I’ve been kind of drawn in by the NLCS this year. I’m a long time Giant’s fan and seeing them actually winning games and having a good chance at going all the way is a bit of an uplifting experience. On top of this, baseball is a highly strategic game. The choice of pitcher in one game will limit the choice of pitcher in the next game. Offensive lineups must sometimes be changed to optimize a variety of scoring strategies. My team winning, combined with a chance to think about strategic behavior, is always a good way to get me interested.
So, all this got me to start thinking about the so called pitcher’s duel. My question is: Why open a series with two ace pitchers? This is a pretty common practice that two top pitchers go head to head, but is there a reason that coaches would want to do this? Couldn’t a team put a lousy pitcher up against a great pitcher and throw the first game in order to give themselves a better shot at later games by wearing out the top pitcher early? Of course, if one team knew that the other team was going to do this, nothing would stop them from switching their pitching strategy, and on and on like that.
The choice of pitchers is a simultaneous game. For my thought process, we’ll assume that pitchers are chosen in secret by each coach. I start with a simple model. There are two teams and they will play one game against each other. Each team can either play their ace pitcher or their second-tier pitcher. If an ace plays and ace, there’s a 50% chance that either will win. If a second plays a second, the same thing will happen. If an ace plays a second, the ace will have a 75% chance of winning the game.
I could draw the payoff diagram, but I’m a bit lazy so I’ll just spoil the punchline. In this single game, both teams have a strongly dominant strategy to play their ace pitcher. The game, however, gets quickly complicated if you consider multiple plays. First off, to avoid the possibility of ties, you’d have to play three games. You’d give each team a first, second, and third ranked pitcher, and assign probabilities based on each pitcher facing each other. The complication then comes in even harder because each opening match up leads to its own sub-game for the next two games.
While working out what would happen in this three game situation would be difficult, there are a few things that can be thought about. First, the probability of winning the three game series, given that you already lost the first game, is lowered even if you do have a pitching advantage in later games. If you think about a seven game series, where ace pitchers may re-enter in later games, then a dominant strategy for all seven games might not exist.
Given that it is almost impossible then to find a dominant seven game strategy, it is probably a useful rule of thumb to simply try to win every game as if that game stands on its own. Winning game one does have a dominant strategy (choose the ace pitcher). So, it’s entirely reasonable then that we often see the opening of any given series start off with a pitcher’s duel between two ace pitchers. A pitching strategy that gives the highest probability of winning each individual game as if they are played one at a time is likely the optimal strategy for winning a series. Proving this is a bit beyond my scope of practice at the moment, but perhaps I might follow up on this idea when I have more skills and time.
