In which I try to sort out game theory
Something about this post at The Book Blog didn’t sit right with me, but because I never studied economics and don’t know anything about game theory I figured I’d run it by TedQuarters resident maverick economist and former roommate extraordinaire Ted Burke.
He also struggled to grasp what MGL was trying to get at when he writes:
If batters and pitchers adjust their approaches according to what the other has done in the past during a game, then one or the other is NOT performing (in terms of their approach/strategy) optimally! Game theory tells us that.
Baseball is a game of adjustments only in terms of learning – a player improving upon his skills and strategy and learning new things about his opponent. It should NOT be about the kind of adjustments during a game that you hear from commentators all the time.
My gripe was that, though certainly it makes sense that a player should rely on the largest sample he has to determine an opponent’s tendencies, there are myriad minor in-game factors that might impact those tendencies (most obviously the pitches a pitcher has the best feel for on any given day) and so it seems reasonable that an opponent should be adjusting to the in-game sample.
Ted Burke’s issues ran a bit deeper, and were perhaps more semantic. Our conversation went like this (apologies in advance for the cheapshot at Jersey drivers):
Burke: He’s assuming that a Nash Equilibrium exists for this situation, which pretty clearly isn’t the case.
Berg: You’re assuming that I have any idea what a Nash Equilibrium is.
Burke: It’s a nerd name for a concept you fully understand as a sports fan and human in society. In a situation where multiple parties each have to make a choice or series of choices, its Nash Equilibrium is the outcome where each party made the right choice for them given the choices of the other parties.
Say the two of us are driving on two different roads and we’re approaching the intersection of the two. Your road has the green light, and my road has a red light. We each have a choice to make: stop at the intersection or continue through it. We’re both licensed drivers, so we have certain expectations about how other licensed drivers will behave in this situation. You figure that if you keep going through that intersection, there’s a slim chance that I will plow into you, but there’s a much greater chance that you’ll pass through the intersection safely.
If you stop at the green light, you eliminate the chance of getting smashed by me, but you realize that doing so will almost certainly result in you being rear-ended by the truck behind you. So you choose the safer expected outcome and drive through the intersection. Similarly, I know that stopping at the red light carries a very small possibility that I will be rear-ended, but there’s a much greater chance that I’ll smash into another car if I attempt to drive through the intersection. So I choose to stop on red. Our paths cross safely, like two handsome ships passing in the night.
So every time we get in that situation, I stop and you keep driving. That’s the Nash Equilibrium.
Berg: And you’re saying that doesn’t exist in baseball (or in some parts of Jersey).
Burke: There are plenty of situations in baseball that have a Nash Equilibrium, but this isn’t one of them. Actually, if there were a Nash Equilibrium in this situation, the game of baseball wouldn’t be nearly as interesting as it is. One primary reason for throwing different pitches at different speeds in different locations is to prevent the batter from knowing if, when, and where to swing. In that sense, game theory would fit perfectly with the adjustment process the announcers described.
If the Yankees are swinging aggressively at first pitches, Pavano would want to start at-bats with a breaking ball or something out of the zone, which would in-turn lead to the Yankees choosing to be more patient, which leads to Pavano throwing fastballs over the plate on first pitches. And the cycle repeats itself.
Berg: It seems like his point is based on the fact that the Yankees should know going into the game that Pavano is going to throw fastballs over the plate and should base their expectations on those tendencies.
Burke: But if Pavano went through the whole game throwing first-pitch strikes and the Yankees spent the whole game swinging at them, the Twins would have to be idiots.
Game theory tracks the decisions people make over time to maximize their own utility. It would never suggest that the Twins continue making a decision that’s getting them pounded when a simple “adjustment” would at least give them a chance of not getting pounded.
Berg: Yeah, which is a big part of why it made no sense to me.
Burke: What the announcers were describing is basically the essence of game theory.
