Monday, July 6, 2009

The Game Theory of 'Get to the Front!'

Game theorist and legendary professor Ben Pollack gives his students the following assignment on the first day of class:

Guess the number between 1 and 100 that is 2/3 of the average guess of everyone else in the room.


Think about it for a minute.


Really. Think about it.


Ok, so you've probably determined the "theoretical" answer. It's a race to the bottom. For any good candidate answer, n, if it's a good answer, everyone will guess it. You're all in the same position, after all. But then if everyone guesses n, you should be at 2/3's of n.

But then, you're in the same position as everyone else, so they'll guess 2/3's of n too. So you should guess 2/3's of 2/3's of n.

Rinse and repeat. Multiply 2/3 by 2/3 by 2/3 by 2/3 until you are infinitesimally close to zero. (For those of you into nonstandard analysis just saying 'infinitesimal' is, of course, not good enough, because there are infinitely many of them.)

That's the theory. What about practice? Ben told me the typical winning number, and I can't remember it for sure, but it's something like 13. Interesting, huh? People who determined the theoretical answer and wrote down "1" or "1 / infinity" didn't get the right answer in practice. They didn't win.

I actually got pretty close in my own guess. My reasoning was that most people would take the o to 100 range and anchor an average at 50. Then they'd take 2/3's of 50, which would be 33. Rather than stop there, most people would think: everyone else probably just did what I did. So the average will be 33. I'll be extra clever and put down 2/3 of that! And they'd answer "22".

My guess was that most people are just too lazy to think beyond two steps. If you think beyond two, really, won't you see it all the way through and throw up your hands? Of course, given the institution, there were some bright people, who would see it through. They'd be the smartasses talking about nonstandard analysis and bringing the average down to lower than 2/3 of 22. And then there would be the smartasses like me who see through it, but think not everyone will, and then try to simulate them and go 1/3 lower. Those people will also push the number down. Competing against a room full of people JUST like me, I will, by definition, lose! So the question, for me, was really about the mix of people at the university.

Turns out, most people just go the two iterations after anchoring at 50.

Get to the Front!

Everyone knows the advantages of being near the front of a group:
  • no accordion effect requiring more acceleration out of corners and up hills;
  • fewer crashes;
  • the ability to see and respond to moves.
On nearly every "coach's corner" course prep I've seen, the advice is the same: get to the front. You saw it in stage 3 of the TDF. There was Lance at the front when the field split in the crosswinds. Everyone praised his racing savvy and heaped scorn on the Evanses and Menchovs who were caught napping.

But saying get to the front is *exactly* like saying guess the number that is two thirds of everyone's guess. If everyone took the advice to heart, every race would be a sprint from the start until we all blew up and fell off our bikes.

If Evans, Menchov, Sastre, Kloden, Armstrong, Leipheimer, Contador, Kirchen, and all other GC contenders were ALL in the front of the field, then they would either be without any domestiques or else they wouldn't be in the front of the field. They would be the field.

Ok, I know this is obvious, but it bears keeping in mind:

It is logically impossible for the whole field to be at the front of the whole field. The very point of being at the front of the field is that there are only a handful of people in it at any given time.

Getting into the front 1/3 of the peleton is just like guessing 1/3 of the average guess.

So much for theory.

Now... what actually happens? You do the guessing with your legs. Some people have superior fitness and get a solid warmup. They finish the initial push for the front well within their limits; then they cruise in great position throughout the race, monitoring breaks, helping teammates and such. Afterwards they credit the sage advice of going to the front for their results: "Really, it is so much easier up there, guys! Everybody should do it!."

Other riders feel themselves plunging deeply into anaerobic territory in the first few minutes of a race until they slide towards the back of the pack where they seek shelter from the wind, trying to pedal smoothly and recover, because it's either that or blow up and drop out. These riders will suffer from the accordion effects. They're more likely to be gapped, more likely to crash. They'll be lucky to finish as pack fodder. Afterwards, they'll tell themselves: "I've got to get to the front next time!"

Bike racing is like my favorite Biblical parable, the parable of the ten talents. I won't rehearse the story but the lesson is something like this: to he who has much, more will be given; to he who has little, even what little he does have will be taken from him.

"Get to the front!" should be "Have, and use, superior power over the first five to ten minutes of the race." In other words: be faster than everyone else.

Really, what it means is that bike races are, in fact, shorter than their advertised distances. Because the rear of the pack is such a punishing place to be, there is a much shorter race within the race to determine who is not stuck there. So really, the advice boils down to saying: hey, train for and race the short race within the longer one.

And that is good advice.

The other thing is that sometimes races just start slow and stay slow, because no one wants to actually be on the front taking on the wind. In these scenarios, getting to the front is not at all hard. If everyone heeded the advice it would be incredibly hard, of course. But they don't.

So here, the advice is: get to the front if the opportunity presents itself.

That's also good advice. Take advantage of the irrationality of others.

Surprise Quiz Paradox

One more example of the departure of theory and practice, just for fun. On Friday, the teacher announces there will be a surprise quiz one day next week. The students reason as follows:

The quiz can't be on Friday and still be a surprise. Because after Thursday, if we haven't had a quiz, we'll know it's coming on Friday. Hence, no surprise.

But now that we've ruled out Friday, we also know it won't happen Thursday. If we go home from school on Wednesday without a quiz, we'll know it's Thursday, because we've already ruled out Friday.

And so the students reason until they've ruled out Wednesday and Tuesday and Monday too.

End of story: on Wednesday there's a quiz and everyone's surprised.

4 comments:

Calvini said...

Awesome, awesome post. I say this objectively, in a completely non-nepotistic way. Absolutely beautiful. Better than anything on Pez or Velo. Except maybe Stephen Cheung, and he's doing the whole training/science thing, and that's different.

Seriously, brilliant. Milky.

Tim Rugg said...

Is this why you weren't at Greenbelt Troy... sitting around writing Thursday morning, invigorating, start my day off thinking way too much, intelligent, metaphorical mumbo jumbo!

Sam said...

good post but it'd be better with more pictures!

qualia said...

Calvini: milk strong, brother! Self congratulation is part of the milkstrong identity. We are on an awesome team, we are riding awesome, our blog is awesome, and we're raising awareness for an awesome cause. Awesome awesome awesome!

Rugg: those who cannot ride, think too hard about it and then get dropped. I skipped Greenbelt because my legs were killing me. I rode over to Hains instead and ended up pushing way too hard so as not to be shown up by some GW cycling guy and an NCVC 4 rider named Doug. WTF?? My legs are once again useless while Hagerstown and Dawg Days await.

Sam: I will aim for a 10 to 1 ratio of words to pictures. I hear that is the law of the interwebs. Speaking of tactics... you are the only guy I've seen who actually chooses to tailgun for periods of the race, I guess when you sense a crash coming or find a steadier pace back there through rollers, and then you move up at will when you sense a move coming or want to kill the field yourself. Very unusual. How do you do that?