I am tripping over the notation a little bit. I was representing eps times 1 as (0,1)(1,0) so (eps, 1)(1, 0) and (1, 0)(eps, 1) both evaluate to 2eps in my mind which would make the tail relevant.
Surely this would be your representation for infinitesimals, not for eps? By eps, I mean epsilon in the sense of standard real numbers, i.e. a small but positive real number. The issue is that in the real world, there will always be complexity, noise and uncertainty that will introduce small-but-positive real gaps, and these gaps will outweigh any infinitesimal preference or probability that you might have.
ah, I think I am starting to follow. It is a bit ambigious whether it is supposed to be two instances of one arbitraliy small finite or two (perhaps different) arbitrarily small finites. If it is only one then the tails are again relevant. “Always” is a bit risky word especially in connection with infinite.
I guess the basic situation is that modelling infinidesimal chances has not proven to be handy. But I don’t see that task would be shown to neccesarily frustrate. One could assume that while in theory one could model something in lexiographic way in reality there is some exchange rate between the “lanes” and in that way the blurryness could aid instead of hinder in applicability. Somebody that really likes real-only probablities could insist that unifying the preferences should be done early but there might be benefits in doing it late.
ah, I think I am starting to follow. It is a bit ambigious whether it is supposed to be two instances of one arbitraliy small finite or two (perhaps different) arbitrarily small finites. If it is only one then the tails are again relevant.
I don’t see what you mean. It doesn’t make a different whether you have only one or two small-but-finite quantities (though usually you do have on for each quantity you are dealing with), as long as they are in general position. For instance, in my modification to the example you gave, I only used one: While it’s true that the tail becomes relevant in (0, 1)(1, 0) vs (1, 0)(0, 1) because 0*1=1*0, it is not true that the tail becomes relevant in the slightly modified (eps, 1)(1, 0) vs (1, 0)(0, 1) for any real eps != 0, as eps*1 != 1*0.
So the only case where infinitesimal preferences are relevant are in astronomically unlikely situations where the head of the comparison exactly cancels out.
I thought we were comparing (eps, 1)(1, 0) and (1, 0)(eps, 1). if eps=eps strict equality. if it was (a,1)(1,0) and (1,0)(b,1) and its possible that a!=b it is unsure whether there is equality.
We were comparing epsilon to no-epsilon (what I had in mind with my post).
Anyway, the point is that strict equality would require astronomical consequences, and so only be measure 0. So outside of toy examples it would be a waste to consider lexicographic preferences or probabilities.
Surely this would be your representation for infinitesimals, not for eps? By eps, I mean epsilon in the sense of standard real numbers, i.e. a small but positive real number. The issue is that in the real world, there will always be complexity, noise and uncertainty that will introduce small-but-positive real gaps, and these gaps will outweigh any infinitesimal preference or probability that you might have.
ah, I think I am starting to follow. It is a bit ambigious whether it is supposed to be two instances of one arbitraliy small finite or two (perhaps different) arbitrarily small finites. If it is only one then the tails are again relevant. “Always” is a bit risky word especially in connection with infinite.
I guess the basic situation is that modelling infinidesimal chances has not proven to be handy. But I don’t see that task would be shown to neccesarily frustrate. One could assume that while in theory one could model something in lexiographic way in reality there is some exchange rate between the “lanes” and in that way the blurryness could aid instead of hinder in applicability. Somebody that really likes real-only probablities could insist that unifying the preferences should be done early but there might be benefits in doing it late.
I don’t see what you mean. It doesn’t make a different whether you have only one or two small-but-finite quantities (though usually you do have on for each quantity you are dealing with), as long as they are in general position. For instance, in my modification to the example you gave, I only used one: While it’s true that the tail becomes relevant in (0, 1)(1, 0) vs (1, 0)(0, 1) because 0*1=1*0, it is not true that the tail becomes relevant in the slightly modified (eps, 1)(1, 0) vs (1, 0)(0, 1) for any real eps != 0, as eps*1 != 1*0.
So the only case where infinitesimal preferences are relevant are in astronomically unlikely situations where the head of the comparison exactly cancels out.
I thought we were comparing (eps, 1)(1, 0) and (1, 0)(eps, 1). if eps=eps strict equality. if it was (a,1)(1,0) and (1,0)(b,1) and its possible that a!=b it is unsure whether there is equality.
yeah (eps,0) behaves differently from (0,1)
We were comparing epsilon to no-epsilon (what I had in mind with my post).
Anyway, the point is that strict equality would require astronomical consequences, and so only be measure 0. So outside of toy examples it would be a waste to consider lexicographic preferences or probabilities.