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# Daily Box Score 8/21: Game Theory

Game theory is a very trendy field of study. It blends aspects of mathematics, economics, philosophy, and psychology to give insight into a specific class of interactions. Specifically, game theory is best at casting light on situations where the number of participants is limited, information is asymmetric (not everyone knows the same things), and each individual's outcome depends not only on her own decision but also those of the other participants. It may not seem especially relevant to baseball, but it has more use than you might think.

A Simple Case

The canonical, and most simplistic, example of game theory in practice is called the Prisoner's Dilemma (that link is to the Stanford Encyclopedia of Philosophy, though if you find it overly technical there is also a good description on Wikipedia). In that case, there are only two actors (the prisoners), and they only have to make one decision (whether or not to rat out one's co-conspirator). So it's a simple game, and can be represented by a two-by-two grid, implying just four possible outcomes.

You might think the application of game theory to baseball would only work with much more complicated models--after all, baseball has a great many variables and dozens of possible outcomes. But there is one situation in baseball that is relatively simple: whether or not a batter should swing when facing a full count. Clearly, the batter's outcome depends not only on his decision (whether or not to swing), but also on the decision of the pitcher (whether or not to throw a pitch in the strike zone). If pitchers always threw in the strike zone, batters would catch on and swing every time. At that point, however, pitchers would start throwing outside the zone. Because the batter isn't always better off swinging or not swinging, we say that there is no dominant strategy. But do they perhaps swing too often? Dave Allen does the math and says yes:

In the blue region batters swing over 75% and for most of it over 90% of the time. So batters do a good job of swinging at pitches they need to. In the red region just outside the break even batters swing between 75 and 50% of the time. So they swing at a large number of pitches they should take, they do not do a good job of taking pitches they should take.

And of course the colored regions he refers to come from some characteristically beautiful heat maps.

What about the case where a pitcher throws the same pitch in the same location a few times in a row--say a fast ball high and tight, or a breaking ball low and away? What does game theory say the pitcher should do next? MGL asked that very question:

My guess is also that the best pitchers, at least the crafty ones who are good because they mix up their pitches, recognize that they should be about just as likely to throw a certain pitch no matter how many times in a row they just threw that same pitch, given the count and game situation [...] The only qualification to that is what he thinks the batter might be thinking (e.g., if he thinks that batter is thinking like the announcer, he might be MORE likely to throw the same pitch), or whether the batter might have that high fastball imprinted on his brain such that he would be less likely to keep throwing that pitch.

This last part is essential and I think MGL is onto something. In many games (such as the beauty contest game), it is essential to have an idea of the level of sophistication of the other participants. When everyone in the game has fully thought through all the possibilities, and everyone knows everyone else has done this , we say that there is "common knowledge" of that fact. But clearly, MGL has shown that we fall far short of common knowledge when it comes to situations like these.

In fact, it's reasonable to believe that many if not most players do not understand that complete unpredictability is the best strategy. If that is the case, the best move is to assume that your opponent (in this case, the batter) believes you wouldn't POSSIBLY throw the same pitch again. That means you should, of course, do just the opposite.

The Draft

What about the draft? Certainly in that venue there is a limited number of participants (30 teams), asymmetric information (no one can be sure how good each prospect will turn out), and the outcomes for each team are influenced by their own decisions as well as the decisions of the other 29 teams.

One particular question of contemporary import is whether first round draft picks are overpaid. Of course, I highly recommend Sky Andrecheck's piece on the subject (which I linked to previously and makes explicit use of game theory). But Erik Manning has done some good work at FanGraphs on the subject. He finds values for buckets of draft picks, using WAR:

Picks 1 though 5 on average gave their teams \$32M of production.

Picks 6 through 10, \$22.4M

11-15, \$17.6M

16-20, \$18.9M

21-30 \$6.6M

And Tom Tango adds his analysis here.

One relatively recent development, which involves game theory and does not seem to be accounted for in these studies, is the benefit of signing a player who falls in the draft due to signing concerns. Once information gets out that a player will not sign unless he is offered a large signing bonus, many of the top teams (who also often are small market teams) are scared off. As a result, players whose talent would suggest they would be early first round picks can often fall to much lower than that (think Rick Porcello at 27th overall). Clearly, there is a strong incentive for teams to break slot recommendations to sign such players, but often small market teams fear retribution from the league more than big market teams, and the result is that such players end up breaking the draft.

It also means averages like the ones Erik came up with looking back may not be as predictive looking forward. Of course, if we get a whole new system (as Dave Cameron suggests), we'd have no such data at all.

Bean Balls

We all know about the primitive logic of beanings in baseball. It's the equivalent of a five year-old on the playground defending himself by yelling, "but he hit me first!" Except that instead of hitting the kid back, you go hit his friend in retaliation. ("Gee, Tommy, how do you really feel?")

But if we really want to do something about beanings, then we have to, you know, do something. So here's a suggestion from Harvard Law professor Alan Dershowitz:

The time has come for Major League Baseball to ban the bean ball. The only way to do this is for baseball to adopt a zero tolerance policy and to impose draconian sanctions not only on pitchers who throw at the heads of batters but, more importantly, on the managers who instruct them to do so. A manager cannot order a pitcher to accidentally hit a batter. Anytime a manager instructs a pitcher to throw at the head of a batter, he has committed the serious crime of reckless endangerment or assault with a lethal weapon. Baseball cannot tolerate such criminality.

The minimum penalty for a manager must be suspension for an entire season, perhaps even for life. For the pitcher, suspension for the season should be mitigated only if the pitcher turned in the manager. There should also be penalties for any baseball player who hears the manager or coach order the beaning of a player without reporting it.

Sound a little draconian to you? I think Dershowitz is correct to say that without penalties that properly disincentivize beanings, they will not stop. So if you're serious about stopping them, you must take drastic action. Over the Monster thinks Dershowitz does not fully realize the consequences that such a strict policy would bring about. 0157H7 writes:

Dershowitz's penalties, which he claims are necessary for proportionality (i.e. "let the punishment fit the crime"), in fact defy it. It would create situations where one player would have a bruised hand or back, and another would be out of baseball for a year. The only defense for a player would be to claim the manager ordered him to do it. [...]

Another issue is that these harsh penalties would so severely impact players that umpires / the Commissioner may be reluctant to enforce them to the fullest extent. If Halladay or another high-profile pitcher threw at someone, the powers that be would think twice about suspending him all season. Yet if that were the only penalty, they might choose not to prosecute at all. Thus, by implementing Dershowitz's penalties we might be trading poor sentences (5-7 days) for none at all.

I am torn on this issue. Sometimes it is legitimately difficult to determine whether a beaning was intentional. But we have harsh penalties for things that are difficult to determine all the time (murder, for example).

Discussion Question of the Day

Do you agree with Alan Dershowitz that penalties for intentional HBPs ought to be significantly more severe? Or would penalties on the scale he is discussion be overly draconian? Why?