If you assume.… [y]ou are, in effect, stipulating that that outcome actually has a lower utility than it’s stated to have.
Thanks, that focuses the argument for me a bit.
So if we assume those curves represent actual utility functions, he seems to be saying that the shape of curve B, relative to A makes A better (because A is bounded in how bad it could be, but unbounded in how good it could be). But since the curves are supposed to quantify betterness, I am attracted to the conclusion that curve B hasn’t been correctly drawn. If B is worse than A, how can their average payoffs be the same?
To put it the other way around, maybe the curves are correct, but in that case, where does the conclusion that B is worse come from? Is there an algebraic formula to choose between two such cases? What if A had a slightly larger decay constant, at what point would A cease to be better?
I’m not saying I’m sure Dawes’ argument is wrong, I just have no intuition at the moment for how it could be right.
A point of terminology: “utility function” usually refers to a function that maps things (in our case, outcomes) to utilities. (Some dimension, or else some set, of things on the x-axis; utility on the y-axis.) Here, we instead are mapping utility to frequency, or more precisely, outcomes (arranged — ranked and grouped — along the x-axis by their utility) to the frequency (or, equivalently, probability) of the outcomes’ occurrence. (Utility on the x-axis, frequency on the y-axis.) The term for this sort of graph is “distribution” (or more fully, “frequency [or probability] distribution over utility of outcomes”).
To the rest of your comment, I’m afraid I will have to postpone my full reply; but off the top of my head, I suspect the conceptual mismatch here stems from saying that the curves are meant to “quantify betterness”. It seems to me (again, from only brief consideration) that this is a confused notion. I think your best bet would be to try taking the curves as literally as possible, attempting no reformulation on any basis of what you think they are “supposed” to say, and proceed from there.
Thanks, that focuses the argument for me a bit.
So if we assume those curves represent actual utility functions, he seems to be saying that the shape of curve B, relative to A makes A better (because A is bounded in how bad it could be, but unbounded in how good it could be). But since the curves are supposed to quantify betterness, I am attracted to the conclusion that curve B hasn’t been correctly drawn. If B is worse than A, how can their average payoffs be the same?
To put it the other way around, maybe the curves are correct, but in that case, where does the conclusion that B is worse come from? Is there an algebraic formula to choose between two such cases? What if A had a slightly larger decay constant, at what point would A cease to be better?
I’m not saying I’m sure Dawes’ argument is wrong, I just have no intuition at the moment for how it could be right.
A point of terminology: “utility function” usually refers to a function that maps things (in our case, outcomes) to utilities. (Some dimension, or else some set, of things on the x-axis; utility on the y-axis.) Here, we instead are mapping utility to frequency, or more precisely, outcomes (arranged — ranked and grouped — along the x-axis by their utility) to the frequency (or, equivalently, probability) of the outcomes’ occurrence. (Utility on the x-axis, frequency on the y-axis.) The term for this sort of graph is “distribution” (or more fully, “frequency [or probability] distribution over utility of outcomes”).
To the rest of your comment, I’m afraid I will have to postpone my full reply; but off the top of my head, I suspect the conceptual mismatch here stems from saying that the curves are meant to “quantify betterness”. It seems to me (again, from only brief consideration) that this is a confused notion. I think your best bet would be to try taking the curves as literally as possible, attempting no reformulation on any basis of what you think they are “supposed” to say, and proceed from there.
I will reply more fully when I have time.