The maximality rule is too demanding. Under maximality, we can’t even say that [1, 10 000] is better than [-10 000, 1.0001]. Are there any good reasons to avoid more permissive rules even in cases like this?
Of course, if your UEV intervals overlap and you choose one of them, you may make a mistake if your idealized version would choose the other one. But choosing at random is no better in this regard.
What seems important is how costly such mistakes are, i.e., how the overall performance of your choice rule compares with that of other rules—such as choosing at random.
Can these performance measures be defined without additional assumptions about how values are distributed within UEV intervals?
Under maximality, we can’t even say that [1, 10 000] is better than [-10 000, 1.0001].
That’s not quite right. Maximality says an action A is impermissible when some alternative B has higher EV on every probability function in your representor. And that can be true even when A’s and B’s EV ranges overlap.
Example:
If A had the same range but sloped the other way, then it would be permissible by maximality:
So to figure out what’s permissible under maximality, we can’t just look at ranges. We need to look at the representor.
Yes, but this means that you know something additional about the structure of the representor, not just its range. I’m asking whether we can do better than maximality even for intervals alone, without adding any further details.
(By the way, pictures are broken.)
UPD: Though, you probably mean that we do know some additional structure for EV if we look at how it is constructed from probabilities and utilities, for which we have just intervals without structure.
And yes I think often we know more than just intervals of EVs. For example, we know whether the EV of some action increases or decreases with the probability of some proposition X.
The maximality rule is too demanding. Under maximality, we can’t even say that [1, 10 000] is better than [-10 000, 1.0001]. Are there any good reasons to avoid more permissive rules even in cases like this?
Of course, if your UEV intervals overlap and you choose one of them, you may make a mistake if your idealized version would choose the other one. But choosing at random is no better in this regard.
What seems important is how costly such mistakes are, i.e., how the overall performance of your choice rule compares with that of other rules—such as choosing at random.
Can these performance measures be defined without additional assumptions about how values are distributed within UEV intervals?
That’s not quite right. Maximality says an action A is impermissible when some alternative B has higher EV on every probability function in your representor. And that can be true even when A’s and B’s EV ranges overlap.
Example:
If A had the same range but sloped the other way, then it would be permissible by maximality:
So to figure out what’s permissible under maximality, we can’t just look at ranges. We need to look at the representor.
Yes, but this means that you know something additional about the structure of the representor, not just its range. I’m asking whether we can do better than maximality even for intervals alone, without adding any further details.
(By the way, pictures are broken.)
UPD: Though, you probably mean that we do know some additional structure for EV if we look at how it is constructed from probabilities and utilities, for which we have just intervals without structure.
Thanks.
Oops, pictures should be fixed now.
And yes I think often we know more than just intervals of EVs. For example, we know whether the EV of some action increases or decreases with the probability of some proposition X.