It’s an appealing idea (and one that has been informally around in LW-space for many years). But I wonder how useful it really is. Consider two classes of infinite-utility scenario.
The first is the sort considered in this paper: some outcome is merely decreed to be infinitely good or bad (e.g., because Christians contend that eternal salvation is a good infinitely superior to anything earthly). In this case, an obvious question is how to map this alleged infinite goodness or badness to a concrete surreal value. Are the glories of heaven worth exactly ω utility? How do we know it’s that rather than √ω or 3ω1/ω or something?
The second (and to my mind more interesting) is where the infinite utilities arise from combining infinitely many finite utilities. Rather than just decreeing that heaven is infinitely good, perhaps we should consider it as an infinite succession of finitely good days (though theologians would quibble with that on multiple grounds). Or perhaps the universe is spatially infinite and contains (e.g.) infinitely many exact copies of our earth, and we need to model that somehow. Or perhaps we’re contemplating an Everett-style quantum multiverse and the underlying Hilbert space is too big for the measures we care about to be finite-valued. (Note: this one may be bullshit; I haven’t thought about it carefully.) This sort of scenario seems like a better prospect for formalization: we can calculate which infinities we need just by adding up the finite ones. Except that we can’t, because there doesn’t appear to be a Right Way to compute infinite sums in the surreal numbers. For instance, consider the sum 1+1+1+⋯ with ω terms. That’s gotta be ω, right? It certainly looks like it should be—but note, e.g., that ω certainly isn’t the least upper bound of the finite sums we encounter on the way; for instance, ω−1 and 3√ω are smaller upper bounds.
Let’s suppose we somehow have a solution to these problems. Are we ready to start using surreal numbers (or, who knows?, some other number system bigger than the reals) to solve infinite-utility decision problems? Nope. Consider e.g. the following problem, which if it isn’t one of the motivational examples in the paper under discussion here is at least of the same type. There are infinitely many people. Infinitely many are really happy (utility +1000) and infinitely many are really unhappy (utility −1000). We have the choice between (1) leaving them all alone, (2) making a million unhappy people happy, and (3) making a million happy people unhappy. Naive real-valued decision theory is no good here because all the utilities are undefined (infinity minus infinity). But, even if we suppose we’ve got a way of computing infinite sums of surreal numbers, and it works kinda like the infinite sums we already know how to compute, we’re still screwed, because those infinite sums are order-dependent. If we line our people up as ++−++−++−++−⋯ then we “obviously” get infinite positive utility. If we line them up as +−−+−−+−−+−−⋯ then we “obviously” get infinite negative utility. But there’s no obvious way to choose the ordering, and what do we do if that action that makes a million unhappy people happy also rearranges them to make the second order more natural somehow when the first was more natural before?
Nothing in the Chen&Rubio paper seems to me to shed any light on these issues, and without that it seems to me we’re not really any better off with surreal utilities than we were with real utilities: the only problems we can solve better than before are ones artificially constructed to be solvable with the new machinery.
“Are the glories of heaven worth exactly ω utility? How do we know it’s that rather than √ω or 3ω1/ω or something?”—We don’t know unless it is specified. However, it’s not a bug, but a feature.
“But there’s no obvious way to choose the ordering, and what do we do if that action that makes a million unhappy people happy also rearranges them to make the second order more natural somehow when the first was more natural before?”—Yep, this is exactly the issue I’m currently working on. But my ideas aren’t quite ready to share yet.
In order to apply surreal arithmetic to the expected utility of world-states, it seems we’ll need to fix some canonical bijection between states of the world and ordinals / surreals. In the most general case this will require some form of the Axiom of Choice, but if we stick to a nice constructive universe (say the state space is computable) then things will be better. Is this the gist of what you’re working on?
Not quite. I don’t think there’s a unique canoncial bijection—I embrace there truly being multiple countable infinities. Although I do want to insist on some regularity. And computability is relevant here, as it makes it much easier to show that certain consistent labellings exist
Infinite sums of finite terms and finite sums of infinite terms might be different and the latter are quite easy. With A= ω * 1000 + ω * −1000, B= ω * 1000 + (ω-1000) * −1000 + 1000000*1000, C= (ω-1000) * 1000 + (ω) * −1000 + 1000000*-1000, its clear that B>A>C
To my belief normal utility funtions can be scaled to remain essentially the same. That is if one explicit version gives numbers 1, 10, 100 to the options then a tenfold function that gives 1, 100 and 1000 to the same options is equally valid. I would expect this to hold in the transfinites in that a function giving 1, ω and ω * ω would be as good as one giving ω , ω * ω and ω * ω * ω.
I am not sure that surreals neccesarily invoke infinite sums and their orderings. ω can be defined without sums and it becomes a separate thing to prove for example that 1+1=2 (that is, this is a genuine claim about how addition works in relation to already existing numbers, it’s not a restatement of the definition of 2). There is the issues that just because a value is transfinite you don’t know how big it is and some problems might be sensitive to get the magnitudes right. Say that you have pascal wager options of not having a life or afterlife, having a life for another day, living one day in heaven and living indefinetely in heaven. The correctish values would be 0, 1*1 , ω * 1 and ω * ω, the fourth option being clearly better than the third rather than equally good. Also there is no natural number N so that 1 * N >= ω but 1* ω = ω. “repeatedly +1” migth only refer to the first. Surreals deals with actual infinites not infinities as a limit of finite processes. In a way both ω abd ω * ω would appear as a series of “++++++...” so decomposition into a plus ordering can’t be their distinguising mark.
It’s an appealing idea (and one that has been informally around in LW-space for many years). But I wonder how useful it really is. Consider two classes of infinite-utility scenario.
The first is the sort considered in this paper: some outcome is merely decreed to be infinitely good or bad (e.g., because Christians contend that eternal salvation is a good infinitely superior to anything earthly). In this case, an obvious question is how to map this alleged infinite goodness or badness to a concrete surreal value. Are the glories of heaven worth exactly ω utility? How do we know it’s that rather than √ω or 3ω1/ω or something?
The second (and to my mind more interesting) is where the infinite utilities arise from combining infinitely many finite utilities. Rather than just decreeing that heaven is infinitely good, perhaps we should consider it as an infinite succession of finitely good days (though theologians would quibble with that on multiple grounds). Or perhaps the universe is spatially infinite and contains (e.g.) infinitely many exact copies of our earth, and we need to model that somehow. Or perhaps we’re contemplating an Everett-style quantum multiverse and the underlying Hilbert space is too big for the measures we care about to be finite-valued. (Note: this one may be bullshit; I haven’t thought about it carefully.) This sort of scenario seems like a better prospect for formalization: we can calculate which infinities we need just by adding up the finite ones. Except that we can’t, because there doesn’t appear to be a Right Way to compute infinite sums in the surreal numbers. For instance, consider the sum 1+1+1+⋯ with ω terms. That’s gotta be ω, right? It certainly looks like it should be—but note, e.g., that ω certainly isn’t the least upper bound of the finite sums we encounter on the way; for instance, ω−1 and 3√ω are smaller upper bounds.
Let’s suppose we somehow have a solution to these problems. Are we ready to start using surreal numbers (or, who knows?, some other number system bigger than the reals) to solve infinite-utility decision problems? Nope. Consider e.g. the following problem, which if it isn’t one of the motivational examples in the paper under discussion here is at least of the same type. There are infinitely many people. Infinitely many are really happy (utility +1000) and infinitely many are really unhappy (utility −1000). We have the choice between (1) leaving them all alone, (2) making a million unhappy people happy, and (3) making a million happy people unhappy. Naive real-valued decision theory is no good here because all the utilities are undefined (infinity minus infinity). But, even if we suppose we’ve got a way of computing infinite sums of surreal numbers, and it works kinda like the infinite sums we already know how to compute, we’re still screwed, because those infinite sums are order-dependent. If we line our people up as ++−++−++−++−⋯ then we “obviously” get infinite positive utility. If we line them up as +−−+−−+−−+−−⋯ then we “obviously” get infinite negative utility. But there’s no obvious way to choose the ordering, and what do we do if that action that makes a million unhappy people happy also rearranges them to make the second order more natural somehow when the first was more natural before?
Nothing in the Chen&Rubio paper seems to me to shed any light on these issues, and without that it seems to me we’re not really any better off with surreal utilities than we were with real utilities: the only problems we can solve better than before are ones artificially constructed to be solvable with the new machinery.
“Are the glories of heaven worth exactly ω utility? How do we know it’s that rather than √ω or 3ω1/ω or something?”—We don’t know unless it is specified. However, it’s not a bug, but a feature.
“But there’s no obvious way to choose the ordering, and what do we do if that action that makes a million unhappy people happy also rearranges them to make the second order more natural somehow when the first was more natural before?”—Yep, this is exactly the issue I’m currently working on. But my ideas aren’t quite ready to share yet.
In order to apply surreal arithmetic to the expected utility of world-states, it seems we’ll need to fix some canonical bijection between states of the world and ordinals / surreals. In the most general case this will require some form of the Axiom of Choice, but if we stick to a nice constructive universe (say the state space is computable) then things will be better. Is this the gist of what you’re working on?
Not quite. I don’t think there’s a unique canoncial bijection—I embrace there truly being multiple countable infinities. Although I do want to insist on some regularity. And computability is relevant here, as it makes it much easier to show that certain consistent labellings exist
Infinite sums of finite terms and finite sums of infinite terms might be different and the latter are quite easy. With A= ω * 1000 + ω * −1000, B= ω * 1000 + (ω-1000) * −1000 + 1000000*1000, C= (ω-1000) * 1000 + (ω) * −1000 + 1000000*-1000, its clear that B>A>C
To my belief normal utility funtions can be scaled to remain essentially the same. That is if one explicit version gives numbers 1, 10, 100 to the options then a tenfold function that gives 1, 100 and 1000 to the same options is equally valid. I would expect this to hold in the transfinites in that a function giving 1, ω and ω * ω would be as good as one giving ω , ω * ω and ω * ω * ω.
I am not sure that surreals neccesarily invoke infinite sums and their orderings. ω can be defined without sums and it becomes a separate thing to prove for example that 1+1=2 (that is, this is a genuine claim about how addition works in relation to already existing numbers, it’s not a restatement of the definition of 2). There is the issues that just because a value is transfinite you don’t know how big it is and some problems might be sensitive to get the magnitudes right. Say that you have pascal wager options of not having a life or afterlife, having a life for another day, living one day in heaven and living indefinetely in heaven. The correctish values would be 0, 1*1 , ω * 1 and ω * ω, the fourth option being clearly better than the third rather than equally good. Also there is no natural number N so that 1 * N >= ω but 1* ω = ω. “repeatedly +1” migth only refer to the first. Surreals deals with actual infinites not infinities as a limit of finite processes. In a way both ω abd ω * ω would appear as a series of “++++++...” so decomposition into a plus ordering can’t be their distinguising mark.