This is an exciting time of year in baseball for a number of reasons. First and foremost, it is exciting because it is when pennant races for the ages are decided. But, as BPro's playoff odds report shows, no team has a percentage chance of making the playoffs that is greater than 28% or less than 80%. That is to say, most of the fun part of September baseball just doesn't look like it's going to happen this year. That's a bummer.

But there are other things about September baseball that are exciting. Obviously, even out-of-contention teams inject excitement with roster expansion call-ups. But what of the Falling Axe Alerts? You know, managers who are in danger of losing their jobs (or not). Too often managers are scapegoats, so I thought we'd spend today talking about one of the few things managers have control over: in-game tactics. And we'll re-tread the fertile ground of game theory.

Table of Contents

Strategy
Bunting for a Hit
Overkill
Discussion Question of the Day

Strategy

When we last talked about game theory, we discussed some of the most simple games, most notably the Prisoner's Dilemma. The Prisoner's Dilemma is most useful because of its simplicity. It's relatively straightforward, and it should be pretty easy to see what the equilibrium will be. 

To summarize, two suspected criminals are caught and placed in separate rooms. Each has two options: snitch on their compadre or remain silent. If they snitch but don't get snitched on, they receive a small punishment. If they don't snitch but do get snitched on, they receive a large punishment. If both criminals snitch, both get medium-sized punishments. However, if NEITHER suspect snitches (that is to say, both remain silent), they get the smallest possible combined punishment. Here's a chart:

In each cell, the first number represents the number of years of punishment for suspect #1 and the second the number of years for suspect #2. 

Two things should become immediately obvious. First, the best case scenario for the criminals is if neither snitches. That is, if they can cooperate somehow, they will be best off. However, if you look at each column individually, you should notice that the incentives for each individual do not reward silence.

Let's assume we're suspect #1 and trying to decide what to do. If our old buddy #2 snitches, we'd be better off if we snitched too (reducing our punishment from 10 to 5 years). If dear comrade #2 does right by us and stays silent, it is still in our best interest to snitch (because we go from two years to one)!

So no matter what our friend does, we should snitch. In game theory, we call a strategy which is always leaves us better off (no matter what else other players do) a dominant strategy. Dominant strategies are great because they make decision making easy. How much do you want to bet that there are simple, easy, dominant strategies in baseball?

Bunting for a Hit

Over at The Book blog, game theory enthusiast MGL has applied game theory to the strategy of bunting for a hit. It's good stuff, really, but it is also a tad dense if you are not familiar with game theory. In fact, my little primer above is all just a way to get up to speed in tackling MGL's ideas. Here we go.

He begins with an assumption:

First the defense sets up and then the offense gets to decide what to do.  So what SHOULD happen?

(I'm not entirely convinced this is the case, but I'll get to that in a moment). So right away, we can tell that we have left the neat, ordered universe of the Prisoner's Dilemma.

MGL's answer is refreshing for its simplicity:

The defense should set up in a way that it doesn’t matter what the offense does.

If the defense is playing optimally, and there is no reason that it shouldn’t [...], then it does not matter what the offense does.  They can bunt 0% of the time or 100% of the time and it won’t change their WE!

That SHOULD be the case in baseball.  But it is not of course.

Okay, let's deal with this the same way you eat a bicycle (one bit at a time).  Right off the bat, it's important to note that we're talking about bunting for a hit, not sacrifice bunting, here. 

Next, MGL is arguing that, given that the defense has to choose where to play before the batter has to choose whether to bunt, the defense wants to minimize the batter's ability to take advantage of the fact that he gets to choose second. How to do this?

Well, if you've ever tried to make even portions between two people, you already know the logic. It is commonly known as "I Cut, You Choose." The beauty of method when it is applied to divvying portions is that the incentives tend toward a perfect 50-50 cut. Of course the chooser will choose the biggest piece (or at least the cutter must assume that he will), and so the cutter should try to cut the thing as close to in half as he possibly can. Once the cutter has sufficient knife prowess (which it is now in his interest to have!), we should see 50-50 cuts each time.

In the bunting scenario, the defense cuts and the batter chooses. The defense should set up in such a way that (and this will be different for each individual batter) the batter is just as well off bunting as he is not bunting. Each step the defense takes toward home plate reduces the win expectancy of bunting and it raises the win expectancy of not bunting. When a marginal step in is worth the same as a marginal step out, the defense has reached its equilibrium.

This, in a certain sort of way, is a dominant strategy. It does not particularly matter what the batter chooses (especially since the batter may not have chosen yet), the defense will always be best off by making the batter completely indifferent between a bunt attempt and swinging away.

Now, if all this is true, it shouldn't matter what the batter does. But do teams really do this? I never saw any teams playing the Reds come all the way in on the infield grass against Willy Taveras this season, so I would guess that not all teams do this all the time.

Given that defenses are (from a game theory perspective) non-rational, how should batters respond? MGL says:

If the defense is not playing optimally in any game situation (for the bunt hit or bunt sacrifice), then the batter is supposed to bunt or not bunt 100% of the time!

There is no game theory involved for the batter (and his manager) in terms of "mixing it up." That would only be true if the batters had to make his decision before the defense set up.  That is never the case. 

Again, we have a dominant strategy. If the defense has cut the cake unequally, the batter ought to go right for the big piece and scarf it down. 

But of course you almost never see a player bunt more than about twice in a row. In real life, defenses and batters never reach the equilibrium we expect that they would. Why?

I have a few ideas. The first goes all the back to the original premise, namely that defenses must position themselves before the batter decides whether to bunt. This may true in terms of starting position, but it is very frequent to see teams have their defenses come in either with or slightly before the pitch. Anecdotally, Chipper Jones in his sprier days used to do this quite often. That means that there is a degree of unpredictability.

When there is unpredictability, often the equilibrium created by dominant strategies begins to break down. In fact, the best possible outcome under conditions of uncertainty is often a "mixed strategy," which is either random or appears random to other decision makers. Mixed strategies, unlike dominant strategies, involve many different possible moves. Their primary benefit is to keep the other players guessing. 

Of course, this is what conventional wisdom tells us batters (especially batters who are fleet of foot) ought to do. You constantly hear announcers saying that Player X's speed "keeps the defense on its toes" or that Player Y is "always a threat to bunt." 

MGL goes on to contemplate (many) other scenarios, and I recommend you read the whole piece if you remain interested. If you've lost your appetite for the hard cases, I don't blame you. What about the easy ones?

Overkill

We like problems that are complex because they can be difficult to figure out. Take, to pick a timely example, Ichiro, who just collected his 2000th hit. He raps a tremendous number of his hits in the infield (12.7% on his career, 16.8% on the year). Should teams play him closer?

Well, not necessarily. There's always the possibility that Ichiro has such tremendous bat control that even that wouldn't work, because he would just hit the balls over the defense's head if they moved in. And maybe some of those balls that went over the heads of the infielders would end up being doubles, which would be even more costly in the end.

As I said, hard cases are fun to think about because they stretch our reasoning. But every once in a while, you come across something like this, and it makes you remember that we definitely do not have the conditions for common knowledge. And for that reason, we have to be very careful applying the tenets of game theory to the decisions of individual players in a baseball game.

Discussion Question of the Day

Can you think of examples of dominant strategies in baseball? 

How about cases where there clearly is no dominant strategy?