In the first entry of this series, I looked at some of the math behind the most popular tabletop simulation of real life baseball, Strat-O-Matic Baseball. Today, I’m going to look at a new and simpler tabletop game, Bottom of the 9th.
The premise of the game is pretty simple in baseball terms. The game is tied. The home team is up to bat. The inning should be evident from the name of the game. The talent gap is wide between the two teams, and so favors the visitors that it is considered a miracle the home team even has a chance to win. It is also presumed the visitors will win should the game go to extras.
The game plays out the inning pitch-by-pitch. Each pitch is a turn in the game, and is resolved with 4 or 5 steps. The heart of the game is in the Stare-Down, where the pitcher picks an area of the zone to throw to while the batter tries to guess where the pitch is going. This is simplified into 2 choices: a red disc for height of the pitch (High or Low) and a white disc for which half of the plate (Inside or Away). The batter gets some benefits for guessing right, while the pitcher gets some benefits when the batter guesses wrong.
After this, the pitcher makes The Pitch by rolling 2 dice, one which determines whether the pitch is outside the zone, inside the zone, or paints the corner, and a standard six-sided die for control, where higher numbers are better. This impacts the swing, where the batter rolls one standard six-sided die. The benefits granted to each player from the Stare-Down are applied here before comparing the results of the swing and control numbers.
The die rolls are the simple part to break down mathematically. The pitch dies shows a pitch outside the zone to occur 1/2 of the time, inside the zone 1/3 or the time, and on the corner 1/6 of the time. A ball is called when the Swing result is less than or equal to the Control result. The ball is put in play when the Swing result equals the Control result for a pitch on the corner or when the Swing result is greater than the Control result for a pitch in the zone. All other combinations result in strikes.Based on these dice alone, a ball in play occurs on 16.6% of pitches, a ball is occurs on 29.2% of pitches, and the strike occurs 54.2% of the time.
How does this compare to actual major league rates? It’s a little off. In the 9th inning of games in 2015 (via Baseball Savant), balls occurred on 34.8% of pitches and balls in play occurred on 17.4%of pitches. This is because of the simplified system used in Bottom of the 9th, which values speed of play over simulated accuracy.
Another example of how the game values speed is what happens on a Contact result. The first player to roll a 5 or 6 tips the result in their favor. This requirement can be modified by the player at bat or pitching, and it depends on speed of rolling a die as well as the players involved.
Clearly, this is not a real simulation. That’s not a bad thing. Bottom of the 9th was designed to appeal to board game fans as well as baseball fans. It represents two of the hotter trends in board gaming: it was funded via Kickstarter, and it’s considered a “micro-game”, a simple game that can be played in around 15-30 minutes. It’s a pretty good game, and definitely recreates the feel of the batter-pitcher confrontation. I picked up a copy from my local game store, but I now wish I had funded the Kickstarter.
I happen to be a man of many interests. Besides baseball, one of my other primary interests is gaming, especially tabletop gaming. My interest in games is rooted in my love of competition and my explorations of the world via mathematics and statistics. It’s no coincidence those are traits inherent to following baseball as well.
Today, I’m starting a series exploring various games that attempt to simulate baseball. My focus will be more of the math that underlies each game and how closely it helps replicate the on-field experience, though I’m sure some game play commentary will filter in. Leading off is perhaps the most well-known of the baseball table top simulations, Strat-O-Matic Baseball.
In the book Curve Ball, Jim Albert and Jay Bennett open the book with a dissection of how various baseball tabletop games model the actual action of a baseball game. Naturally, Strat-O-Matic Baseball was covered, in which they explain some of the math behind the model and how it assigned credit to the batter, pitcher, and defense. I want to focus more on the game design and the probabilities involved.
The basic mechanics of the game are relatively simple, though there are optional levels of complexity that can be added to the game now that were not a part of the original edition. There are batter cards and pitcher cards, and each card contains a table of possible results that are determined by the roll of 3 six-sided dice. One die, typically white, determines which card and which column the result comes from, with the result corresponding to the the sum of the 2 other dice, typically red, in the designated column. In many instances, a result then requires the roll of an additional 20-sided die. This provides 4,320 different possible outcomes.
Unfortunately, there is no master database of SOM player cards that is available to fully analyze this model. However, a massive Strat-O-Matic Baseball fan by the name of Bruce Bundy put together a bunch of formulas to forecast how a player’s card would be created. My impression is that he created these formulas by looking at a bunch of player card sheets. I’ll use it here because it’s the best publicly available information about the game model that I can find.
Looking at the formulas provides insights into a number of assumptions made about baseball by Strat-O-Matic. Player cards are customized based on their statistics, but this customization is achieved using some assumptions about the probabilities of certain events occurring that are built into the game model.
Consider the old fashioned base-on-balls, the least sexy of the Three True Outcomes. The Walk formulas for both Batters and Pitchers both are adjusted by a constant of 9. In terms of SOM, this means that the batter and pitcher cards are designed with the assumption that 9 out of the 108 results from the other card will result in a walk. Thus, the game implies an unintentional walk occurs about 8.3% of the time in baseball, with the credit being split between the pitcher and the batter. While the latter claim is not possible to investigate prior to pitch-by-pitch data being available, the former is. Here’s the overall major league non-Intentional walk rate year-by-year since 1952, using the event logs courtesy of Retrosheet
You see that for most seasons here, the actual MLB non-intentional walk rate (in red) is slightly less than the the estimated rate modeled by SOM (in blue). The average across these seasons is that non-IBB walks occur in 7.81% of the plate appearances, which is about 17/216. Since 17 is an odd number, it can’t be divided equally between the batter and pitcher, a key component of the Strat-O-Matic model. Thus, it seems that the walk rate implied by Bundy’s formulas is reasonable, though a bit high.
Here are the implied rates from Bundy’s formulas and their actual instance rates from the same Retrosheet data for a few other events:
- Doubles – SOM rate of 180/4320 = 4.2%, Actual = 4.1%
- Triples – SOM rate of 30/4320 = 0.7%, Actual = 0.6%
- HRs – SOM rate of 100/4320 = 2.3%, Actual = 2.3%
This replication of a generic baseball reality is why Strat-O-Matic has been so beloved for over 50 years. Hal Richman, the game’s inventor and mastermind, has created a game model that is flexible enough to work across eras. This enables SOM to sell new sets based on every season and specially designed sets, all of which can be mixed and matched as the gamer sees fit. If you’ve never played the game, find a way to do so at least once.