I imagine you would be able to solve this by replacing real-valued utilities with utilities in a numbering system that contains infinitesimals. However, it seems to me that it would not matter much in practice, since if you had even a tiny chance of affecting the leading term, then that chance would outweigh all of the other terms in the calculation, and so in practice only the leading term would matter for the decisions.
Or to give an argument from reflection, there’s a cost to making decisions, so an agent with a lexicographic ordering would probably want to self-modify into an agent that only considers the leading term in the preference ordering, since then it doesn’t have to spend resources on the other terms.
The “leading term will dominate” effect is broken if there are infinidesimal chances around.
It might be sensible for an agent for some purposes to assume away some risks ie treat them as 0 chance. However it might come about that in some circumstances those risks can’t be assumed away. So a transformation in the other direction of turning a dirty hack into an actual engine that can accurately work on edgecases might be warranted.
The trouble is that for some combinations of infinidesimal and transfinite their product can be an ordinary finite real. One can’t thus keep them strictly separated in different “lanes”.
Oh, you meant literal infinitesimal (non-real) probabilities. I thought you meant very small probabilities.
The argument against lexicographic ordering is also an argument against infinitesimal probabilities. Which is to say, suppose you have some lotteries l1, l2 that you are considering choosing between (the argument can be iterated to apply to a larger number of lotteries), where each lottery ln can have payouts rn1, rn2, rn3, … with probabilities pn1, pn2, np3, ….
Suppose that rather than being real numbers, the probabilities and payouts are ordered lexicographically, with payout m being of shape (rnm1, rnm2, rnm3, …) and probability m being of shape (pnm1, pnm2, pnm3, …).
In order for the infinitesimal tail of the sequence to matter, you need the sum over r1m1\*p1m1 to be exactly precisely 100% equal to the sum over r2m1\*p2m1 (that is, after all, the definition of lexicographic orderings). If they differ by even the slightest margin (even by 10^-(3^^^^^^3)), this difference outweights anything in the infinitesimal tail. While you could obviously set up such a scenario mathematically, it seems like it would never happen in reality.
So while you could in principle imagine lexicographic preferences, I don’t think they would matter in reality, and a proper implementation of them would act isomorphic to or want to self-modify into something that only considers the leading term.
You need the product to be exactly equal and you don’t neccesarily need to do it factor by factor. (0,1)(1,0) can equal (1,0)(0,1) that is neglible chance of a finite reward is as good as a certainty of a neglible reward. Because they lanecross in this way knowing your rewards doesn’t mean you can just take the most pressing factor and forget the rest as the propabilities might have impacts that make the expected values switch places.
In application it is not straightforwar what things would be well attributed to small finite chances and what would be well attirbuted to infinidesimal chances. As a guess say that you know that some rocks would break under one meteor impact and other rocks would break under two meteor impacts. But you don’t know how likely meter impacts are an assume them to be 0. It kinda still remains true that the hard rocks need to have twice as good valuables in them in order to justify to wait around them rather than the soft rocks. If they might contain valuable of immense value at some point it starts to make sense to be more curious about the unknown risks and rewards rather than the known and modelled risks and rewards. Some of the actions and scenarios that are not assumed to be 0 will lean more heavily to the unmodeled parts such as a long plan might have triple the chance of meteors, what ever it is, compared to a short plan.
If one insists that everything needs to be real then making up arbitrary small finites for parts of the model that you have little to reason with might get very noisy. With keeping up several archimedean fields around one doesn’t force to squish in to a single one. That is if your ordinary plans have an expected value difference of 0.0005 then you can estimate that if meteor impactsw have less effect than that you know your assumtions are effectively safe. However if the differences are 0.000000000000002 then you migth be more paranoid and start to actually look whether all the “assumed 0” assumptions should actually be made.
In your example, r1m1*p1m1 is exactly equal to r2m1*p2m1; 0*1=0=1*0. The point is that if in the former case, you instead have (eps, 1)(1, 0), or in the latter case, you instead have (1, 0)(eps, 1), then immediately the tail of the sequence becomes irrelevant, because the heads differ. So only when the products of the heads are exactly equal do the tails become relevant.
However if the differences are 0.000000000000002 then you migth be more paranoid and start to actually look whether all the “assumed 0” assumptions should actually be made.
This requires you to not use infinitesimal probabilities. 0.000000000000002 is still infinitely bigger than infinitesimals.
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.
If we just have pure lexiographics then it can be undefined whether we can product them together. In my mind I am turning the lexiographic to surreals by using them as weights in a Cantor normal form and then using surreal multiplication.
So in effect I have something like (a,b)(c,d)=(ac,ad+bc,bd).
I guess I know about “approximately equal” when two numbers would round out to the same nearest real number. The picture might also be complicated on whether immense chances exist. That is if you have a coin that has a finite chance to come up with something and I give you more than finite amount of tries to get it there is only a neglible chance of failure. Then ordinarily if a option has no finite chances of paying out it could be ignored but an immense exploration of a “finite-null” coin which has neglible chances of paying out could matter even at the finite level. And I guess on the other direction are the pascal wagers. Neglible chances of immense rewards. So there are sources other than aligning the finite multipliers to get effects that matter on the finite level.
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.
I imagine you would be able to solve this by replacing real-valued utilities with utilities in a numbering system that contains infinitesimals. However, it seems to me that it would not matter much in practice, since if you had even a tiny chance of affecting the leading term, then that chance would outweigh all of the other terms in the calculation, and so in practice only the leading term would matter for the decisions.
Or to give an argument from reflection, there’s a cost to making decisions, so an agent with a lexicographic ordering would probably want to self-modify into an agent that only considers the leading term in the preference ordering, since then it doesn’t have to spend resources on the other terms.
The “leading term will dominate” effect is broken if there are infinidesimal chances around.
It might be sensible for an agent for some purposes to assume away some risks ie treat them as 0 chance. However it might come about that in some circumstances those risks can’t be assumed away. So a transformation in the other direction of turning a dirty hack into an actual engine that can accurately work on edgecases might be warranted.
No, when there are infinitesimal chances, you take the ones which favor your leading term.
The trouble is that for some combinations of infinidesimal and transfinite their product can be an ordinary finite real. One can’t thus keep them strictly separated in different “lanes”.
Oh, you meant literal infinitesimal (non-real) probabilities. I thought you meant very small probabilities.
The argument against lexicographic ordering is also an argument against infinitesimal probabilities. Which is to say, suppose you have some lotteries l1, l2 that you are considering choosing between (the argument can be iterated to apply to a larger number of lotteries), where each lottery ln can have payouts rn1, rn2, rn3, … with probabilities pn1, pn2, np3, ….
Suppose that rather than being real numbers, the probabilities and payouts are ordered lexicographically, with payout m being of shape (rnm1, rnm2, rnm3, …) and probability m being of shape (pnm1, pnm2, pnm3, …).
In order for the infinitesimal tail of the sequence to matter, you need the sum over r1m1\*p1m1 to be exactly precisely 100% equal to the sum over r2m1\*p2m1 (that is, after all, the definition of lexicographic orderings). If they differ by even the slightest margin (even by 10^-(3^^^^^^3)), this difference outweights anything in the infinitesimal tail. While you could obviously set up such a scenario mathematically, it seems like it would never happen in reality.
So while you could in principle imagine lexicographic preferences, I don’t think they would matter in reality, and a proper implementation of them would act isomorphic to or want to self-modify into something that only considers the leading term.
You need the product to be exactly equal and you don’t neccesarily need to do it factor by factor. (0,1)(1,0) can equal (1,0)(0,1) that is neglible chance of a finite reward is as good as a certainty of a neglible reward. Because they lanecross in this way knowing your rewards doesn’t mean you can just take the most pressing factor and forget the rest as the propabilities might have impacts that make the expected values switch places.
In application it is not straightforwar what things would be well attributed to small finite chances and what would be well attirbuted to infinidesimal chances. As a guess say that you know that some rocks would break under one meteor impact and other rocks would break under two meteor impacts. But you don’t know how likely meter impacts are an assume them to be 0. It kinda still remains true that the hard rocks need to have twice as good valuables in them in order to justify to wait around them rather than the soft rocks. If they might contain valuable of immense value at some point it starts to make sense to be more curious about the unknown risks and rewards rather than the known and modelled risks and rewards. Some of the actions and scenarios that are not assumed to be 0 will lean more heavily to the unmodeled parts such as a long plan might have triple the chance of meteors, what ever it is, compared to a short plan.
If one insists that everything needs to be real then making up arbitrary small finites for parts of the model that you have little to reason with might get very noisy. With keeping up several archimedean fields around one doesn’t force to squish in to a single one. That is if your ordinary plans have an expected value difference of 0.0005 then you can estimate that if meteor impactsw have less effect than that you know your assumtions are effectively safe. However if the differences are 0.000000000000002 then you migth be more paranoid and start to actually look whether all the “assumed 0” assumptions should actually be made.
In your example, r1m1*p1m1 is exactly equal to r2m1*p2m1; 0*1=0=1*0. The point is that if in the former case, you instead have (eps, 1)(1, 0), or in the latter case, you instead have (1, 0)(eps, 1), then immediately the tail of the sequence becomes irrelevant, because the heads differ. So only when the products of the heads are exactly equal do the tails become relevant.
This requires you to not use infinitesimal probabilities. 0.000000000000002 is still infinitely bigger than infinitesimals.
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.
If we just have pure lexiographics then it can be undefined whether we can product them together. In my mind I am turning the lexiographic to surreals by using them as weights in a Cantor normal form and then using surreal multiplication.
So in effect I have something like (a,b)(c,d)=(ac,ad+bc,bd).
I guess I know about “approximately equal” when two numbers would round out to the same nearest real number. The picture might also be complicated on whether immense chances exist. That is if you have a coin that has a finite chance to come up with something and I give you more than finite amount of tries to get it there is only a neglible chance of failure. Then ordinarily if a option has no finite chances of paying out it could be ignored but an immense exploration of a “finite-null” coin which has neglible chances of paying out could matter even at the finite level. And I guess on the other direction are the pascal wagers. Neglible chances of immense rewards. So there are sources other than aligning the finite multipliers to get effects that matter on the finite level.
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.