Game theory is the next Moneyballby Dave Studeman
March 14, 2013
I had a great time at the second annual SABR Analytics Conference last week. The SABR folks put together another fine program, with several excellent speeches, panels, analytic presentations, and sessions set aside to discuss the business of baseball (Ben Lindbergh has some thoughts about the latter at Baseball Prospectus). I didn't attend everything, but I want to mention some of the things I did.
The Diamond Dollars case competition For the second year, Vince Gennaro created a baseball business case as a teaching opportunity for students interested in the business of baseball. Students at 11 schools read the case, analyzed their options and traveled to Phoenix to present their recommendations to a panel of judges. I was one of those judges and, as always, I learned something in the process.
This particular case was set one year from now, in the Angels' front office. The Angels are considering whether to sign Mike Trout to a long-term contract or to let him run through his arbitration years, and the students have been asked to present their recommendations to General Manager Jerry DiPoto. (Footnote: I look nothing like Jerry DiPoto, though I've been told that I sound like Jeff Luhnow and look like Bill James). We're talking bullet points, graphs, financial numbers, baseball statistics, Powerpoint and the whole nine yards.
I'm always particularly intrigued by the ways different students account for the risk of long-term contracts but, truth be told, there's not a lot of risk to signing Mike Trout to a seven-year deal (which is what most teams recommended). The primary issue was the amount of "arbitration discount" the students would be willing to concede for a long-term deal. Congratulations to NYU and Pepperdine for winning the undergraduate and graduate divisions, respectively.
The Rawlings announcement There was a bit of cloak and dagger suspense concerning a "special announcement" on Friday's agenda. SABR and Rawlings used the time to announce that they will be changing the voting rules of the Gold Glove Awards. Moving forward, Gold Glove winners won't be selected based solely on voting by major league managers and coaches. Rawlings also will include a statistic in determining the winners (though voting will still account for the majority impact) to be called SDI (Saber Defensive Index).
A panel of folks (anonymous at this point) will come together to construct the SDI. I assumed that this will be an open-source stat, but the press release indicates otherwise. If the stat is open-source, this is a very good thing (and kudos to SABR for making it happen). If the stat isn't open-source, then it's a bad thing. There shouldn't be any closed doors when it comes to handing out awards. Let everyone know how the players are being judged, and be willing to take your lumps from those who disagree. That's how we improve.
The presentations I wanted to go to all the presentations, but many were scheduled against each other. I did see Graham Goldbeck present some really interesting data regarding how deep in the strike zone batters tend to hit the ball. It turns out that the optimal place to hit a ball for a home run is about a foot in front of the plate and that a couple of batters, such as Alfonso Soriano and Alexei Ramirez, typically hit the ball nearly two feet in front of the plate.
It also turns out that pitchers can be measured by how deep in the zone batters make contact (hint: in general, the faster the fastball, the deeper in the zone the ball is hit), though there are some notable exceptions—enough exceptions to make us want more data. Alas, this was HITf/x data (I think. Maybe it was the FIELDf/x data?) and it will be hard to come by.
The most important thing
What really opened my eyes, however, were the two Friday presentations about baseball and Game Theory. Game theory is a branch of economics in which economists and other mathy types study how competitors should compete against each other, given certain expectations and parameters. It's complicated stuff, but the payoff can be sublime.
The penalty kick in soccer is an excellent example of game theory in sports. Let's say you are a right-footed kicker and, given the way the ball comes off the foot, you typically kick the ball better when you kick it to your left (off the inside of your right foot). So you tend to kick the ball in that direction. However, the goalie knows you're going to do this, so he (or she) anticipates your direction and dives to his (or her) right. However, you know the goalie is going to do this, so you kick to your right instead. And so on and so on.
Kind of like that scene from the Princess Bride, right? The one where the Dread Pirate Roberts (Wesley to you and me) puts poison in one of two cups and Vizzini tries to choose the un-poisoned one by "out-thinking" him? Like Vizzini, you may think there is no way out of this endless cycle of potential soccer kicks; that the answer is to just randomly pick one side or the other (or to switch goblets when the other guy isn't looking).
You'd be half right. You do want your actions to be random and unpredictable. But you don't want to choose your options 50/50, because you are more likely to score when kicking to your left. You know it and the goalie knows it. So how often should you randomly kick to your left? And how often should the goalie randomly dive in that direction?
The answer, given typical success rates in professional soccer leagues, is 61 percent of the time. And the goalie should anticipate your kicking in that direction 58 percent of the time. Don't believe me? Here's the mathematical proof. In game theory terms, this is called a "mixed strategy," in which you pursue different strategies but at optimal rates.
The truly fascinating thing is that these percentages will lead to the best outcome for both sides. Neither the kicker nor the goalie will be able to improve their success by varying from these percentages over the long term. Game theorists say that this is the point at which both sides are "indifferent" to the other's strategy. The extra cool thing is that professional soccer players actually fall in line with these percentages. Reality mirrors theory.
Naturally, the percentages will vary by player and goalie, depending on their relative strengths and weaknesses. Most kickers have different underlying strengths and weaknesses, as do most goalies. Good players will adjust their percentages according to the nature of the opposition. Which brings us to pitchers and batters.
In one presentation, Middlebury sophomore Kevin Tenenbaum (subbing for Dave Allen, who is now a professor at Middlebury after several years of publishing PITCHf/x analysis) applied game theory to pitchers and batters and used complicated mathematical models to determine where pitchers should locate pitches in 0-2 counts. I thought the presentation was excellent and the fundamental conclusions made sense, but the math was beyond me. I'm not going to try to explain it here.
There is an easier way to become comfortable with game theory in baseball. Last December at THT, Matt Swartz published a five-part series on baseball and game theory. I considered them the most important sabermetric articles of 2012, though I admit that I'm biased. In the series, Matt laid out an entirely new way of thinking about what pitchers should throw on specific counts.
The basics of game theory
Applying basic game theory to pitch selection and finding that batters should take more often on 2-2 counts than 3-2 counts
Taking it a step further and showing that pitchers should throw their best pitch less often on 2-2 counts
Adding Bayes probability to factor in the batter's decision process and calling him Willie Bayes
Relaxing some of the assumptions and pointing to future research
At the SABR Analytic Conference, Matt picked right up with that future research thing. He presented data from 2-2 and counts and 3-2 counts and found that batters actually do swing more often at 3-2 pitches than 2-2 pitches (on pitches both in and out of the strike zone). He also expanded his analysis and found that pitchers and batters follow predicted behavior (throwing strikes and swinging at pitches) across all counts. Reality reflects theory.
Then Matt really dove into the data and found that baseball players aren't always maximizing their opportunities. He assigned run values to specific pitches and ball/strike situations by pitch type and compared the run value of fastballs across all counts to the run value of non-fastballs across all counts. According to game theory, the relative value of these two type of pitches should negate each other. Added together, they should equal zero.
Doesn't happen. It's the key table from Matt's presentation, showing the relative difference in fastballs and non-fastballs by count:
Most importantly, Matt found that fastballs are underused with no strikes on the batter and overused with two strikes on the batter. On the other hand, he found that batters swing too often with two strikes on them. Reality is no longer reflecting theory.
Matt took it even further by examining different types of pitcher specialists—those with the most effective change-ups, sinkers, sliders and curveballs. I'm not going to review all of his findings because they were basically consistent with the above table. Plus, you can download Matt's Powerpoint slides from the References section below. The bottom line, however, is that all types of pitchers follow the same type of behavior.
There's gold in this information. I know I'm purposely overselling things when I say that game theory is the new Moneyball, but given all of the resources and data available to major league teams, how can they ignore this extra level of analysis that can directly yield results? Many teams have three or four (sometimes more) data analysts on staff. How about hiring someone with game theory experience as well?
During one of the panel sessions—the one with Brian Kenny, Bill James and Joe Posnanski—the panel talked about the trend toward breaking up a manager's job into different roles. One of those roles was a bench/game strategy coach; someone to worry about the next steps in the game, get the bullpen ready, select the pinch hitters, decide when to bunt (if ever!), etc. It's a good idea, and it screams for someone who can think in terms of game theory.
I don't know how it will play out, but smart major league teams are going to get on top of this. Count on it.
References and Resources
Here are Matt's Powerpoint slides. The link is to a Dropbox file, but you shouldn't need a Dropbox account to view the presentation or download it.
Dave was called a "national treasure" by Rob Neyer. Seriously. Comments about this article can be sent to him through the miracle of e-mail.
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