Charlie Brown and game theory

I grew up watching Charlie Brown and the many Peanuts holiday cartoon specials. At holiday time I am always reminded of the Great Pumpkin, Snoopy’s antics, and the cheesy phone conversations. But most of all I am reminded of Charlie Brown and game theory.

Most of you will recall the running gag. Charlie Brown would be taunted by his friend Lucy to kick a football. Lucy would hold the football and Charlie Brown would attempt to kick it. Charlie would enthusiastically run up and wind up for a powerful kick. Then, at the last moment, Lucy would withdraw the ball, Charlie Brown would whiff, and he would fly up into the air and land on his back humorously.

The funny part was how the dynamic continued year after year. Charlie Brown seemed to learn that Lucy was untrustworthy. He would often protest that he could not trust her, and that the only reason she offered to hold the ball was to pull it away and mock him. But Lucy was crafty, and time and again she came up with new reasons why Charlie Brown should trust her. Lucy would appeal to anything and everything to get Charlie Brown to attempt the kick. And inevitably, Charlie would get duped and land flat on his back.

There is a nice Youtube video about one of the attempts. Here is an incident where Lucy gives a seemingly legally binding promise to Charlie. (skip to about 7 seconds in the video to get to the clip)

Link to Youtube video Charlie Brown

Link to Youtube video Charlie Brown

Transcript of video (starting from 7 seconds)

Lucy: Say Charlie Brown, I’ve got a football. How about practicing a few placekicks? I’ll hold the ball, and you come running and kick it.

Charlie Brown: Oh brother. I don’t mind your dishonesty, half as much as I mind your opinion of me. You must think I am stupid [to fall for this gag again].

Lucy: Oh, come on Charlie Brown.

Charlie Brown: No.

Lucy: I’ll hold it steady.

Charlie Brown: No.

Lucy: Please!

Charlie Brown: You just want me to come running up to kick that ball so you can pull it away and see me lying flat on my back and kill myself.

Lucy: This time you can trust me. See, here is a signed document testifying that I promise not to pull it away.

Charlie Brown: It is signed! It’s a signed document. I guess if you have a signed document in your possession, you can’t go wrong. This year I am really going to kick that football.

[runs up to kick the ball, but Lucy pulls it away and he falls flat on his back]. AARG!

Lucy: Peculiar thing about this document–it was never notarized.

The game theory interpretation

(I became familiar with the game theory interpretation from Avinash Dixit and Barry Nalebuff’s classic lay text Thinking Strategically (this is also discussed in the revised version The Art of Strategy). Here is my take, which is largely based on Dixit and Nalebuff’s idea.)

The incident was one of my first introductions to game theory, albeit unknowingly.

The problem is that Lucy’s promises always appeared credible but they never were. There are ways to make your claims sound credible, and in this respect, Lucy was a game theory expert.

A rational player needs to think about the loopholes and incentives outside the stated claims. And amusingly, Lucy would always explain in victory the loophole to her original promise–like how a non-notarized note may be invalid and not binding.

To think about the situation more generally, we can draw out the following game tree:

Charlie can either reject Lucy’s offer or accept it. If he rejects, then the game is over and he walks away. But if he accepts, then Lucy has a choice. Lucy can either hold the ball or pull it away. What will happen?

We can use the idea of thinking ahead (backwards induction) to solve the game. If Charlie accepts, then we think about what Lucy would choose. Based on experience, it is obvious that Lucy prefers to pull the ball away. This will happen whenever Charlie accepts. And so, Charlie must reject the offer if he wants to be pain-free.

But Charlie never appropriately uses backwards induction. He is always tricked into thinking that Lucy will somehow hold the ball. He unfortunately never learns.

One can only take solace that perhaps Charlie’s fictitious misfortune can be an example that helps us avoid the same mistakes.

Have a happy Thanksgiving!