Going forward, analysts could use FanGraphs’ o-swing and z-swing statistics to actually calculate some more exact payoffs. Variation in pitch usage by count could be explored differently, too, using these payoffs. Additionally, discussions with players could determine ability to detect pitches. Variations in counts could be used to adjusted x+y and z frameworks. Other pitches could be added to complicate the equilibrium for pitchers with more than two pitches, too.

The goal of these last four articles on pitch selection (and on these last five articles on game theory in general) was not to issue a final word on the proper way to play perfect baseball, but to instead talk about how much teams could improve their chances by thinking about strategy like real strategists do. There is no reason to assume that typical mechanisms of “free markets” to ensure efficiency would emerge in a baseball setting, simply because there are so few people on the planet with the skills to make it onto the diamond. Now, the opportunity is there for whoever wants to add strategy on top of talent.

Addendum for the advanced reader


Similarly to yesterday's article, the above examples are not true Nash Equilibria, because the pitcher is not indifferent between the fastball and the curveball. However, the results are similarly equivalent like yesterday, whereby the batter randomizes in some situations to make the pitcher actually indifferent. The batter will take occasionally when getting a signal of fastball in those examples above where the pitcher would otherwise minimize fastballs (i.e. making the hitter indifferent between swinging and taking when getting a signal), and he will swing occasionally when getting no signal of fastball in those examples above where the pitcher is maximizing fastballs (i.e. making the hitter indifferent between swinging and taking when getting no signal).

The batter would set q(s) = p[z(2-x-y) + (x+y)] / [2(x+y)] when q(n) = 0, and would set q(n) = [z(2-x-y) - (x+y)] / [2z(2-x-y)] when q(s) = 1.

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2012-12-21T05:48:15+00:00 Game theory and baseball, part 4: if batters were Bayesians http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-4-if-batters-were-bayesians/ http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-4-if-batters-were-bayesians/#When:10:51:15 applications of game theory to baseball, continuing with the trend of exploring pitch selection. In the previous two articles, we broke down a series of payoffs and found some surprising results. Perhaps the most surprising was that pitchers with better out-pitches should be throwing them less frequently in two-strike counts than pitchers with mediocre out-pitches. This followed basic game theory, whereby players who would only succeed when their opponent could not predict their actions must select strategies that keep their opponents indifferent.

We considered the following pair of games structured below, where batters are always better off when the actions by the pitcher and batter are strike/swing or ball/take, and pitchers are always better off when the actions are strike/take and ball/swing. The two examples differed in that a pitcher who had a great curveball had an even higher expected payoff to ball/swing.

Table 1

















Pitcher\Batter Swing Take
Strike (Fastball) -1,1 1,-1
Ball (Curveball) 1,-1 -1,1


Table 2

















Pitcher\Batter Swing Take
Strike (Fastball) -1,1 1,-1
Ball (Curveball) 1.5,-1.5 -1,1



In Table 1, the pitcher threw a strike 50 percent of the time and the batter swung 50 percent of the time. In Table 2, the pitcher threw a strike 56 percent of the time and the batter swung 44 percent of the time. The pitchers with better out-pitches threw more fastballs to encourage the batters to actually swing at their out-pitches when they do throw them.

The most obvious objection to this set-up is that the batter usually has some sense of whether a pitch will land in the strike zone when he sees it thrown. While a batter is often misled by a good pitcher, he at least has some idea what’s coming once it’s been released.

Bayes at the plate


To put this into a proper game theoretic framework, we allow the batter to observe a “signal” after the pitcher throws the ball. The signal will basically imply “this looks like a fastball for a strike”; absence of a signal will basically imply “this looks like a curveball for a ball.” Of course, the signal does not mean that there is a 100 percent chance of a strike; its absence does not imply that there is a zero percent chance of a strike. Instead, let’s call the probability of observing an accurate signal when the pitcher throws “fastball-for-strike” as “x” and the probability of observing an inaccurate signal when the pitcher throws “curveball-for-ball” as “y.” Without loss of generality, we can assume x > y.

So the set-up is that the pitcher must first select a strategy “p,” which represents his probability of throwing a fastball (a strike). The batter must have a two-pronged strategy: q(s) and q(n), depicting his probability of swinging with probability q(s) as a function of receiving a signal, and swinging with probability q(n) as a function of receiving no signal. Unlike the games from the previous article, there is no reason for q(s) to be equal to q(n). Presumably, our batter will set q(s) > q(n), and potentially set q(s)=1 and q(n)=0. This would mean that he swings when he gets a signal that the pitch looks like as a strike.

The batter will behave like a Bayesian, so let’s call him “Willie Bayes.” We’ll call our hurler “Chuck.”

Since this game is sequential, it will require using the extensive form to visualize, which is shown below. The first player to make a decision is Chuck, the pitcher. Decision Node I, at the top, is circled. Chuck must select either a fastball or a curveball to throw.

At this point, the next mover is not the batter, because after Chuck throws a fastball or a curveball, something else must happen. Willie Bayes needs to know whether he has observed a signal or not, something Chuck can’t control. It is typical to write out the next move as being made by “Nature” at this point. Nature is not an agent with preferences—she just follows rules. When Chuck has thrown a fastball, she sends a signal with probability “x” and fails to send a signal with probability “1-x.” When Chuck has thrown a curveball, she sends a signal with probability “y” and fails to send a signal with probability “1-y.”

At this point, Willie Bayes has to formulate two strategies: q(s) for when he gets a signal, and q(n) for when he does not. Decision Node II occurred when he gets a signal, but Willie does not know if this is because Chuck has thrown a fastball and Nature has sent a correct signal, or if Chuck has thrown a curveball and Nature has sent an incorrect signal. Decision Node III, occurs after no signal has been received.

Figure 3A

.image

Batter's strategy


Fortunately, Willie knows Bayes’ Theorem and constructs probabilities that he is seeing a fastball or a curveball, conditional on receiving a signal. Since we are using backwards induction to solve this problem, we will first develop responses by Willie conditional on Chuck’s strategy “p” and then move backwards to decide how Chuck will respond knowing this.

Willie knows the following probabilities exist:

Fastball and signal: p*x
Curveball and signal: (1-p)*y
Fastball and no signal: p*(1-x)
Curveball and no signal: (1-p)*(1-y)

Therefore, he knows that the probability that Chuck has thrown a fastball when he observes a signal is simply:

Pr(fastball|signal) = px / [px + (1-p)y]

He also knows the probability that Chuck has thrown a fastball when he does not observe a signal:

Pr(fastball|no signal) = p(1-x) / [p(1-x) + (1-p)(1-y)]

He can also construct his payoffs to swinging and not swinging in these scenarios as well:

His payoff to swinging when he receives a signal is equal to the inferred signal-based probability of each pitch multiplied by the value of swinging in each case:

V(s) = { px / [ px + (1-p)y ] } * (1) + { 1 – px / [ px + (1-p)y ] } * (-1)

His value of taking is:

V(t) = { px / [ px + (1-p)y ] } * (-1) + { 1 – px / [ px + (1-p)y ] } * (1)

He swings when the value of swinging, V(s), exceeds the value of taking, V(t). Mathematically, he swings when:

px / [px + (1-p)y] > 0.5

(As a check, we can remember that the original non-signal scenario is just a generalization of this set-up where x=y, which yields the p>0.5 rule we found before.)

We can also do the same math to determine when the batter swings if he receives no signal. His value of swinging is the probability of getting a fastball when no signal has been received (calculated above) times the payoff of getting a curveball when no signal has been received times the value of swinging in that case:

V(s) = { p(1-x) / [ p(1-x) + (1-p)(1-y) ] } * (1) + { 1 - { p(1-x) / [ p(1-x) + (1-p)(1-y) ] }} * (-1)

His value of taking is:

V(t) = { p(1-x) / [ p(1-x) + (1-p)(1-y) ] } * (-1) + { 1 - { p(1-x) / [ p(1-x) + (1-p)(1-y) ] }} * (1)

He swings when V(s) > V(t), which is equivalent to saying when:

p(1-x) / [p(1-x) + (1-p)(1-y)] > 0.5

Pitcher's Strategy



To solve the whole equilibrium, we want to figure out what strategy “p” is optimal for Chuck the pitcher. We know Willie Bayes will set q(s) = 1 when the term { px / [ px + (1-p)y ] } is greater than 0.5, will set q(s)=0 when the term is smaller than 0.5, and will be indifferent between all q(s) values when receiving a signal when the term is equal to 0.5. We also know that Willie Bayes is going to set q(n)=1 when the term { p(1-x) / [ p(1-x) + (1-p)(1-y) ] } is greater than 0.5, will set q(n)=0 when the term is less than 0.5, and will be indifferent between all q(n) values when the term is equal to 0.5.

We know that if Chuck has a strict preference for throwing a fastball when q(s)=q(n)=0 and we know that Chuck has a strict preference for throwing a curveball when q(s)=q(n)=1. In other words, if Willie never swings, Chuck would just always throw fastballs—in which case Willie would always swing, and if Willie always swings, Chuck would always throw curveballs—in which case Willie would never swing. So any equilibrium is going to require Willie swinging at least some of the time and taking at least some of the time. It’s only natural that we need him to prefer swinging when he gets a signal (or at least being indifferent between swinging and taking), and we need him to prefer taking when he gets no signal (or being indifferent).

Knowing this, Chuck will pick a strategy “p” between 0 and 1, reflecting the probability that he throws a fastball, and p will need to be low enough that q(n)=0 is optimal for Willie and high enough such that q(s)=1 is optimal for Willie.

The outcomes could look like any of the following:

Figure 3B

image

We’ll ignore mixed strategies by Willie conditional on signals because they are not necessary here.

Let’s play with some numbers. Let’s set x=0.9 and y=0.4. In other words, let’s assume that the batter is more likely to fail to notice spin on a curveball than he is to mistakenly notice spin on a fastball. This assumption is important to yield the answers below (and will later be generalized).

For Willie to actually prefer to swing when he gets a “signal” that a pitch is a fastball, we need:

px / [px + (1-p)y] > 0.5

When x = 0.9 and y = 0.4, this is equivalent to saying p > .31.

For Willie to prefer to take when he gets no signal, we need:

p(1-x) / [p(1-x) + (1-p)(1-y)] < 0.5

which is when p < .86.

Therefore, whenever p is between 31 percent and 86 percent, Willie will respond by swinging if and only if he receives a signal.

No slouch with Bayes Theorem himself, Chuck must determine his best response to this, having done the requisite backwards induction. Chuck knows that he has to select fastball at least 31 percent of the time and no more than 86 percent of the time. Chuck also know that if he throws a fastball, Willie will swing 90 percent of the time (giving Chuck a payoff of -1), and will take 10 percent of the time (giving Chuck a payoff of 1). Chuck knows that if he throws a curveball, Willie will swing 40 percent of the time (giving Chuck payoff of 1), and will take 60 percent of the time (giving him a payoff of -1).

Therefore he must select a strategy “p” that maximizes his expected value:

MAX [ (p) * (-1*.9 + 1*.1) + (1-p) * (1*.4 + -1*.6) ]

Simplifying this, we know he selects “p” to maximize:

MAX [ -0.2 – p ]
Subject to .31 < p < .86.

Therefore is must pick “p” that maximizes -0.2 - p, constrained to choosing between 31 percent and 86 percent. Obviously, that’s 31 percent.

Therefore, we have solved the game! Chuck throws 31 percent fastballs, and Willie swings when he observes a signal and takes when he does not observe a signal.

Sneakier curveballs


In the last article, we interpreted a better curveball as one that gives the batter a very low payoff when he swings at it—we changed that -1 into a -1.5. In today’s article, let’s re-interpret a better curveball as “looking more like a fastball” but keep with the -1 payoff to swing/ball.

So, in the previous example, we found out what would happen if Willie observes signals after 90 percent of fastballs and 40 percent of curveballs. Now, let’s make 50 percent of curveballs look like fastballs. How will Willie respond to this sneaky pitcher? In the spirit of having a very good curveball, let’s call this pitcher “Uncle Charlie”; or maybe, let’s just call him Charlie.

When Charlie’s on the mound, Willie must still want to swing when he gets a signal, so he requires that px / [px + (1-p)y] > 0.5, which is when p > .36.

To get Willie not to swing when he gets no signal, we require that p(1-x) / [p(1-x) + (1-p)(1-y)] < 0.5, which is when p < .83.

Therefore, we need Charlie to throw a fastball with probability “p” such that “p” is between 36 percent and 83 percent. We want Charlie to optimize “p” conditional on that constraint. Using the same approach as above, we can find his goal to be picking the best “p” such that:

MAX [ p*(-1*.9 + 1*.1) + (1-p)*(-1*.5 + 1*.5) ]
Subject to .36 < p < .83

This is equivalent to finding:

MAX [-0.8p]
Subject to .36 < p < .83

In other words, Charlie must pick the optimal condition “p” such that he maximizes -0.8p, which means the smallest “p” possible as long as p is between .36 and .83. Obviously, that means he chooses p=0.36, which means he throws 36 percent fastballs.

Recall that the weaker-curveball-throwing Chuck above threw 69 percent curveballs, while Charlie throws his superior curveball only 64 percent of the time. As we found yesterday in the simpler example, the pitcher with a better curve should throw his curveball less often. This will actually entice batters to swing at it more often, making it more potent when it does come, all the while taking advantage of the batter’s reluctance to swing by throwing more fastballs over the plate for strike three.

This result is pretty generalizable, but as we will see tomorrow, there was important assumption that we had to make along the way. I will show what that assumption was, relax it to find exceptions, and also begin to make a more general solution to the pitch selection problem. If nothing else, however, the above should clearly demonstrate that batters and pitchers both could gain an edge by improving their approaches.

Addendum for the advanced reader


It is worth noting that the solution above is not quite a Nash Equilibrium, because the pitcher will strictly prefer to throw curveballs, conditional on the batter's strategy. The actual Nash Equilibrium is very similar, though, and the results are qualitatively the same.

For the pitcher to be indifferent between throwing a fastball and a curveball, the batter must have a mixed strategy when he gets a signal of fastball. This requires that the pitcher actually selects a strategy that keeps the batter indifferent between swinging and taking when getting a signal, which is the same px / [px + (1-p)y ] = 0.5 condition above.

The solutions of 31 percent and 33 percent above hold for the pitcher's strategy. Setting the pitcher's value of throwing a fastball to a curveball, the batters achieves this when = 1/(x+y). Therefore, the batter actually does swing a little bit less often when getting a signal from the superior pitcher (77 percent vs. 71 percent).

Tomorrow: Generalizing the pitch selection model


Click here to learn about THT's download subscriptions.]]> Matt Swartz 2012-12-20T10:51:15+00:00 Game theory and baseball, part 3: more pitch selection http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-3-more-pitch-selection/ http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-3-more-pitch-selection/#When:05:41:15 most recent article of my multi-part series on game theory and baseball, I applied mixed strategies to pitch selection. I considered a situation that I will use for several articles—a full count, with a pitcher deciding to throw a ball or a strike, and the batter simultaneously deciding whether to swing or take.

In the previous article, I constructed an approximate normal form set-up of payoffs under each situation, which I have duplicated below. The batter is equally well off when the outcome is strike/swing as when the outcome is ball/take, and the pitcher is equally well off when the outcome is strike/take as when the outcome is ball/swing.

Table 1

















PitcherBatter Swing Take
Strike (Fastball) 1,-1 -1,1
Ball (Curveball) -1,1 1,-1


In this situation, we found the equilibrium by constructing players’ payoffs conditional on the other player’s strategies. We defined V(s) as the batter’s value of swinging and taking, respectively, conditional on the pitcher selecting a probability “p” of throwing a strike:

V(s) = (p)*(1) + (1-p)*(-1) = 2p – 1
V(t) = (p)*(-1) + (1-p)*(1) = 1 – 2p

The pitcher’s value of throwing a strike and a ball, respectively, conditional on the batter swinging with probability “q” were:

V(g) = (q)*(-1) + (1-q)*(1) = 1 – 2q
V(b) = (q)*(1) + (1-q)*(-1) = 2q – 1

The only equilibrium occurred when the batter set q=0.5 and the pitcher set p=0.5, leading them both to have no better response to the other’s strategy. The batter selected “swing” 50 percent of full count pitches, and the pitcher selected “strike” 50 per cent of full count pitchers.

Great curveballs


Now, let’s tweak that scenario: Suppose that the pitcher has an amazing curveball, which is nearly unhittable out of the strike zone. We will re-interpret the strike/ball decision as a choice between throwing a fastball for a strike or a curveball for a ball. The difference between the new situation and the one above is that the payoff to the batter of swinging at a ball is even worse than -1: it’s now -1.5.

The question that we want to ask is not “what would happen if the payoff is 50 percent worse from strike/ball?” but rather is “what differences should we expect to see between pitchers with great out-pitches and pitchers with average out-pitches?” The answer may surprise you.

The normal form is as follows:

Table 2

















PitcherBatter Swing Take
Strike(Fastball) -1,1 1,-1
Ball (Curveball) 1.5,-1.5 -1,1




Let’s re-solve the equilibrium. The batter’s payoff to swinging, conditional on the pitcher’s probability “p” of throwing a strike is:

V(s) = (p)*(1) + (1-p)*(-1.5) = 2.5p – 1.5

The value of taking, conditional on “p” is:

V(t) = (p)*(-1) + (1-p)*(1) = 1 – 2p

The batter will swing all the time if 2.5p – 1.5 > 1 – 2p, which is when p > (5/9). He will take all the time when p < (5/9). So the pitcher’s strategy that will make the batter indifferent between swinging and taking (thus being willing to randomize) is when p = (5/9).

The pitcher’s value of throwing a strike, conditional on a batter’s probability of swinging “q” is:

V(g) = (-1)*(q) + (1)*(1-q) = 1 – 2q

His value of throwing a ball is:

V(b) = (1.5)*(q) + (-1)*(1-q) = 2.5q – 1

The pitcher will prefer to throw a fastball for a strike when 1 – 2q > 2.5q – 1, which is when q < (4/9). The pitcher will prefer to throw a curveball for a ball when q > (4/9). The pitcher is indifferent when q = (4/9).

Therefore, the only equilibrium is when the batter swings 44 percent of the time and the pitcher throws fastballs 56 percent of the time. In other words, the pitcher with the fantastic out-pitch should throw it less often than the pitcher with the average out-pitch in full counts!

This is probably surprising. After all, pitchers have a tendency to rely more on their great pitches when they reach two strike counts. However, if batters were responding optimally, they would know how unhittable these pitches were and keep the bat on their shoulder more often. To entice the hitters to be willing to take a hack, pitchers should throw more fastballs. When they do throw curveballs, batters will be caught off guard.

The batters, on the other hand, should take more pitches, knowing that swinging too much in these counts will only encourage the pitchers to throw the hook. If they keep the bat on their shoulder more often, the pitcher is likely enough to give them a fastball when they do swing. Therefore, pitchers with great curveballs may actually get more called third strikes on fastballs than pitchers with mediocre curveballs.

Extrapolating this to other two-strike counts, the frequency of failed 0-2 and 1-2 pitches to land for strikes in major league baseball suggests that pitchers are probably over-relying on their great pitches, so much so that they are losing their potency.

Discussion


It is not actually clear who is behaving sub-optimally in real baseball. It could be that batters are swinging at too much junk in an effort to save face, choosing to go back to the dugout knowing that they at least went down swinging. Of course, it could be that pitchers are afraid to get beaten when the count is in their favor, and instead lose their advantages by predictably throwing pitches out of the strike zone in two-strike counts.

Does all of this mean that batters and pitchers are foolish? Not necessarily—I made significant simplifications to formulate these set-ups. It is totally plausible that players may be strategizing incorrectly. Consider the usual reasons why you would trust an expert in other arenas. For instance, if I try to determine an optimal price for a big business in a competitive market, and come up with a price that is significantly different than what the business is charging, I am probably wrong. In a free market, a challenger start-up that profitably undercuts the market leader’s price will quickly begin to dominate the market. One who can wisely over-price the incumbent firm may be able to rapidly gain enough market share to hire its competitor’s employees away. The free market mechanism of “free entry” leads to an optimal price.

In fact, one would not need to use calculus to determine optimal pricing strategies. A business could simply be following a “rule of thumb” such as a 20 percent mark-up that is equivalent to an optimal mathematical derivation. For example, using calculus, the "rule of thumb" mark-up rule is equivalent to setting the derivative of the revenue function equal to the derivative of the cost function, under conditions of a certain constant elasticity demand curve. Effectively, business executives behave like economists without knowing it.

Why doesn’t this same line of reasoning apply to baseball? The answer is that there is no realistic free entry. I can do all the math and declare that Aroldis Chapman should be throwing fewer fastballs in an effort to blow hitters away when he does throw them, but I can’t enter the market. I don’t have a 103 mph fastball. If he’s throwing too many fastballs, there is no savvy new executive ready to step in and steal his customers. Limited entry of pitchers with fantastic fastballs means that the free market won’t solve this one. Aroldis Chapman throwing too many fastballs is still a better pitcher than me throwing the perfect mix of batting practice fastballs and hanging curves.

In the next article we will relax another assumption—we will allow hitters to take a look at the ball while it is on the way, and react accordingly. Let’s give the hitter a chance to take an educated guess, and then let’s give him a little more education.

Tomorrow: If batters were Bayesians

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2012-12-19T05:41:15+00:00 Game theory and baseball, part 2: introduction to pitch selection http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-2-introduction-to-pitch-selection/ http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-2-introduction-to-pitch-selection/#When:08:37:15
Obviously doing this analysis requires extreme oversimplification, but even this basic framework will suggest that pitchers and hitters are not maximizing their chance of winning. Using this framework may very well be able to provide teams with a solid opportunity to improve their odds of winning. See the previous article in this series for some definitions and concepts that I will use below.

If you ever hear a commentator talk about knowing what pitch should be thrown in any given situation, he is probably misinformed. If batters knew what pitches were coming in any given count, they would change their strategies. If it’s easy enough for the commentator to guess, then it’s easy enough for opponents to guess. This is especially true in two-strike counts where you often hear commentators proudly declare that the pitcher should have thrown the ball out of the strike zone to get the batter to chase. If pitchers never threw into the strike zone in two-strike counts, batters would react by simply taking the free ball and improving the count in their favor. Pitchers do sometimes throw strikes in these counts, and that is why hitters swing at balls in the dirt.

In reality, what we usually observe when watching games is an equilibrium outcome. Pitchers and hitters randomize their approaches to given situations to be less predictable, and we only see one such outcome during each pitch.

Dominant Strategies



Of course, sometimes there are dominant strategies. Take an example of a 3-0 count where the batter is a relatively average hitter. The pitcher has a pretty good chance of throwing a strike if he intends to, but it is not guaranteed. In 2012, batters making contact on 3-0 counts actually had wOBAs of just .390. When the count reached 3-1, batters collectively had a wOBA of .417 for the rest of the count. Of course, there are some selection biases present in these numbers, but it suggests that batters may be better off taking 3-0 pitches. This simplified example is an easy introduction to pitch selection.

We can rank the four outcomes from the hitter’s perspective, knowing that the pitcher’s perspective would obviously rank them in reverse order. The hitter’s preferences are:

(a) Ball/Take
(b) Strike/Take
(c) Strike/Swing
(d) Ball/Swing.

In this example, we can just call the payoffs to the hitter 4, 3, 2, and 1 for (a)-(d) above, and call the pitcher’s payoffs the negatives of these numbers.

Table 1A
















Pitcher\Batter Swing Take
Strike -2,2 -3,3
Ball -1,1 -4,4


We can solve the equilibrium by finding the best response of each player, conditional on each of the other player’s strategies:

(a) When the pitcher chooses to throw a strike, the batter gets a payoff of 3 by taking, which is greater than his payoff of 2 if he chooses to swing. So, we bold and underline the “3” under Swing/Take.
(b) When the pitcher chooses to throw a ball, the hitter gets a payoff of 4 by taking, higher than his payoff of 1 if he swings. We bold and underline the “4” under Ball/Take.
(c) When the batter chooses to swing, the pitcher gets a payoff of -1 by throwing a ball, which is greater than -2 by throwing a strike. So, we bold and underline the “-1” under Ball/Swing.
(d) When the batter chooses to take, the pitcher gets a payoff of -3 by throwing a strike, which is greater than -4 by throwing a ball. So, we bold and underline the “-3” under Strike/Take.

Table 1B
















Pitcher\Batter Swing Take
Strike -2,2 -3,3
Ball -1,1 -4,4


The equilibrium is obvious when we realize that the batter has a dominant strategy of taking, so knowing this, the pitcher’s preference is to throw a strike. Getting batters to actually guarantee taking 3-0 pitches in real life, challenging the pitcher to throw a strike, is obviously a tough sell. Effectively, the reason that this equilibrium exists is that the odds are high enough of a ball on a 3-1 or 3-2 count thereafter that it’s not worth letting the pitch move closer to a strikeout on a 3-0 pitch.

Mixed Strategies in Full Counts



There are not many situations above, where a player has a dominant strategy. In most cases, pitch selection and hitter responses will be solved by mixed strategies. As I demonstrated in the previous two articles, a mixed strategy entails a player selecting two (or more) strategies with probabilities between 0 and 1, and the Nash Equilibrium requires the player be indifferent between two (or more) strategies, conditional on his opponent’s (potentially mixed) strategy.

So let’s consider a new situation—a full count. The batter is better off not swinging if the pitcher throws a ball, since he will walk by taking, but will probably be out if he swings at a ball out of the strike zone. If the pitcher throws a strike, the batter is better off trying to make contact, rather than striking out, however. The pitcher naturally prefers the opposite of what the better prefers.

Let’s simplify the payoffs such that the payoff is equally good for the hitter when the outcome is strike/swing and when the outcome is ball/take. Similarly, the hitter is equally worse off when the outcome is ball/swing and when the outcome is strike/take.

Here is the normal form of this game:

Table 2

















Pitcher\Batter Swing Take
Strike -1,1 1,-1
Ball 1,-1 -1,1


As we discussed in the last article, it is not hard to notice that no one has a dominant strategy here.

To solve the game, we consider the batter’s value from swinging and not swinging, conditional on a pitcher’s given strategy “p,” the probability of throwing a strike. The hitter’s value from swinging will be his payoff from Strike/Swing (1) times the probability of this outcome occurring if he swings (p), added to the product of his payoff from Ball/Swing (-1) times the probability of this outcome occurring if he swings (1-p). Therefore, we define the hitter’s value from swinging as:

V(s) = (1)*(p) + (-1)*(1-p) = 2p – 1

The batter’s value from taking is:

V(t) = (-1)*(p) + (1)*(1-p) = 1 – 2p

Conditional on the batter’s strategy “q,” his probability of swinging, the pitcher’s value from throwing a strike (we’ll call this V(g) to differentiate it from the hitter’s value V(s) above) is:

V(g) = (-1)*(q) + (1)*(1-q ) = 1 – 2q

The pitcher’s value of throwing a ball when the batter swings with probability “q” is:

V(b) = (1)*(q) + (-1)*(1-q) = 2q – 1

The batter will always prefer to swing if 2p – 1 > 1 – 2p (i.e. when p > 0.5), so in this case, his strategy is to always swing (i.e. set q = 1). Similarly, the batter will always take when p < 0.5, so his strategy would be to always take (i.e. set q= 0).

However, the pitcher will always prefer to throw a strike when 1 – 2q > 2q – 1 (i.e. when q < 0.5), so in this case, his strategy is to always throw a strike (i.e. set p = 1). Similarly, the pitcher will always throw a ball when q > 0.5, so his strategy would be to throw a ball (i.e. set p = 0).

Taking these together:

(a) When p<0.5, the hitter will set q=0; but when q=0, the batter would set p=1. Since 1 is not less than 0.5, we know that there are no equilibria where p<0.5.
(b) Equivalently, if we set p>0.5, the hitter will set q=1; but when q=1, the pitcher would set p=0. Since 0 is not greater than 0.5, we always know there are no equilibria where p>0.5.
(c) However, when p=0.5, the pitcher is equally happy with any “q.”

So all equilibria will require p=0.5.

It’s not hard to see where this is going.
(a) When q<0.5, the batter would set p=0; but when p=0, the pitcher would set q=1, so there are no strategies where q<0.5.
(b) When q>0.5, the batter would set p=1, but when p=1, the pitcher would set q=0, so, there are no strategies where q>0.5.
(c) Therefore, the only possibility is for q=0.5, in which the hitter is equally happy with all “p”.

Therefore, the only Nash Equilibrium that exists is when p=0.5 and q=0.5. In words, the only possible outcome is for the pitcher to throw strikes in 50% of these full count situations, and the batter to swing in 50% of these situations. The expected values for the hitters and pitchers are both equal to 0 when this happens. No one can pick a strategy that increases their expected value above 0.

Mixed Strategies in 2-2 Counts



We do not know the exact payoffs for any hitter or pitcher above, so we cannot say that the exact probabilities that they should choose are 50% each. However, we can compare situations to figure out if the theory matches real baseball strategies, so let’s consider a 2-2 count. We will figure out what the optimal strategies would be in this situation and compare them to the 3-2 count. Are the probabilities of throwing strikes higher or lower, and are the probabilities of swinging higher or lower? Do these compare well with what we observe in real baseball?

We can pretty easily construct payoffs that are consistent with the ones above. We know that the payoffs of swinging at a strike on a 2-2 count should be the same as swinging at a strike on a 3-2 count. Either you strike out or you make contact, and whether there were two or three balls at the time won’t matter.

We also know that the payoffs of swinging at a ball in a 2-2 and 3-2 count should be the same. Taking strike three with a 2-2 count and a 3-2 count obviously are equivalent two, since both are strikeouts. The only difference is taking a ball, since the payoff in that case will be a 3-2 count—which means the expected value should be the 0 expected value of the equilibrium above.

Here is the normal form of the 2-2 count game then:

Table 3

















Pitcher\Batter Swing Take
Strike -1,1 1,-1
Ball 1,-1 0,0


We can see that there are no pure strategy equilibria in this game as well. We can compute the value functions in each of these cases. The batter’s value of swinging is:

V(s) = (1)*(p) + (-1)*(1-p) = 2p – 1

The batter’s value of taking is:

V(t) = (-1)*(p) + (0)*(1-p) = -p

The pitcher’s value of throwing a strike is:

V(g) = (-1)*(q) + (1)*(1-q) = 1 – 2q

The pitcher’s value of throwing a ball is:

V(b) = (1)*(q) + (0)*(1-q) = q

The pitcher will prefer to throw a strike when 1 – 2q > q, which is when q < (1/3). The pitcher will prefer to throw a ball when 1 – 2q < q, which is when q > (1/3). The pitcher is indifferent when q = (1/3).

The batter will prefer to swing when 2p – 1 > p, which is when p > (1/3). The batter will prefer to take when 2p – 1 < p, which is when p < (1/3). The batter is indifferent when p = (1/3).

The only equilibrium is going to occur when p = (1/3) and q = (1/3). In words, that means that the batter is swinging at a third of 2-2 pitches (less than he swings in 3-2 counts), and the pitcher will throw a strike a third of the time on 2-2 pitches (less than he throws over the plate in 3-2 counts). The pitcher’s decision to throw fewer strikes on 2-2 counts does seem to follow what we observe in real life, but I was surprised that the hitter should swing less often in 2-2 counts. It seems to me that batters smell a free base and take more pitches in 3-2 counts than 2-2.

This framework strongly suggests that batters are not behaving optimally. They should be expanding the zone more on 3-2 pitches due to the pitcher’s increased willingness to throw a pitch that finds the corner, while they should be skeptical of pitchers with a ball to spare on 2-2 counts.

In the next article, I will begin to explore what happens when we adjust for quality of pitchers, and consider whether pitchers with extraordinary skills should behave differently. The results may surprise you.

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2012-12-18T08:37:15+00:00 Game theory and baseball, part 1: concepts http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-1-concepts/ http://www.hardballtimes.com/main/article/game-theory-and-baseball-part-1-concepts/#When:08:37:15
Using a game theoretic framework will justify some common baseball strategies but often will suggest that teams and players should behave differently. However, before any of the specific applications can be discussed, it’s important to get some concepts together. Today’s article will talk about some basic game theory concepts that we’ll need going forward and some basic applications associated with each to illustrate them. This gives us a broader framework for discussing pitch selection for a few articles after this.

Strategies, best responses, Nash Equilibrium, and dominant strategies


The principal difference between game theory and other economics is the incorporation of other people’s actions into determining one’s own. A strategy is defined as an option that a player can choose at any point in time, conditional on the information that he has at the time. In a simultaneous move game, the strategies are just the allowable actions.

In baseball, a set of strategies could be all possible lineups that a manager could choose at the beginning of the game. Choosing lineups is a simultaneous move game that mangers play at the beginning of the game. A sequential move game involves strategic substitutions with a dynamic aspect.

The most important concept in game theory to understand is “best responses,” as incorporated in a Nash Equilibrium. A Nash Equilibrium is an outcome to a game such that all players have “best responded” to each other’s strategies. In other words, no one would regret his strategy, based on the revelation of other players’ strategies.

In previous work, I have discussed the concept of why slotting was an effective way to keep bonuses down for drafted players, but let’s use a one-shot game (with no future drafts to follow) of this as an example and explain why slotting would not work without future drafts.

Suppose that the Dodgers and the Giants, two high-spending division rivals, are fighting for the National League West, and they are considering either:

(A) going over slot for a highly touted first-round pick out of high school and paying a large bonus
(B) paying the recommended slot bonus for a college senior in the first round

There are a few similar players at each spot, too, so the Dodgers and Giants can each select a high school or college player as they see fit. And let’s suppose for simplicity that all contracts are secretly hammered out in advance (i.e. decisions are not actually sequential). Both teams must decide whether to work out a contract with a highly touted high school prospect well in advance without knowing what the other team will do.

Assuming all else equal, the added advantage in future division championships leads to payoffs as follows:

Table 1A
















Dodgers\Giants College draftee High-school draftee
College draftee 5,5 1,7
High-school draftee 7,1 3,3


In the above table, we have the sets of strategies available to each team represented by the rows (for the Dodgers) and by the columns (for the Giants). In other words, the Dodgers get to pick the row that we end up in (making the Dodgers the “row player”) and the Giants pick the column (making them the “column player”).

The payoffs listed are the Dodgers (in blue), followed by the Giants (in orange) for each outcome. So, for instance, the payoff to the Dodgers of picking a high-school draftee is seven if the Giants pick a college draftee, but the Giants' payoff would be just one.

To figure out what the teams will do, we need to figure out what the best response would be to each pick. Let’s be the Dodgers here. If the Giants pick a college draftee, which payoff is higher—choosing a college draftee or a high-school draftee? The payoff of seven for a high-school draftee exceeds the payoff from drafting a college player of five—so the Dodgers would prefer a high-school draftee if the Giants pick a college draftee.

What about if the Giants pick a high-school draftee? In that case, the Dodgers’ payoff to a college draftee would be just one, while a high-school draftee would be worth a payoff of three. The Dodgers would be better off picking a high-school draftee in either case.

This means that the Dodgers have a dominant strategy, which means their best response is the same for any Giants action. The Giants' decision is symmetric, so they also have a dominant strategy. Both teams pick high-school draftees and get payoffs of three and we have a Nash Equilibrium, since both teams would not regret their strategies upon learning the strategy of the other player.

Sometimes, it’s easiest to underline and bold the best responses so that you can find the best answer. An equilibrium occurs when both numbers are underlined and bolded in a cell.

Table 1B

















Dodgers\Giants College draftee High-school draftee
College draftee 5,5 1,7
High-school draftee 7,1 3,3


Extensive form vs. normal form


The table format above is called a “normal form,” but there is another way of displaying games, which is called the extensive form. This is most useful for sequential games because it shows the order of decision-making.

Let’s revise some assumptions. Say the Dodgers pick first, and then the Giants choose after observing the Dodgers' pick. In the normal form, simultaneous-move game above, I fudged this assumption by pretending that everything can be negotiated in advance, but let’s take that back.
Figure 1A

image

Let’s break down the extensive form above. There are three decision nodes at which decisions can be made:

(I) Dodgers choose high school or college.
(II) Giants observe Dodgers' choice of high school and choose high school or college.
(III) Giants observe Dodgers' choice of college and choose high school or college.

If the Dodgers choose high school, and the Giants see this and choose high school, they both get a payoff of three (blue for the Dodgers, orange for the Giants). If the Dodgers choose high school, and the Giants choose college having observed this, the Dodgers get a payoff of seven and the Giants get a payoff of one. And so on. Now, let’s solve this game.

Backwards induction


Another important concept in game theory is backwards induction. This is a concept in sequential games that describes how players will solve the equilibrium by working out all decisions backwards from the end of the game to the beginning.

To find the equilibrium using backwards induction, you figure out all of the “final decision” moments during the extensive form game. That means all of the “decision nodes” where a player has a new “information set” for the last time. These are Decision Node II (after Dodgers have picked high school, when the Giants must pick high school or college) and Decision Nodes III (after Dodgers have picked college, when the Giants must pick high school or college). In each of these instances, the Giants will pick high school, for the same reasons as before.

Knowing this, we backwards induce to the Dodgers' decision. The Dodgers know that if they pick high school, they will end up with a payoff of three, and if they pick college, they will end up with a payoff of one.

Sometimes it is easiest to draw over top of the extensive form of the game with bright-colored lines to show the decisions that will be made. In the end, the equilibrium will be such that you can follow a red line from the top to the bottom.

Figure 1B

image

Below, you can see how we draw the extensive form of a simultaneous-move game. This is not always useful, but it is another way of looking at the game. If the Giants do not know what the Dodgers have selected by the time they have to make their decision (as in my original example), we draw a dashed line between the two possible decision nodes at which the Giants might reside. We label the nodes and the dashed line between them “II.”

Figure 2A

image

We’ll use red lines to solve this problem again, only we require that the Giants pick the same decision regardless of what the Dodgers picked … because, after all, they don’t know what the Dodgers picked! Here is the solution: as above, both pick high school draftees and they get payoffs of three.

Figure 2B

image

This is just the tip of the iceberg when it comes to “prisoner’s dilemma” games and various assumptions that can be made, but it is sufficient for now. Next, let’s consider mixed strategies.

Mixed strategies


Nash Equilibria are named after John Nash, because he proved that there are solutions to all games that meet certain criteria. However, the “solutions” sometimes entail “mixed strategies.” Mixed strategies are characterized by selecting probabilities of taking a given action. For example, a (dumb) mixed strategy above could entail choosing “college” 70 percent of the time and “high school” 30 percent of the time.

So, let’s suppose that a batter is up and is deciding whether to swing or take on a full count, and let’s suppose the pitcher is deciding whether to throw a ball or strike. If the player swings at a strike, he’ll get a hit and win the game; if he swings at a ball, he’ll miss and strike out, losing the game. Naturally, taking strike three will also lose the game, and let’s just say that taking a ball will effectively win the game because Superman is on deck. Thus, the effect on each player’s team's record is reflected in the below table:

Table 2A

















Pitcher\Batter Swing Take
Strike -1,1 1,-1
Ball 1,-1 -1,1

For the sake of brevity, I will save the calculations to this for another article, but this game requires mixed strategies to find the Nash Equilibrium. Why is this true? You can see it when you realize that for every cell on the table above, one player wishes he selected the other action—no cell involves both players “best-responding.” This may be easier to see by underlining and bolding the correct best responses for the batter and the pitcher.
Table 2B

















Pitcher\Batter Swing Take
Strike -1,1 1,-1
Ball 1,-1 -1,1

The way to view this game properly is to define strategies by “p” and “q” such that the batter selects “Swing” with probability “q,” and the pitcher selects “Strike” with probability “p.”

Table 2C

















Pitcher\Batter Swing (with probability "q") Take (with probability "1-q")
Strike (with probability "p") -1,1 1,-1
Ball (with probability "1-p") 1,-1 -1,1

In the end, the only strategies that will work will be when the batter selects a strategy of having a 50 percent chance of swinging and a 50 percent chance of taking, and the pitcher selects a strategy of 50 percent strikes and 50 percent balls. They will each win half the time, and neither player could be any better off by selecting a different strategy. The batter knows that the pitcher is throwing 50 percent strikes, so he’ll win half the time whether he always swings or never swings.

Conclusion


The pitch-selection example above is an important one, and it is one of the areas of baseball analysis that has barely been analyzed. As I will show in the coming articles, players are probably not picking optimal strategies in terms of pitch selection, and they could improve their winning percentages by following a more strategic framework.

Tomorrow: Introduction to pitch selection

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2012-12-17T08:37:15+00:00 Big, long deals http://www.hardballtimes.com/main/article/big-long-deals/ http://www.hardballtimes.com/main/article/big-long-deals/#When:09:05:15 Cole Hamels signed a six-year deal with the Phillies for $144 million last month, forcing the Phillies to put a lot of eggs in one basket. The Phillies' success over the next few years undoubtedly will be tied to the aging of a single southpaw hurler.

Is that really such a wise risk? It’s tough to say. If some of the rumors are true, chances are the man might be a pretty safe bet, but what about big, long deals in general? We all know that players are betrothed to their organizations for their first six major league seasons, but how wise is it for teams to offer the same long commitment to players?

Everyone remembers disastrous long-term deals like the ones dished out to Ken Griffey Jr., Mike Hampton, and Gary Matthews Jr., but few people realize just how productive Andruw Jones, Scott Rolen, and Alex Rodriguez were during their contracts.

However, when you put these big contracts together, you find that on average they are just about as good as other contracts. Adjusting older salaries upwards to their 2011 equivalent, the 24 deals that I looked at cost $6.59 million per WAR, just four percent more than the $6.35 million per WAR that all deals cost in 2011.

To do an unbiased analysis, I limited my scope to all deals ending at some point between 2007 and 2011 that bought out at least five years of free agency. For example, Jones’ six-year deal lasting from 2002 through 2007 was considered a five-year deal for $65 million (the share of his salary coming once he reached six years' service time after 2002), while Albert Pujols’ deal that guaranteed seven years from 2004 through 2010 did not count, because he did not reach free agency until after 2006, when only four years were left.

If a deal had an option that was declined, I treated the buyout as part of the salary in the previous seasons. If an option was picked up, I treated the extra year and salary as part of the contract. However, contracts where the fifth year was an option year (such as David Ortiz’s deal from 2007 through 2010 with a 2011 option) were not included in the analysis.

To adjust for the growth in salaries, I used my estimates of dollars per WAR for 2000 to 2010 from this article and scaled the average annual salary upwards for each year to $6.35 million/WAR. (For example, the 2003 estimate of $4.0 million per WAR meant that any salary in 2003 would be multiplied by $6.35/$4.00). On top of that, I used adjustments for draft picks foregone as I have done in previous contract analyses.

The table below lists all of the deals in order of effective price per WAR. The best rate on any deal was on Jones’ contract with the Braves. He accumulated over six WAR per season while only making $13 million per season during the free agent portion of his contract. Even adjusting for inflation to current salary levels, the deal would have cost only about $19 million per season. Rolen’s deal was a steal. as well.

As we would expect based on some of my recent findings, infielders were overrepresented in the best contracts, while outfielders were quite common among the worst contracts. The worst deal (by rate) was Matthews, since he was the only player to actually produce negative WAR on a five-year deal. However, in terms of cost, the worst deals were those signed by Griffey and Hampton. Relative to current salaries, those contracts would cost over $20 million per season, while they each produced 1-2 WAR per season on average.










































































































































































































Name Years Deal (excluding pre-FA years) Total Inflation-Adjusted Millions of $ Total fWAR Effective Price ($MM/fWAR adj. for draft picks)
Andruw Jones 2003-07 5/$65 $93.53 30.1 $3.11
Scott Rolen 2003-10 8/$90 $119.41 37.8 $3.52
Miguel Tejada 2004-09 6/$72 $94.41 25.3 $3.95
Lance Berkman 2006-10 5/$74.5 $90.06 22.1 $4.60
Carlos Beltran 2005-11 7/$119 $144.40 32.2 $4.70
Alex Rodriguez 2001-07 7/$171 $260.52 57.1 $4.72
Vladimir Guerrero 2004-08 6/$82 $107.82 22.1 $4.88
Adrian Beltre 2005-09 5/$64 $80.30 16.7 $5.00
Kevin Millwood 2006-10 5/$60 $72.16 15.1 $5.21
J.D. Drew 2007-11 5/$70 $79.90 13 $6.15
Aramis Ramirez 2007-11 5/$75 $85.75 16 $6.20
Derek Jeter 2002-10 9/$176 $242.90 43.5 $6.35
Juan Pierre 2007-11 5/$44 $49.49 7.5 $6.60
Magglio Ordonez 2005-09 6/$90 $112.51 16.8 $6.70
Paul Konerko 2006-10 5/$60 $72.16 11.5 $7.00
Todd Helton 2004-11 8/$130.9 $160.28 25.7 $7.64
Manny Ramirez 2001-08 8/$160 $236.04 34.2 $7.74
Jim Thome 2003-08 7/$95 $128.29 19.4 $7.98
Jason Giambi 2002-08 7/$120 $172.74 21.9 $9.01
Eric Chavez 2005-10 6/$66 $81.92 7.9 $20.38
Mike Hampton 2001-08 8/$121 $177.89 10.7 $26.40
Ken Griffey Jr. 2000-08 9/$116.5 $179.08 11.7 $31.76
B.J. Ryan 2006-10 5/$47 $56.12 3 $35.66
Gary Matthews Jr. 2007-11 5/$50 $56.51 -0.6 -


There were very good deals and very bad deals included in this analysis. For teams giving out large contracts, the best approach is not to avoid them altogether, but to pick out the right ones. Picking out the right ones is tricky, but the importance of information is crucial.

As I have discussed in the past many times, teams re-signing their own players (whom they know well) fare far better than teams who sign deals for “Other People’s Players.” As we can see, this is true for very long deals as well. The cost per WAR was 39 percent higher for Other People’s Players (OPP) than re-signed players (RSP).
























Category N Total Adj. $MM Total fWAR Effective Price
RSP 7 $785.73 168.9 $5.22
OPP 17 "$2 168.47 " 331.8 $7.27


Looking through the deals above, you might notice that it’s hard to find something comparable to Cole Hamels’ pact, since only three of the above contracts went to pitchers. Over the next few years, as we watch deals like CC Sabathia’s and Cliff Lee’s play out, we may have a better sense of the wisdom of long deals given to pitchers.

Another important factor to consider when signing long-term deals is that, by their very nature, they will look better during the early years than the latter years. I found that these 24 players averaged 4.1 WAR per season in the first half of their contracts and 2.5 WAR per season in the second half. When you look only at the first year and the last year, the difference is more extreme: 4.6 WAR in the first season and 2.3 WAR in the last season.

That information neither makes the deals look strong or weak, but it is important to remember that if you are not getting a lot of WAR per dollar on average in the first few years of a deal, you probably will be in trouble at the end of it. In fact, one third of these players provided less than 1.0 WAR during their last year while all players produced at least 1.0 WAR in their first year.

The point of this study is that it is unreasonable to sign a long contract with the expectation that you will have a player on top of his game the whole time. Instead, you should sign a player for as long as you expect him to merit a roster spot, and price accordingly. Make sure it’s a bargain at the beginning, since these contracts are effectively loans. The end of the contract is the repayment period for elite seasons banked at the beginning.

Overall, this analysis should dispel the myth that long-term contracts are usually bad. Instead, they are usually about average, but with a lot of risk in either direction. Having some risks with favorable outcomes is a prerequisite for a championship, while unfavorable outcomes can remove a team from contention for years at a time.

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2012-08-06T09:05:15+00:00 Free agent value and building teams from within http://www.hardballtimes.com/main/article/free-agent-value-and-building-teams-from-within1/ http://www.hardballtimes.com/main/article/free-agent-value-and-building-teams-from-within1/#When:05:32:15 marginal payroll per marginal win. The idea is simple: if you fill a team with fringe players available on the free agent market for the league minimum salary, you would win about 43 games and spend about $12 million. Therefore, a team should not be evaluated only on how many wins it can get beyond that baseline of 43, but on how efficiently it can use its resources to exceed that number.

For example, the Rangers averaged 85 wins and $74 million from 2007-2011, so they spent $62 million above league minimum to get 42 wins above a replacement-level team. They spent about $62 million to get about 42 wins, which equals $1.47 million in marginal payroll per marginal win.

The idea changed baseball analysis—efficiency is an essential concept for team construction. However, if we look at how teams rank in this crucial statistic, we start to see its limitations.

Looking at the last five years, the Marlins are unsurprisingly on top and the Yankees are unsurprisingly at the bottom. However, despite the fact that the Marlins have been able to average about 79 wins per season (about 36 wins beyond what a team of retreads would cobble together) at a rate of $0.78 million per marginal win, that does not mean that the Yankees could have realistically achieved their 17 extra wins per season at the same efficient clip. To win 17 more games, the Yankees used mostly free agents, who cost an average of $5.4 million per fWAR (the Fangraphs’ version of Wins Above Replacement) over the last five seasons.

Here are the rankings for 2007 through 2011. This is just the top five and bottom five. The full table is at the bottom of the article.

image

Unlike winning pennants, the way to win the marginal-payroll-per-marginal-win prize is to not spend money on free agents. About 70% of all major league WAR comes from players who are not eligible for free agency in the first place, so the average team would go 69-93 by retaining all players in its system until they were eligible for free agency.

But it’s neither good business nor good baseball to sit on your hands at that point. Opening up your checkbook may put you over the top and into the playoff race, which could be far more profitable than sitting tight near the 70-win mark.

Performance from players yet to reach free agency


There are really two different sources of per-dollar efficiency when we look at marginal payroll per marginal win:
(1) How well a team gets production from players not yet eligible for free agency
(2) How efficiently a team spends on free agents

So, for the following analysis, I will use two classifications of players that are particularly important.
(1) NM = Non-Market Players, who are either bound to their team by the reserve clause or eligible for arbitration
(2) AM = Auction-Market Players, who are eligible for free agency or are at least eligible for auction by being professional amateurs from countries like Japan and Cuba.

Taking the analysis further, I considered two different definitions of a Non-Market player:

{exp:list_maker}NM Drafted & Signed: where the signing team gets credit for the player’s performance, even if that team had traded him to another team. For example, Neftali Feliz’s WAR is credited to the Braves (who signed him as an amateur).
NM Players: where the team that holds the player on its roster gets credit for the player’s performance. In this case, Neftali’s Feliz WAR is credited to the Rangers (where he played). {/exp:list_maker}
Players who are playing for the team that originally signed them are included in both definitions.

Both versions exclude any player who had six years of service time (or was signed as a professional out of Japan, Cuba, or elsewhere). To estimate the impact of a specific player, I calculated the difference between the team’s record and the player’s WAR.

The Red Sox, for example, would have won an average of 79 games per season by keeping players from their system who had not yet reached free agency, but traded away about 11 wins, leaving them with 68 wins from NM Players still on the team. They then supplemented those 68 wins with 25 wins from the free agent market or by acquiring free agents signed by other teams.

On the other hand, the Dodgers’ payroll averaged $110 million over the last five seasons, but they might have been just as good if they had just retained their own draftees and amateur signings. The only cost would have been about $30 million in arbitration and league minimum salaries, and they would have been about as good as they were spending $110 million.

The following table ranks teams by the amount of non-market talent they developed, regardless of whether they held onto that talent of traded it away. The first column sums up the talent they originally signed, the second column sums the wins from non-market players on the team and the third column reflects the team’s actual performance.

In other words, the difference between the second and third columns is a reflection of how much talent each team signed through free agency.

image

Using this metric, we can see which teams had the most productive systems, and gave themselves the best jumping off point, and which teams had the worst head starts.

The Braves, like the Dodgers, traded away a lot of young talent, but they at least supplemented the talent they lost with free agents that made a difference. The Dodgers lost 16 wins of non-market talent they had in their system and only added 18 wins of auction-market talent back, while the Braves lost eight wins of non-market talent that they had in their system and added 17 wins of auction-market talent back on to their total.

The Diamondbacks and Rockies were good at scouting and acquiring amateur talent, and good at retaining it too, but they both did a relatively poor job supplementing this talent with free agents necessary to put them into the playoffs continuously.

The Padres, Royals, and Astros have been unsuccessful in recent years primarily due to a dearth of in-system amateur talent. While none were particularly spendthrift when it came to free agents, they had terrible starting points, and would only have been able to cover so much of this gap by opening their wallets.

Bang for your buck


The other element of front office efficiency is getting the most bang for your buck when you do supplemental your non-market talent with free agents. As you can imagine, there were large differences between teams in terms of how efficiently they spent their money. The Cardinals were able to buy wins at a rate of approximately $3.3 million per WAR. On the other hand, the Athletics, despite their prowess in terms of marginal-payroll-per-marginal-win, had to spend over $13 million for every WAR they acquired on the free agent market.

The $/WAR numbers below include an adjustment for the WAR lost from draft picks lost due to free agency signings, as well as the bonus money saved. This is most obvious when we look at the Pirates. The Bucs only managed to purchase 2.5 WAR from free agents, but did so at the expense of a few draft picks that actually were worth about 2.6 WAR.

All salaries are considered relative to the yearly league minimum as well, and then credited to the team that signed the deal (so Manny Ramirez’s WAR in 2009 as a Dodger went to the Red Sox).

Table 3 below lists teams $/WAR from free agents alone. Notice that even though the Cardinals generated a lot of their efficiency by buying Albert Pujols’ wins on the cheap, they had good contracts on a lot of players.

image
Note: The Pirates lost 2.6 WAR from the value of their draft picks, so their net $/WAR was negative.

Putting it all together and building a team


Comparing the rankings in terms of marginal payroll per marginal win in Table 1 to the equivalent statistic for free agents in Table 3, we see stark differences.

The Marlins had the best ranking in the Table 1, getting wins at a rate of $0.78 million, while the Yankees’ $3.71 million puts them at the back of the pack. However, Table 3 shows both teams are square in the middle in terms of how much they spent on free agent talent—the Yankees spending $5.4 million per win, and the Marlins spending $5.6 million per win.

The real difference was that the Marlins averaged 35.4 non-market WAR and only 2.0 auction-market WAR, while the Yankees averaged 18.7 NM WAR and 33.0 AM WAR. The Yankees spent far more than anyone else on free agents, presumably because they got the most value from doing so. Winning in New York is valuable. They were only okay at producing talent from within, but they were not at the bottom of the pack.

Looking at the teams at the top and bottom of the rankings, we can learn a little bit more about what these front offices have done. The Padres were actually quite efficient when they did spend money, but they just didn’t spend that much. Their position in the Table 2 explains why. The production from within their system was so weak that they would have gone broke trying to supplement it with free agents.

The Rays were second on the marginal payroll per marginal win ranking for a few reasons. They were above average with 30.7 WAR from their amateurs who had less than six years of service time, and also made some great trades to supplement this, leading the league with 38.8 non-market WAR for their players. On top of that, they still only spent $3.8 million per WAR for free agents. The reason that they were not running away with the AL East every season is that they just didn’t spend that much money.

Below I provide the full tables for all 30 teams. Each team’s story becomes a little bit clearer when looking at Table 2 and Table 3 than looking at Table 1. Marginal payroll per marginal win is useful, but it does not differentiate between being productive by buying underpriced talent and by being cheap. The Marlins may have looked very efficient from 2007 to 2011, but I bet they’ll make more profit with their newfound approach to spending on the free market.

Appendix of Tables


Table 1A: Marginal Payroll per Marginal Win, 2007-11



























































































































































































Rk Team (2007-2011) Avg. W-L Marginal $/Win
1 Marlins 79-83 $0.78
2 Rays 86-76 $0.88
3 Padres 78-84 $1.16
4 Diamondbacks 80-82 $1.41
5 Rangers 85-77 $1.47
6 Athletics 76-86 $1.52
7 Pirates 65-97 $1.53
8 Indians 78-84 $1.57
9 Rockies 82-80 $1.58
10 Brewers 85-77 $1.61
11 Reds 79-83 $1.80
12 Twins 82-80 $1.80
13 Blue Jays 82-80 $1.84
14 Nationals 68-94 $1.87
15 Braves 84-78 $1.96
16 Royals 69-93 $2.01
17 Cardinals 86-76 $2.02
18 Giants 81-81 $2.08
19 Phillies 95-67 $2.25
20 Angels 91-71 $2.31
21 Dodgers 85-77 $2.36
22 Tigers 85-77 $2.56
23 White Sox 81-81 $2.63
24 Astros 73-89 $2.67
25 Orioles 67-95 $2.75
26 Red Sox 93-69 $2.76
27 Cubs 82-80 $2.95
28 Mariners 72-90 $3.05
29 Mets 81-81 $3.29
30 Yankees 96-66 $3.71


Table 2A: Production from Within, Retention of Young Talent, and Actual Records

































































































































































































































Rk Team NM Drafted & Signed NM Players Average W-L
W-L W-L
1 Dodgers 83-79 67-95 85-77
2 Red Sox 79-83 68-94 93-69
3 Angels 78-84 77-85 91-71
4 Diamondbacks 76-86 77-85 80-82
5 Tigers 76-86 70-92 85-77
6 Phillies 76-86 76-66 95-67
7 Braves 75-87 67-95 84-78
8 Rays 74-88 81-81 87-75
9 Rockies 73-89 75-87 82-80
10 Rangers 73-89 75-87 85-77
11 Mets 73-89 65-97 81-81
12 Brewers 72-90 75-87 85-77
13 Giants 72-90 69-93 82-80
14 Mariners 71-91 61-101 72-90
15 Pirates 70-92 65-97 65-97
16 Twins 70-92 73-89 82-80
17 Athletics 69-93 73-89 76-86
18 Yankees 67-95 62-100 96-66
19 Nationals 66-96 62-100 68-94
20 Blue Jays 66-96 70-92 82-80
21 Marlins 65-97 77-85 79-83
22 Indians 65-97 73-89 78-84
23 Orioles 64-98 61-101 67-95
24 Cardinals 63-99 63-99 86-76
25 Cubs 63-99 57-105 82-80
26 White Sox 63-99 66-96 81-81
27 Astros 63-99 61-101 73-89
28 Reds 61-101 71-91 79-83
29 Padres 61-101 72-90 78-84
30 Royals 57-105 64-98 69-93


Table 3A: Team Dollars per WAR from Free Agents



























































































































































































Rk Team FA fWAR FA $/fWAR
1 Cardinals 24.9 $3.30
2 Padres 6.3 $3.60
3 Braves 14.2 $3.60
4 Rays 4.7 $3.80
5 Rangers 12.7 $4.00
6 Cubs 24 $4.40
7 Blue Jays 12.4 $4.40
8 Twins 9.2 $4.80
9 Brewers 8.4 $4.80
10 Red Sox 22.8 $4.90
11 Indians 7.9 $5.00
12 Nationals 6.5 $5.00
13 Phillies 18.9 $5.40
14 Tigers 15.1 $5.40
15 Marlins 3.5 $5.40
16 Yankees 32.2 $5.60
17 Astros 10.7 $5.80
18 White Sox 14.7 $5.90
19 Dodgers 13.8 $6.30
20 Diamondbacks 3.3 $6.30
21 Giants 12.4 $6.40
22 Mariners 10.2 $6.40
23 Royals 5 $6.50
24 Angels 14.3 $6.90
25 Reds 6.9 $7.50
26 Rockies 5.8 $7.90
27 Mets 13.2 $8.10
28 Orioles 6.4 $9.10
29 Athletics 3.5 $13.20
30 Pirates 0.5 -

Note: The Pirates lost 2.6 WAR from the value of their draft picks, so their net $/WAR was negative.

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2012-04-03T05:32:15+00:00 Adjusting defense efficiency by the quality of pitching http://www.hardballtimes.com/main/article/adjusting-defense-efficiency-by-the-quality-of-pitching/ http://www.hardballtimes.com/main/article/adjusting-defense-efficiency-by-the-quality-of-pitching/#When:09:18:15 Fausto Carmona throws a hard sinker on the outside corner, but Ichiro Suzuki turns it into a well-struck ground ball by going the other way, splitting the defenders on the left side of the diamond. We know who should get credit for the single on the Mariners' side of the box score—there was only one guy with a bat. But who on the Indians will take the blame for the single? Is it Carmona who made the pitch, or the defenders who could not get to the ball fast enough?

Bill James invented Defensive Efficiency, measuring the percentage of balls in play that a defense turns into outs. It became apparent just how useful this would be for evaluation of team defense when Voros McCracken famously concluded that, "There is little if any difference among major-league pitchers in their ability to prevent hits on balls hit in the field of play." A natural corollary to this thesis says that to measure team defense, one should use Defensive Efficiency rate.

However, since McCracken’s original thesis, the community has determined with certainty that while there is little difference between pitchers, there definitely are some major "little" differences. Following on work by J.C. Bradbury and others, I have shown that a pitcher’s ability to control the number of hits he surrenders on balls in play is well correlated with strikeout rate, walk rate and ground ball rate, the so-called “DIPS” (Defense Independent Pitching Statistics) that are not determined by the defense behind a pitcher. In fact, a pitcher’s BABIP in a given season correlates more with his DIPS the previous season than with his BABIP the previous season. In other words, DIPS predicts BABIP better than BABIP itself does.

As I close in on how to measure a pitcher’s ability to control BABIP without actually using what happened on balls in play, I have realized that I can actually see how much of team defensive efficiency is the fault of hurlers. It turns out that a large portion of defensive efficiency is pitching after all. I have shown the following to be true:

A) Pitchers who strike more hitters out give up fewer hits on balls in play.
B) Pitchers who induce fewer ground balls give up fewer hits on balls in play.
C) Pitchers who walk fewer hitters give up fewer hits on balls in play.

Using this information, I have found that the variance in BABIP among starting pitchers who pitch over 150 innings can be attributed approximately as follows:

A) 12 percent pitching skill
B) 13 percent team defense skill
C) 75 percent luck

Of the fraction that pitchers do control, you can predict about 10.4 of those 12 percent using DIPS. Yes, pitchers do exhibit some control over their BABIP, but in an entirely estimate-able way. I think this passes the smell test, too, because if I try to imagine a pitcher who you expect to limit hits on balls in play, I picture one who fools hitters into whiffing a lot too, or perhaps one who pops a lot of hitters up.

One of the most underrated aspects of SIERA is that it implicitly computes an “Expected BABIP,” by using regression techniques. Since it looks directly at expected ERA, conditional on strikeout rate, ground ball rate, and walk rate, it does not directly compute the effect of a strikeout on ERA; instead, it computes what pitchers’ ERAs will look like given their strikeout rate (and holding everything else constant). Thus, SIERA expects high-strikeout pitchers to have low BABIPs, and makes similar adjustments for ground ball rate and walk rate as well.

As I considered how individual pitchers' DIPS correlate with expected BABIP recently, I realized that there are considerable differences among whole teams in their strikeout and ground ball rates. The 2010 Giants struck out 21.6 percent of hitters faced; the 2006 Royals struck out only 14.1 percent and unsurprisingly had a team BABIP that was 24 points higher than the 2010 Giants.

Putting this all together, I found that the variance in team defensive efficiency can be attributed roughly as follows:

A) 48 percent team defensive skill
B) 40 percent luck
C) 12 percent pitching skill

With about 4,350 balls in play per team per year, you get rid of most of the luck, so this number shrinks to just 40 percent, and of course, team defense still explains BABIP better than anything else does. However, a very large part (12 percent) of keeping a batted ball from resulting in a hit is pitching. (Put in a mathematically equivalent but different way, there is a .37 correlation between a team’s Expected BABIP based on its pitching peripherals and its actual BABIP.)

To study this more objectively, I redefined “BABIP” to include errors, and ran a regression on all individual pitchers in the majors in 2002-2011 with 80 balls in play or more, weighted by balls in play, and using net ground ball rate ((GB-FB)/PA), strikeout rate, walk rate, all of their squares and interactions, dummy variables for season, and pitcher starter/relief role.

Then I simply applied this to each individual’s pitching statistics, and came up with an expected number of batters reached per ball in play with neutral defense and luck. Then I used that to develop an expected "BABIP" (with errors) for each team.

The lowest expected team BABIP (relative to the rest of their league) belonged to the 2002 Twins, with just a .299 expected rate of reaching on balls in play, below the league average of .307. The actual Twins allowed a .297 BABIP, which means that they were good defensively and also good at pitching, resulting in particularly few hits.

The highest expected team BABIP (relative to the rest of the league) belonged to the 2007 Blue Jays, who had a .321 expected BABIP, as compared with a .316 league average that year. The actual 2007 Jays’ BABIP was a very low .297. Their defense was actually fantastic, and their pitching made it harder and cost them the league best BABIP. Relative to their expected BABIP, their 19-point lower actual BABIP was the best in the league, but they finished millimeters behind the Red Sox. However, the Red Sox had pitchers with more strikeouts and lower ground ball rates, and their defense had a much easier battle to make outs.

Overall, there is pretty high year-to-year correlation in a team’s expected BABIP, .47, which is not so shocking since teams generally do not turn over most of their pitching staff in an offseason. This highlights the fact that one cannot look at aggregate numbers over a longer period of time to determine how teams play defense, hoping other factors will wash out; a defense can look bad for several years, when the pitchers should actually shoulder the blame.

Below I list teams by their 2011 ranking in “adjusted BABIP.” This is done by taking their actual BABIP (again, including errors as hits), and adjusting it for their expected BABIP based on their pitchers relative to the league BABIP. I also include the team’s ranking by actual BABIP surrendered, for comparison.

Of particular note is the Giants, who would have been 10th overall in BABIP, thanks to a somewhat wild pitching staff that was rather groundball prone, but still managed to make a lot of outs. Relative to the high BABIP that would have been expected given their pitching staff, the Giants actually appeared to have the fifth best defense at recording outs per ball in play.

Hurlers like Tim Lincecum, Matt CainJonathan Sanchez simply do not allow hitters to get good wood on the ball, and as a result, the defenders behind them look strong behind them when batters do make contact. On the other hand, the Diamondbacks were ranked above the Giants, at seventh, using BABIP alone, but their high-flyball stuff actually requires an adjustment to bump them down to 10th. (Again, recall that BABIP here includes ROE as hits.)

image

For all of the rankings for 2002 through 2011, see this Google Doc.

There are a number of interesting examples of teams whose defensive efficiency can be reinterpreted based on their pitching stats. The following table gives my favorite examples of teams re-interpreted using this method, some of which I describe below.

image

image

The 2010 Giants were actually on the other end of the spectrum than the 2011 Giants. They had a similar high strikeout rate and walk rate, but their groundball rate was much lower, making their expected number of outs much higher, since fly balls are easier to catch.

This was partly due to Matt Cain’s groundball rate going up from 36.2 to 41.7 percent. It was also due to replacing Barry Zito’s 33 starts with a 36.1 percent groundball rate in 2010, with just nine Barry Zito starts at a 39.8 percent groundball rate in 2011, and 45.6 percent ground balls in Ryan Vogelsong’s 28 starts in 2011. They also got 15 more starts out of Madison Bumgarner, whose groundball rate was 45.1 percent in 2010 and 46.0 percent in 2011, instead of Todd Wellemeyer’s 33.5 percent groundball rate in 11 starts as they received in 2010.

In both seasons, the Giants had fantastic strikeout rates that we know correlate with less hittable pitches, and more catchable balls in play, but the groundball rate was very different in 2010 and 2011.

The 2003 Mariners were an interesting story of run prevention. A large part of their league-leading defensive efficiency was fantastic defense. They had an outfield of Ichiro Suzuki in eight (21.1 UZR), Mike Cameron in center (19.6 UZR), and Randy Winn in left (4.3 UZR), combined with an infield that featured John Olerud at first base (11.0 UZR), Bret Boone at second (10.4 UZR).

But they also had an excellent flyball staff that kept the ball catchable in the first place. Jamie Moyer had 215 innings pitched with only a 38.3 percent groundball rate, Freddy Garcia had a 41 percent groundball rate in 201.1 innings, Gil Meche had a 36.8 percent groundball rate in 186.1 innings, and Ryan Franklin had a 34.3 percent groundball rate in 212 innings. The only starter who was not particularly flyball prone was Joel Pineiro, who had only a 45.4 percent groundball rate himself.

None of these starters were particularly good at missing bats, but their extreme flyball tendencies made up the difference. When combined with their fantastic defense, the 2003 Mariners were fantastic at making outs.

The 2007 Rangers relied on their 46.5 percent groundball rate to keep opponents from scoring, which has the side effect of permitting a lot of singles. On the down side, they struck out only 15.3 percent of hitters faced. As a result, they were 22nd in the league in preventing hits on balls in play.

However, they would have been 17th if they had an average staff in terms of BABIP skill. Pitchers like Kameron Loe, Kevin Millwood and Vicente Padilla contributed to the high groundball numbers without striking enough hitters out to shorten swings and reduce BABIP.

The Nationals trailed the league at striking hitters out in 2009, whiffing only 14.3 percent of hitters. Unsurprisingly, the Nationals were 24th in defensive efficiency in 2009, but they would have been right near the middle at 19th if you adjust for their staff. John Lannan, Craig Stammen and Shairon Martis are hittable in all the ways you would expect—they do not strike hitters out and hitters make better contact with the ball as well.

The Indians took away the dubious crown for worst strikeout staff in the league in 2010 from the Nationals, and they allowed a lot of hits too. Their defensive efficiency was .316, definitely below average, but their pitching numbers suggest that it should have been .310 anyway, reapportioning most of the blame from the defense to the pitchers.

Disentangling credit between pitching and defense appeared to take a great step forward with McCracken’s discovery about pitcher BABIP control (or lack thereof), and this is assuredly one of the most important findings of sabermetrics. However, as analysts collectively step back from the extreme position that a pitcher should never be blamed or credited for his BABIP, we should also reinterpret team defensive rankings as well. A full 12 percent of variance in team defensive efficiency is directly attributable to pitching. As we always knew, there are many factors in play once the ball hits the bat.

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2011-12-15T09:18:15+00:00 You shall know our velocity http://www.hardballtimes.com/main/article/you-shall-know-our-velocity/ http://www.hardballtimes.com/main/article/you-shall-know-our-velocity/#When:09:33:15 a book by Dave Eggers, but it could more aptly be named after an earlier work by Eggers entitled, "A Heartbreaking Work of Staggering Genius." The work of genius, however, was not my own, by derived a brilliant hypothesis put forth by John R. Mayne in 2010. Mayne emailed to alert me of this piece earlier this year, after my release of SIERA (Skill-Interactive ERA) at FanGraphs, and I recently tested it. Despite initial pessimism, I was shocked by what I found.

Everyone now knows how important velocity is for a pitcher. For years, pitching coaches scolded amateurs about over-reliance on velocity. Prepubescent pitchers are lectured about Greg Maddux, told that movement and location are more important than a couple digits on a radar gun.

I’m no PITCHf/x expert, but everything I’ve read by those capable of studying that data says that velocity is actually very important, perhaps more important than movement and location after all. It’s hard to throw a 100 mph fastball that is easy to hit, and you have to be Jamie Moyer to get away with an 80 mph, lukewarm heater. Even a few ticks in the ones column of a radar gun can make world of difference.

However, until very recently, I believed that a proper study of a pitcher’s peripherals could tell you which of two guys with a 92 mph fastball has the superior arm, and I also believed that two pitchers with the same SIERAs with different fastball speeds were no different in future skill level.

When discussing SIERA’s ability to adjust for pitcher control of BABIP, Dave Cameron once noted that velocity may explain some of the missing pieces of the puzzle that correlated with both strikeout and BABIP skills. However, I found that if you control for peripherals, age, year, and role, then knowing a pitcher’s velocity is not useful.

In fact, running a regression on all of these, you will actually get an insignificant and positive coefficient of .00035 on velocity; in other words, a 3.0-mph increase in velocity with the same characteristics will correspond with a BABIP that is a full point higher!

When Mayne emailed me with this suggestion, I expressed my skepticism, but I was thinking about the idea the wrong way. Mayne was talking about projections in that article—predicting the future. What I now found was that knowing a pitcher’s velocity tells you about his potential to improve the statistics that express skill level better.

If you just run a regression of a pitcher’s ERA next season on his ERA from the current season, you get the following equation:

ERA_next = 2.76 + .368*ERA

Include velocity, and you get:

ERA_next = 9.49 + .327*ERA - .073*velocity

This formula says that, of two pitchers with the same ERA last season, the one who threw faster is more likely to improve. That’s not surprising. We know that a pitcher was probably more capable if he threw faster, so he probably had better peripherals and worse luck if he had the same ERA and more velocity. Right?

Actually, let’s take a closer look at the pitcher’s true skill level and replace his ERA with his SIERA to see what happens. If you run a regression of a pitcher’s ERA next season on SIERA from the current season, you will get the following equation:

ERA_next = 1.21+ .733*SIERA

However, if you run a regression of ERA next season on SIERA and velocity, you get the following result:

ERA_next = 4.52 + .677*SIERA - .034*velocity

Both coefficients are statistically significant at the 99.9 percent level. In words, this means that a 2.9-mph increase in velocity will correspond with a 0.10 lower ERA, even if you know the pitcher’s SIERA from the previous season.

What’s going on here? Well, the pitchers who throw faster are doing something better than others with the same peripherals. What is that? I looked at various components of pitcher performance to find the answer and found why Mayne’s hypothesis was accurate.

Suppose you know a pitcher’s strikeout rate. In this case, you can predict his future strikeout rate next year very well:

K%_next = 3.87 + .764*K%

However, once you know that pitcher’s velocity, you have a lot more information.

K%_next = -16.1 + .701*K% + .233*velocity

Verbally, this mean that if you have two pitchers with the same strikeout rate the previous year, the pitcher who throws 4.3 mph faster will strike out one percent more batters the following year than the pitcher who throws slower.

What about walks?

BB%_next = 2.864 + .644*BB%
BB%_next = 0.237 + .638*BB% + .296*velocity

In the case of walks, more velocity actually portends an increase in free passes.

However, if you start to include more terms, its significance disappears. Higher velocity is just correlated with other variables that are related to increases in walk rates, such as relief role, age, and strikeout rate itself!

Including strikeout rate in the regression on next year’s walks renders the velocity coefficient insignificant (p = .224), while it remains very significant (p = .000) in the regression on next year’s strikeouts:

BB%_next = 1.04 + .0151*K% + .6361*BB% + .0179*velocity
K%_next = -15.9 + .6984*K% + .0752*BB% + .2254*velocity

Controlling for both rates, more speed foreshadows an improvement in strikeout rate. Including a slew of other variables (results omitted for brevity) did not alter this conclusion.

If you look at BABIP, you start to see more of an effect of a good fastball. If you try to predict BABIP next season using only this season’s BABIP, and then try to do so with BABIP and velocity, you can create a clearer picture:

BABIP_next = .238 + .191*BABIP
BABIP_next = .283 + .191*BABIP - .00050*velocity

Velocity helps predict next season’s BABIP pretty well, though this effect is somewhat minimized when considering the effect on other variables.

The rate of home runs per fly ball is another metric that is mostly determined by luck but incorporates some skill as well. Velocity actually corresponds well with a decreased rate of home runs per fly ball, even in the same season.

Running a regression of home runs per fly ball while incorporating peripherals with interactions, season, year, age, and role, we will still get a coefficient of -.00066 on velocity. This means that a pitcher who gives up 3.0-mph in velocity will yield one fewer home run every 500 fly balls. It’s not a big deal, but it’s statistically significant.

It also matters because the coefficient only goes down to -.00063 when changing the dependent variable to next year’s home run-per-fly ball rate. The skill is something that shines through over time, revealing an ability to get hitters out that gets behind the luck mashed in with other statistics.

However, if we simply check how much velocity adds to HR/FB itself in predicting next year’s HR/FB rate, we can see that:

(HR/FB%)_next = 8.39 + .186*(HR/FB)
(HR/FB%)_next = 20.17 + .173*(HR/FB) - .129*velocity

Knowing velocity is important for this as well.

Velocity is an even bigger deal than we thought, and Mayne hit the nail on the head. Not only do pitchers who throw faster succeed more often, but they improve more as well. It foretells a higher strikeout rate, lower BABIP, fewer home runs per fly ball, and a subsequently lower ERA than other pitchers with similar yearly statistics.

Incorporating velocity into projection systems would appear to be not only a useful tool, but perhaps a pivotal one in better understanding the importance getting the ball to the batter sooner has on getting him out.

Click here to learn about THT's download subscriptions.]]> Matt Swartz 2011-12-09T09:33:15+00:00