Last time we looked at the dilemma of Charlie Brown attempting to kick a football and Lucy pulling the football away from him (to this day he still tries). In this scenario I modified the game tree to where only Charlie knows his payoffs but not Lucy’s. I introduced probability distribution into the game to determine the probability of Charlie choosing to kick the football based on the potential payoffs. Let’s apply probability distribution to other instances of sequential games.
A great example comes from the humanities. One of William Shakespeare’s most famous plays was Hamlet. In Hamlet, the title character seeks revenge against his uncle King Claudius who, according to the ghost of Hamlet’s father, murdered Hamlet’s father and took the throne for himself. We can represent a sequential game as follows:
Hamlet has the choice of either living or killing himself (to be or not to be). If he chooses to kill himself (not to be) then he will receive a payoff of -50 since he will be unable to avenge his father’s killer. Now if he chooses “to be” then Claudius can either let Hamlet live or kill him.
If Hamlet is allowed to live he will receive a payoff of 200 since he will avenge his father’s death by killing Claudius. However, if Claudius kills Hamlet then Hamlet’s payoff is -100 since he was unable to avenge his father and he was killed by his father’s killer (double negative).
Since we do not know the payoffs of Claudius, we can only determine the potential outcome through probability distribution. First let p represent the probability that Claudius kills Hamlet and 1-p be the probability that Claudius does not kill Hamlet. The probability distribution of this game would be:
-100p + 200(1-p) = -50 =>
-100p + 200 – 200p = -50 =>
-300p = -250 =>
p = 5/6
1-p = 1/6
So Hamlet will choose “to be” if the probability of Claudius killing Hamlet is less than 5/6 or if the probability of Claudius letting Hamlet live is greater than 1/6. Of course, things don’t look good for Hamlet in this situation. More than likely Claudius would want to kill Hamlet. But we cannot make that assumption since we do not know Claudius’s payouts.
Try to solve the following game below. In this scenario, you are currently at work and have a tendency to shirk (neglect your job) instead of doing your job (untrue of course). Now if you decide to shirk your boss would either write you up or not (more than likely write you up). Solve for p, 1-p, and explain the probability of you choosing to shirk.