Problem Solving in Fake Baseball

Today’s article is about a form of baseball that doesn’t exist in real life. Let that be a disclaimer upfront — if you’re looking for an interesting middle reliever or 10 reasons why Christian Yelich is great against fastballs (number six will shock you!), this isn’t the article for you. That’s not to say those don’t exist — I wrote about Miguel Castro literally yesterday, for example. But today’s piece simply won’t be one of them.

With that out of the way, let’s set the stage. I’ve been playing a lot of Out Of The Park Baseball 21 lately. Some of that is for an article series, but some of it is because I want baseball in my life, and the game absolutely delivers. One mode in particular has been a great avenue for the analytical rabbit holes I love diving down: tournaments.

Without going into the specifics of the way the game is built, I’ll give a rough outline of the problem that first interested me. The standard tournament format is a 32-team bracket that plays best-of-seven series in each round. There’s no reseeding, no fancy gadgets; just 16 seven-game series, followed by eight, then four, then two, and then the finals.

In almost all other ways, the game approximates baseball. You field a 26-man roster, players need rest, and starters and relievers have realistic stamina. There are all kinds of fun roster choices to make — mid-career Tony Gwynn or young Pete Rose in right field, peak Ichiro or 1997 Bernie Williams in center, to name two — but for the most part, it’s a faithful simulation of baseball.

There is, however, one very specific wrinkle. In these tournaments, there are no off days. The first set of series takes place over seven consecutive days. As soon as the last series is decided, the next round begins the following day. You could, in theory, win the first round in seven games, win the second round in seven games, and end up playing on at least 18 straight days even if you sweep or get swept in the third round.

With no travel days, this ends up looking quite unlike the real-life playoffs. Position players need off days after eight or nine consecutive games, so backups get more starts when a series goes the distance. The game amps up catcher fatigue artificially even beyond that — your catcher will be taking every fourth day off, at best.

Most meaningfully, however, fifth starters are necessary. Unless you sweep every round, you’re guaranteed to play five games in five days at some point. Unless you’ve decided to run some strange all-bullpen roster, pitchers can’t consistently start on three days’ rest. A fifth starter is a must.

On the other hand, a fifth starter is clearly less important than a number two starter. The number two starter will often pitch twice in the first series, whereas the fifth starter will pitch no more than once, and sometimes not at all. What’s more, even if the first series goes seven games, your team’s fifth starter won’t appear twice in the second round — even if that round goes the distance, the 14th game would, naturally, be started by the fourth starter. The fifth starter would get the 15th start, at the start of the third round.

Win a series in less than seven games, and things get even better for you. Take, for instance, this example, a two six-game series followed by a seven-game series:

Starters in Three Hypothetical Series
Game Round 1 Round 2 Round 3
1 1 2 3
2 2 3 1
3 3 4 4
4 4 1 2
5 5 5 5
6 1 2 3
7 1

Over 19 games, your ace goes five times. The two and three starters each get four turns, while fourth and fifth starters get only three tries. A quick note: in this, and all subsequent examples, we’re assuming that every day, you simply start your best rested pitcher. It’s win or go home: why do anything else? That’s only one example, of course, but the overarching point remains: even if every starter had the exact same talent level, their contribution to the team depends on their spot in the rotation.

Oh yeah, one more wrinkle: teams operate under a sort of salary cap. You can’t just splurge at every spot, at least in the mode I play; every point you spend on a fifth starter needs to come from somewhere else on the roster. The question then becomes: how hard should you punt that roster spot?

At first, this seems like a pretty easy question to answer. How many starts does each pitcher get, on average, in the first round? If we assume a 50% winning percentage for our team, it looks like this:

Average Starts in the First Round
Series Length Probability 1 Starts 2 Starts 3 Starts 4 Starts 5 Starts
4 12.5% 1 1 1 1 0
5 25.0% 1 1 1 1 1
6 31.25% 2 1 1 1 1
7 31.25% 2 2 1 1 1
Sum 100.0% 1.625 1.3125 1 1 0.875

From there, we can simply repeat that process based on our rest status going into the second round. For example, 37.5% of the time that we make the second round, we do so with no starter making more than one turn in the first round. In that case, the rotation sets up from the top: the first game of round two goes to our ace, just like round one.

Another 31.25% of the time, our rotation sets up in 2/3/4/1/5 order — if the first series ends in six games, the first starter made two turns and won’t be ready to pitch again until Game 4 of the second round. In the final 31.25% of our second round matchups, the first round went a full seven games. That means that our rotation sets up as 3/4/5/1/2 in the second round — there were no days off that let anyone slide up in the pattern.

Aggregating all of those possibilities, as well as the expected series lengths in that round and the odds of winning the first round (50%, naturally), gives us this set of expected starts through two rounds of play:

Average Starts Over Two Rounds
Pitcher Number 1 2 3 4 5
Starts 2.24 1.95 1.65 1.55 1.33

You can think of those numbers as the expected number of starts each player would take in a two-round tournament with no scheduled off days. Sometimes you lose in five games in the first round and every starter gets only one turn. Sometimes your ace gets three starts. It’s all determined by probability, if not quite random chance.

Rather than continue on in this fashion, however, I’m going to stop there and show a new method. Why? Iterating in this way through all five rounds is certainly possible, but it’s missing a key point. Every game isn’t 50%. That’s the whole point of playing a bad fifth starter! We’ll need to break out individual game win probabilities to actually catch what’s going on.

Here, I’m going to create a simplified model. This isn’t quite how the salary cap works, but let’s assume that you can accept a 1% decrease in winning percentage out of one starter to boost another one by the same 1%. Let’s further cap it at 55%; the game will let you play as bad of a pitcher as you’d like, but most tournaments have a maximum talent level so that you aren’t facing off against an endless loop of peak Greg Maddux, 1999 Pedro, and 1968 Bob Gibson, or whatever the best pitching rotation imaginable would be.

With those rules in place, it’s time to go to Monte Carlo. More specifically, it’s time to use a Monte Carlo simulation to game out what might happen, over and over again, and give us an idea of what will happen, on average, over a long time horizon.

To build one of these scenarios, all we need to do is specify the odds of each potential outcome. Then, we put it into the magic random number generator of Python, spin the wheel, and see what happens. You can see my attempt at doing so here. Please note that I’m reasonably confident in the results, but not in the cleanliness of the code; I know enough to get reasonable outputs but not to optimize the length of time it takes to get those outputs.

First, let’s run the simulation a million times with each starter having a 50% win percentage. We win the tournament, in my simulation, 3.124% of the time; in the limit, it would be 3.125% of the time, but even a million trials leaves room for some small random variance. The pattern of the ace pitching the most and fifth starter pitching the least is clear:

Average Starts, Each Starter 50%
Pitcher Number 1 2 3 4 5
Starts 2.81 2.49 2.17 2.04 1.76
Percentage 24.9% 22.1% 19.3% 18.1% 15.6%

Next, let’s move our ace to a 55% win rate and our fifth starter to a 45% win rate. When we run the simulation again, we now win 3.45% of the time, significantly more often, and every starter gets to start more often:

Average Starts, Take Two
Pitcher Number 1 2 3 4 5
Win Percentage 55% 50% 50% 50% 45%
Starts 2.86 2.55 2.19 2.09 1.79
Percentage 24.9% 22.2% 19.1% 18.2% 15.6%

We can do better. Let’s punt the fifth starter even more: a 40% win probability in their games, and a 55% win probability for the first two starters. Now we’re up to a 3.8% win rate, and more starts, again, for everyone:

Average Starts, Take Three
Pitcher Number 1 2 3 4 5
Win Percentage 55% 55% 50% 50% 40%
Starts 2.91 2.59 2.28 2.14 1.83
Percentage 24.8% 22.0% 19.4% 18.2% 15.6%

Let’s crush that fifth starter one more time. He’s now a 35% true talent pitcher, really the pits. Our top three starters are now 55% true talent. We’ve done it! Our 3.99% tournament win rate is our best yet, and yet again, every single starter is getting more games, though our rate of gain has gone down:

Average Starts, Take Four
Pitcher Number 1 2 3 4 5
Win Percentage 55% 55% 55% 50% 35%
Starts 2.93 2.61 2.30 2.16 1.84
Percentage 24.7% 22.0% 19.4% 18.2% 15.5%

We can even try totally tanking our fifth starter, though a 30% win rate for a pitcher beggars belief. Now all four of our main starters are nails (55%) and our fifth starter is a chump. Tournament win percentage is all the way up to 4.12%, and yet again, the starts marginally tick up:

Average Starts, Take Five
Pitcher Number 1 2 3 4 5
Win Percentage 55% 55% 55% 55% 30%
Starts 2.95 2.63 2.31 2.17 1.85
Percentage 24.8% 22.1% 19.4% 18.2% 15.5%

There are plenty of other directions we could go from here. So far, we’ve been saving money on our fifth starter and putting it into other starters. We could put it into the offense instead, though we’d need a way to stylize that in our simulation. We could allow winning percentages higher than 55% and start tanking our fourth starter as well, though in practice the game isn’t very hospitable to that strategy due to talent caps. We can goof with the mix between relievers and offense, hoping to squeeze a few more drops of optimality out. Every little bit adds up over time.

Honestly, there’s even more than that. In all of these examples, we’re assuming a static punching bag of an opponent. They just sit there and let our aces crush them. They could punt their fifth starters too! They could try any number of other strategies; juice their own offense, or bullpen, or experiment with piggybacks of their best reliever with the fifth starter.

Not only that, but the rules of the game aren’t as clean as my approximation here. You can’t just specify how many wins you’d like to move from one spot to another. There are actual players in those spots, and their idiosyncrasies matter. Platoon matchups matter. There may not be a real player with a 30% win percentage, or four pitchers with a 55% win percentage. I didn’t even consider the fact that sometimes every series finishes before the seventh game, and there are fewer rest days than you planned for, which can throw a wrench in the works.

The truth is, this is the kind of puzzle you can pour an infinite amount of time into, were you so inclined. If your brain is wired like mine, you might even enjoy putting that time in. You might discover a new cult hero, like Nate Cornejo, who is the closest the actual game gets to our mythical awful fifth starter. You might spend hours tinkering with fake lineups hoping for another 1% of sweet, sweet game win percentage — hypothetically speaking, of course.

When real baseball returns, there will be occasions to use this kind of analysis in real games. The ability to take a situation and abstract it into a form that you can simulate is just as useful when the situation you’re abstracting is real baseball, and it’s a heck of a lot more satisfying. But if I can’t have real baseball right now, I’m glad that I can perform the same mental gymnastics that make me so happy on a baseball-like substance. It might not be real life, but it’s the next best thing.

Ben is a writer at FanGraphs. He can be found on Twitter @_Ben_Clemens.

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