So, first a disclaimer: I know very little about actual baseball.

I do, however, love games, numbers, strategy, and game theory. So when Chart Party (a recurring feature on the sports YouTube channel SB Nation hosted by Jon Bois) ran the numbers on what would happen if Barry Bonds, one of the greatest baseball players of all time, played without a bat, I was intrigued. The following is my response to the question posed at the end of the video. I suggest watching it first before continuing.

To answer the first question: yes, I agree Bois’ methodology is correct, and the result is a little bit puzzling. How can the performance of a great batter not be affected by the removal of his bat? To answer that, we will need to abstract a little bit away from baseball as a holistic game and just talk about the interaction between the batter and the pitcher.

In baseball, the pitch qualifies as a sub-game, described in the following payout matrix:

Swing | No swing | |

Strike | ???? | -1/3 |

Ball | -1/3 | 1/4 |

There are two players, the batter and the pitcher, and each has two actions that they could perform. The pitcher acts first and attempts to throw either a ball or a strike; the batter must react to this decision. If a ball is thrown and the batter does not swing, then the batter scores a “ball”, and doing so four times results in a free walk to first base. If the pitcher throws a strike, the batter *must* swing, otherwise the batter scores a strike; three of these results in a strike-out. Swinging at a ball also results in a strike. The remaining situation involves the entire rest of the team and is hard to assign a value to so we will ignore it for the time being.

Now, in the Chart Party experiment, Bois modeled the pitcher as a random number generator. This may seem unfair, because common sense says that professional players shouldn’t be throwing balls randomly; however, this is actually a good way of modeling high-level play. A professional pitcher pitching to a low-level player, like me, would quickly adapt to my inability to hit the ball and strike me out every time. Likewise, a professional batter would just as quickly adapt to my inability to throw a ball and would either launch one out of the park or take the free walk. However, things change when two professional players play each other. In this case, both players would notice and adapt to any pattern exhibited by the other; therefore, the best strategy is to not exhibit any patterns at all, which is the definition of random.

(Note: since the batter reacts to the pitcher, the batter doesn’t need to swing randomly, only avoid indicating to the pitcher what the batter plans on swinging at.)

Finally, we are only interested in ‘On Base Percentage’ (OBP) or the amount of times the batter leaves home base successfully. Runs don’t matter and for this simulation getting to first base is just as good as hitting a home run. So, with all these assumptions and simplifications in place, the pitching sub-game becomes extremely easy to model mathematically. If the batter never swings, we are left with a simple binomial distribution: six pitches at a probability of throwing a ball at 58.7% (as reported in the video). We are interested in the probability of a game resulting in at least 4 balls. So, if we take a quick calculator break….

we arrive at a probability of 51.7%. In terms of baseball, that would be an OBP of 0.517.

Now, this number doesn’t include intentional walks and hit by pitches, which the video sadly lumps together with normal walks, so I cannot accurately calculate their effect. However, the video does report Bonds’ total walk rate to be 0.381 and if even a quarter of those are intentional walks (not a difficult assumption, given the background in the video) that would easily push his OBP to the reported value of 0.608.

The above graph demonstrates the relationship between a random pitcher, their probability of throwing a ball, and the probability of being walked, assuming the batter doesn’t swing. In a meta where pitchers threw balls less than 30% of the time, the effect on the batter’s OBP is minimal. Not swinging would result in getting to base only slightly more than 5% of the time. However, the numbers start changing quickly in metas with higher probabilities. A single 10% jump from 30% to 40% triples the expected OBP of the batter, while each of the next two 10% jumps both double it again. Suffice it to say, tiny changes in the pitching meta can have a massive impact on the expected OBP of the batter.

So where does the batter’s skill come into this?

Allowing the batter to start swinging would likely have a negative effect on their OBP. The simple fact that a batter can swing at balls will always negatively impact their score. Obviously, the better the batter is, the less this effect will be, but unless they play perfectly, inclusion of this option will always drop their OBP by at least a small amount. As noted, swinging at strikes has several possible outcomes. The batter can swing and get a strike, swing and not make it to base, or swing and make it to base. So the outcome of the rest of the entire game will have a variable effect depending on the skill of the team.

For the case of Bonds, we know the outcome. Swinging a bat had no effect on his overall OBP, which means he successfully swung at enough strikes to counteract the negative effect of swinging at balls. Seeing as he is one of the greatest batters of all time, I would assume that this is the upper limit of batting performance and lesser batters would perform a lot worse. In terms of this experiment, I hypothesize–although cannot prove– that for most regular batters, taking away their bat would actually improve their OBP.

Is this bad for baseball? No. In the real world, if Bonds showed up without a bat the pitcher would adapt quickly and strike him out; it’s a dumb strategy. What this is though is a good indicator that OBP is a terrible statistic, and likely shouldn’t be used as a proxy for a batter’s skill.