Solving Sequential Games III

Why does Charlie always kick?

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Today we are going to analyze one of the greatest and puzzling dilemmas in cartoon animation history. Why does Charlie Brown always attempt to kick the football that Lucy is holding for him? In every instance she always pulls the ball away from him and he ends up going airborne from missing the ball. Let’s see if we can understand Charlie’s reasoning in a sequential game.

First we will represent the game in the following game tree:

Charlie 1

In this game Charlie has the option of either kicking the football or not. If he chooses not to kick the football then his payout will be 5 utils (happiness) while Lucy’s will be 0 (no bumps and bruises for Charlie). Now if Charlie chooses to kick the ball then he is at the mercy of Lucy. If Lucy allows Charlie to kick the ball then the payouts will be 15 for Charlie and 5 for Lucy. If Lucy pulls the football then Charlie’s payout is -5 since he will miss the kick and go flying into the air. Lucy will achieve a payout of 10 utils since she enjoys watching Charlie suffer (never liked Lucy).

Let’s solve this game under the assumption of complete rationality:

Charlie 2

The rational choice for Lucy is to not allow Charlie to kick since 10 > 5. Charlie should assume that Lucy is a rational decision maker and will opt to not allow him to kick. As such, Charlie should just walk away knowing that he did not give Lucy the satisfaction of missing the kick. But we know this is not the case as time and time again Charlie attempts to kick the football.

It’s possible to assume that Charlie always makes an irrational decision while Lucy always makes the rational choice. Lucy may know that Charlie is irrational hence her reason for always offering Charlie the opportunity to kick the ball. But what if we didn’t factor in Lucy payouts? Let’s assume Charlie does not know Lucy’s payouts and only knows his own as shown below.

Charlie 3

How would Charlie choose now? Let’s solve this using a probability distribution similar to ¬†mixed strategies. Let p be the probability that Lucy doesn’t allow Charlie to kick and 1-p be the probability that Lucy allows Charlie to kick. The equation would appear as follows:

-5p + 15(1-p) = 5

Now let’s solve for p and 1-p:

-5p + 15(1-p) = 5 =>

-5p + 15 – 15p = 5 =>

-20p = -10 =>

p = -10/-20 or 1/2

1-p = 1 – 1/2 = 1/2

The way to interpret this is Charlie will choose to kick the ball as long as the probability of Lucy not allowing him to kick is less than 1/2. Or we can say Charlie will choose to kick the ball as long as the probability of Lucy letting him kick is greater than 1/2. So Charlie is still looking at a 50/50 split.

Based on Charlie’s history though we can only assume that he makes irrational decisions while Lucy always makes the rational choice. For some reason the desire for Charlie to kick that ball beats out all of the future body aches and pains. Poor Charlie.

 

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