The other assumption made about Prisoner’s Dilemma, that I do not see you allude to, is that the payoffs account for not only a financial reward, time spent in prison, etc., but every other possible motivating factor in the decision making process. A person’s utility related to the decision of whether to cooperate or defect will be a function of not only years spent in prison or lives saved but ALSO guilt/empathy. Presenting the numbers within the cells as actual quantities doesn’t present the whole picture.
Let’s assume that your utility function (which is identical to theirs) simply weights and adds your payoff and theirs; that is, if you get X and they get Y, your function is U(X,Y) = aX+bY. In that case, working backwards from the utilities in the table, and subject to the constraint that a+b=1, here are the payoffs:
a/b=2: (you care twice as much about yourself) (3,3) (-5,10) (10,-5) (1,1)
a/b=3: (3,3) (-2.5,7.5) (7.5,-2.5) (1,1)
a=b: Impossible. With both people being unselfish utilitarians, the utilities can never differ based on the same outcome.
b=0: (selfish) The table as given in the post
I think the most important result is the case a=b: the dilemma makes no sense at all if the players weight both payoffs equally, because you can never produce asymmetrical utilities.
EDIT: My newbishness is showing. How do I format this better? Is it HTML?
Eliezer,
The other assumption made about Prisoner’s Dilemma, that I do not see you allude to, is that the payoffs account for not only a financial reward, time spent in prison, etc., but every other possible motivating factor in the decision making process. A person’s utility related to the decision of whether to cooperate or defect will be a function of not only years spent in prison or lives saved but ALSO guilt/empathy. Presenting the numbers within the cells as actual quantities doesn’t present the whole picture.
Important point.
Let’s assume that your utility function (which is identical to theirs) simply weights and adds your payoff and theirs; that is, if you get X and they get Y, your function is U(X,Y) = aX+bY. In that case, working backwards from the utilities in the table, and subject to the constraint that a+b=1, here are the payoffs:
a/b=2: (you care twice as much about yourself)
(3,3) (-5,10)
(10,-5) (1,1)
a/b=3:
(3,3) (-2.5,7.5)
(7.5,-2.5) (1,1)
a=b:
Impossible. With both people being unselfish utilitarians, the utilities can never differ based on the same outcome.
b=0: (selfish)
The table as given in the post
I think the most important result is the case a=b: the dilemma makes no sense at all if the players weight both payoffs equally, because you can never produce asymmetrical utilities.
EDIT: My newbishness is showing. How do I format this better? Is it HTML?
It’s not HTML, but “markdown” which gets turned into HTML.
http://wiki.lesswrong.com/wiki/Comment_formatting
Thank you!