Sure, I used that as what I take to be the case where the argument would be most easily recognized as valid.
One generalization might be something like, “losing makes it harder to continue playing competitively.” But if it becomes harder to play, then I have lost something useful, i.e. my stock of utility has gone down, perhaps by an amount not reflected in the inferred utility functions. My feeling is that this must be the case, by definition (if the assumed functions have the same expectation), but I’ll continue to ponder.
The problem feels related to Pascal’s wager—how to deal with the low-probability disaster.
I really do want to emphasize that if you assume that “losing” (i.e. encountering an outcome with a utility value on the low end of the scale) has some additional effects, whether that be “losing takes you out of the game”, or “losing makes it harder to keep playing”, or whatever, then you are modifying the scenario, in a critical way. You are, in effect, stipulating that that outcome actually has a lower utility than it’s stated to have.
I want to urge you to take those graphs literally, with the x-axis being Utility, not money, or “utility but without taking into account secondary effects”, or anything like that. Whatever the actual utility of an outcome is, after everything is accounted for — that’s what determines that outcome’s position on the graph’s x-axis. (Edit: And it’s crucial that the expectation of the two distributions is the same. If you find yourself concluding that the expectations are actually different, then you are misinterpreting the graphs, and should re-examine your assumptions; or else suitably modify the graphs to match your assumptions, such that the expectations are the same, and then re-evaluate.)
This is not a Pascal’s Wager argument. The low-utility outcomes aren’t assumed to be “infinitely” bad, or somehow massively, disproportionately, unrealistically bad; they’re just… bad. (I don’t want to get into the realm of offering up examples of bad things, because people’s lives are different and personal value scales are not absolute, but I hope that I’ve been able to clarify things at least a bit.)
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.
Sure, I used that as what I take to be the case where the argument would be most easily recognized as valid.
One generalization might be something like, “losing makes it harder to continue playing competitively.” But if it becomes harder to play, then I have lost something useful, i.e. my stock of utility has gone down, perhaps by an amount not reflected in the inferred utility functions. My feeling is that this must be the case, by definition (if the assumed functions have the same expectation), but I’ll continue to ponder.
The problem feels related to Pascal’s wager—how to deal with the low-probability disaster.
I really do want to emphasize that if you assume that “losing” (i.e. encountering an outcome with a utility value on the low end of the scale) has some additional effects, whether that be “losing takes you out of the game”, or “losing makes it harder to keep playing”, or whatever, then you are modifying the scenario, in a critical way. You are, in effect, stipulating that that outcome actually has a lower utility than it’s stated to have.
I want to urge you to take those graphs literally, with the x-axis being Utility, not money, or “utility but without taking into account secondary effects”, or anything like that. Whatever the actual utility of an outcome is, after everything is accounted for — that’s what determines that outcome’s position on the graph’s x-axis. (Edit: And it’s crucial that the expectation of the two distributions is the same. If you find yourself concluding that the expectations are actually different, then you are misinterpreting the graphs, and should re-examine your assumptions; or else suitably modify the graphs to match your assumptions, such that the expectations are the same, and then re-evaluate.)
This is not a Pascal’s Wager argument. The low-utility outcomes aren’t assumed to be “infinitely” bad, or somehow massively, disproportionately, unrealistically bad; they’re just… bad. (I don’t want to get into the realm of offering up examples of bad things, because people’s lives are different and personal value scales are not absolute, but I hope that I’ve been able to clarify things at least a bit.)
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.