Of all the strategic elements of baseball, few are more fascinating than the poker game between pitcher and hitter. Each participant knows his strengths and those of his adversary, and that knowledge informs both players’ tactics in a complex entanglement of actions and counteractions.That's one thing about the knukleball. It is so unpredictable on its own that a good knuckleball pitcher should throw it every chance he gets. As opposed to Dickey's 88 to 90 mph fastball, I was bringing a 55 mph fastball, so after the first two innings I pitched, I threw one fastball the rest of the season (about 29 more innings). And that was thrown only because the catcher messed up the sign. Sure, a knuckleball pitcher may get pounded some days, but, generally, the more non-knuckleballs he throws, the harder he gets hit. Anyway, I only highlighted the article because of the Dickey shoutout.
If the best pitch in a hurler’s repertoire is his fastball, for instance, he might be inclined to use it really frequently. But batters will pick up on that proclivity, and in time, the fastball will lose its effectiveness if it’s not balanced against, say, a change-up — even if the fastball is a far better pitch on paper.
Eventually, we would expect this pitcher’s arsenal to settle into the optimal mix for retiring opposing hitters: a mix of fastballs and change-ups that’s impossible for a batter to exploit.1 In game theory terms (and assuming the batter adapts accordingly), this is a version of the famous Nash equilibrium, which describes a situation in which neither party in a game has anything to gain by changing his or her strategy.
That’s all, well, theory. But how can we detect which real-life pitchers are closest to their equilibria? One idea is to look for hurlers whose effectiveness is relatively equal on every kind of pitch he throws. And fortunately, Fangraphs tracks not only the frequency with which each pitch type is employed, but also its potency, estimated in terms of runs added or subtracted per 100 pitches. Using that data to find out how balanced a pitcher’s performance is across his entire repertoire, I computed a metric that I’m dubbing the “Nash Score.”....
Here’s how it works: Start by measuring for each pitch type the difference2 between its effectiveness and that of all the pitcher’s other pitches combined. Then weight those differences according to the frequency with which each pitch is thrown. The resulting average is the Nash Score, a sort of variance that measures whether a pitcher is close to his equilibrium (lower score) or could conceivably benefit from varying the distribution of his pitches (higher score).
Take R.A. Dickey as an example. The Blue Jays starter, known for his mesmerizing knuckleball, throws the pitch 87 percent of the time — about as much as any pitcher in baseball uses his No. 1 pitch. Yet Dickey’s Nash Score isn’t especially low, so under the concept of equilibria outlined above, he should be using the knuckler even more. Dickey’s fastball — his No. 2 (and essentially only other3 pitch) — is far less effective than his knuckler, even in its limited use as a complementary, change-of-pace pitch. According to game theory, Dickey could conceivably boost his overall effectiveness by throwing the knuckleball on an even greater proportion of his pitches.
Sunday, August 16, 2015
Game Theory and Knuckleballs
FiveThirtyEight does some pitcher analysis: