Tuesday, January 26, 2010

The Value of the Stolen Base: Two Economic Schools of Thought

Since the release of "Moneyball" in 2003, Sabermetrics has turned conventional baseball wisdom on its head.

Scouts' hunches, batting averages, and other long-cherished means of evaluating player talent have been replaced by complex statistical tools with names like VORP and OPS (and some that are much more obscure), many of which are founded on solid economic principles. The name of the game now is getting on base and hitting for power, while many traditional tactics and metrics have been criticized, including the stolen base.

The most illustrative critique of the stolen base I have seen comes from the Baseball Prospectus's "Baseball Between the Numbers: Why Everything You Know About the Game Is Wrong." A chapter compares the 1982 season of Rickey Henderson (in which he shattered the MLB record for stolen bases) and the 1986 season of Pete Incaviglia (a plodding outfielder). The book points out that while Rickey's steals helped his team, he got caught so often that he did a lot of damage, too:
The run-expectation tables from 1982 show that Henderson added an extra 22.2 runs to the A's offense with his 130 steals. But the 42 times he was caught cost the team 20.6 runs, meaning that for all that running, the A's gained a total of 1.6 runs for the season. In his first season, Incaviglia stole three bases and was caught twice. He cost his team about half a run. Because Henderson got caught so often, the difference between his base-stealing performance in 1982 and Incaviglia's in 1986 added up to about 2 runs.
Every base stolen slightly increases a team's chance of scoring in an inning, but every caught stealing drastically reduces it (for example, the team goes from a runner on first with one out to the bases empty with two outs). The raw number of bases that a player steals is irrelevant; if he steals successfully less than 75% of the time, he is costing his team runs, on average.

Professor Ben Polak, who taught a game theory course through Yale University's Open Yale program in 2007, has a decidedly different approach. The relevant discussion video is here (about 14:45 in) and the transcript is here (do a search for "baseball" to get to the right spot).

He analyzes base stealing in the context of mixed strategies. The theory behinds mixed strategies is rather involved, but it boils down to choosing one \action against an opponent some of the time and another action the rest of the time. For example, if a soccer player always kicked to the right on penalty shots, the goalie would always dive to the right and thus have a good chance of stopping many shots. So the player would score more points by mixing between kicking right sometimes and left others, even if he can kick right more skillfully.

In contrast, if kicking right always had a higher expected value, no matter what the goalie did, the player should always do it. This is known as a pure strategy.

So, back to base stealing, in which Polak talks to a member of the baseball team who is taking his class:

Professor Ben Polak: You're a pitcher, okay. So he's not going to get on base now, so he's not going to answer this. Suppose you did get on base, pitchers don't often get on base. Let's assume that happens, what might you randomize? There you are, you're standing on base, what might you randomize about?
Student: Whether to steal second or not.
Professor Ben Polak: Right, whether to steal or not, whether to try and steal or not. Stay up a second. So the decision whether to try and steal or not is likely to end up being random. If you're the pitcher, what can you do in response to that?
Student: You can either choose to try to pick them off or not.
Professor Ben Polak: What else? So one thing you can try and pick him off. What else?
Student: You can be quicker to the plate.
Professor Ben Polak: Quicker to the plate, what else?
Student: You can pitch out.
Professor Ben Polak: You can pitch out, what else? At least those three things, right?
Student: Yeah.
Professor Ben Polak: At least those three things okay, thank you. I have an expert here, I'm glad I had an expert. So in this case we can see there's randomization going on from the runner whether he attempts to steal the base or not, and by the pitcher on whether he throws the pitch out or whether he tries to throw, to get to the plate faster.
Because the other team is taking so many steps to prevent a player from stealing successfully, they are worse off in other dimensions. Polak provides the punchline (emphasis mine):
Student: The pitcher has to react differently in pitching when he knows that there's a fast guy on base.
Professor Ben Polak: Good, so our pitcher has to react differently. Let's talk to our pitcher again, so one thing our pitcher said was he wants to get to the plate faster. What does that mean getting to the plate faster? –Shout out so people can hear you.
Student: It means just getting the ball to the catcher as fast as possible so he has the best chance to throw out the runner.
Professor Ben Polak: Right, so you're going to pitch from, you're not going to do that funny windup thing, you're not, thank you, you're going to pitch from the stretch, I knew there was a term there somewhere. I'm learning American by being here. And you're more likely to throw a fast ball, there's some advantage in throwing a fast ball rather than a curve ball. Both actions of which, both having to move more towards fast balls and pitching to the stretch are actually costly for the pitcher.
From game theory's perspective, it's no surprise that base stealers don't add or subtract many runs from a team's total, but the costly actions it requires of the pitcher make a base-stealing threat very valuable. It helps players batting with the runner on base, who can expect more fastballs and pitchouts and thus have an advantage.