On the importance of taking limits: Infinite Spheres of Utility

I had a dis­cus­sion re­cently with some Less Wrongers about a de­ci­sion prob­lem in­volv­ing in­fini­ties, which ap­pears to have a para­dox­i­cal solu­tion. We have been warned by Jaynes and oth­ers to be care­ful about tak­ing the proper limits when in­fini­ties are in­volved in a prob­lem, and I thought this would be a good ex­am­ple to show that we can get an­swers that make sense out of prob­lems that seem not to.

The prob­lem is the “In­finite Spheres of Utility.” To quote a de­scrip­tion from Philos­o­phy et cetera,

Imag­ine a uni­verse con­tain­ing in­finitely many im­mor­tal peo­ple, par­ti­tioned into two “spheres”. In one sphere [sphere A], all the in­hab­itants live a bliss­ful ex­is­tence, whereas the mem­bers of the other sphere [sphere B] suffer un­bear­able agony. Now com­pare the fol­low­ing two vari­a­tions:

  1. Every­one starts off in the bliss­ful sphere. But each day, one more per­son gets per­ma­nently trans­ferred across to the agony sphere, where they reside for the rest of eter­nity.

  2. Every­one starts off in the agony sphere. But each day, one more per­son gets per­ma­nently trans­ferred across to the bliss­ful sphere, where they reside for the rest of eter­nity.

At first con­sid­er­a­tion, the prob­lem ap­pears to cause a para­dox:

Which sce­nario is bet­ter? The an­swer, para­dox­i­cally, ap­pears to be “both”.

  • At any mo­ment in time, there will be in­finitely many peo­ple in the origi­nal sphere, and only a finite num­ber who have been trans­ferred across. So op­tion 1 is bet­ter.

  • How­ever, each par­tic­u­lar per­son will spend only a finite amount of time in the first sphere, whereas they will spend an eter­nity in their post-trans­fer home. So op­tion 2 is bet­ter.

Given these rea­son­able but hard-to-rec­on­cile view­points, how do we make a de­ci­sion?

De­cid­ing how to decide

We first need to de­cide which kind of de­ci­sion anal­y­sis we want to use to choose a start­ing sphere. For ex­am­ple, we might sim­ply have an ar­bi­trary prefer­ence for putting ev­ery­one in the bad sphere. Para­dox over. Even if we want to use a form of util­i­tar­i­anism, we have more than one type to choose from. One of the sim­plest and most in­tu­itive is ad­di­tive util­i­tar­i­anism, in which we define a util­ity for each per­son, add them all to­gether, and make the choice with the larger to­tal util­ity. We can think of the para­dox above as re­sult­ing from our in­sis­tence that the solu­tion con­form to this kind of de­ci­sion anal­y­sis: it ap­pears that there should be more than one con­flict­ing solu­tion. In fact, we will see that there are three mu­tu­ally ex­clu­sive solu­tions con­form­ing to ad­di­tive util­i­tar­i­anism, and we can get to any one of them by solv­ing the finite prob­lem and choos­ing how to take the in­finite limit.

The Finite Problem

We will be­gin by cre­at­ing an analo­gous finite prob­lem us­ing ad­di­tive util­i­tar­i­anism.

Define and as the util­ities for one per­son liv­ing for one day in spheres or , re­spec­tively, which carry units of [util­ity/​(day*per­son)]. We might, for ex­am­ple, choose and , but we will cer­tainly choose . If we make “choice 1,” we be­gin with all peo­ple in sphere A. If we make “choice 2,” we be­gin with all peo­ple in sphere . The to­tal util­ity is the sum of and over all peo­ple and all days, or

where is the day, is the num­ber of days each per­son lives, and are the num­ber of peo­ple in spheres and on day , re­spec­tively, and I’ve im­plic­itly summed over the num­ber of peo­ple for brevity. We have as­sumed a lin­ear util­ity func­tion that val­ues ev­ery per­son’s util­ity equally on ev­ery day, but we could eas­ily gen­er­al­ize to other func­tional forms. Define as the peo­ple-trans­fer rate, which the prob­lem dic­tates to be

.

For choice 1, we be­gin with peo­ple in sphere and lose of them per day so

and

,

with these vari­ables only defined for , be­yond which ev­ery­one would be dead. Note that even though is 1, we’ll end up with a unit er­ror if we don’t carry it around. If we go with choice 2 in­stead, these vari­ables are switched, so

and

.

Ac­cord­ing to the prob­lem state­ment, the trans­fer rate is con­stant, but we could again eas­ily gen­er­al­ize to any trans­fer func­tion if we wanted to (even non-mono­tonic ones, or ones that de­pend on the num­ber of peo­ple in ei­ther sphere). Call the to­tal util­ities for each scheme and . From here on out, I will re­place sums with in­te­grals, be­cause the graphs will look bet­ter and the math takes up less space. We take on a small er­ror in the pro­cess, but it won’t af­fect the con­clu­sion.

Then the to­tal util­ities are

for choice 1, and

for choice 2, so that the differ­en­tial util­ity is

In­te­grat­ing over , we have

When this func­tion is pos­i­tive, choice 1 is bet­ter. When it’s nega­tive, choice 2 is bet­ter. Keep in mind that the sec­ond fac­tor is always pos­i­tive, since we set .

First, let’s con­firm that this re­sult makes sense for finite and . Figure 1 shows what hap­pens as we in­crease the num­ber of peo­ple in the prob­lem, but keep their life spans fixed. The differ­en­tial util­ity forms a line, with the first part of the line be­low the hori­zon­tal axis. For a small num­ber of peo­ple, it’s bet­ter to start ev­ery­one in sphere , since ev­ery­one will be quickly trans­ferred to . But for a large num­ber of peo­ple, it’s bet­ter to let them live out most of their lives on av­er­age in sphere be­fore get­ting trans­ferred to sphere . As ap­proaches in­finity, choice 1 is clearly bet­ter, since we have pos­i­tive .

Figure 1: Fixed lifes­pan. As the pop­u­la­tion in­creases, choice 1 be­comes mono­ton­i­cally bet­ter.

In con­trast, Figure 2 shows what hap­pens as a fixed num­ber of peo­ple be­come long-lived. This time, we get a parabola. If ev­ery­one dies quickly, it’s bet­ter to start in sphere , since many of them will die be­fore they get trans­ferred. If they live a long time, it’s bet­ter to start in sphere , and then live out most of their lives af­ter be­ing trans­ferred to sphere . As ap­proaches in­finity, choice 2 is clearly bet­ter, since we want peo­ple liv­ing out their im­mor­tal­ity in a good uni­verse, which cor­re­sponds to be­ing nega­tive.

Figure 2: Fixed pop­u­laiton. As life span in­creases, choice 2 be­comes bet­ter.

The In­finite Limit

What hap­pens if both and go to in­finity, as stated in the prob­lem? If we just plug into our equa­tion, the an­swer is un­defined, and that is the sub­stance of the para­dox. But of course, math­e­mat­ics lets us care­fully take limits of func­tions of mul­ti­ple vari­ables. Figure 3 is a den­sity plot of as a func­tion of both and . The value of is mapped on to color. When we re­fer to a sys­tem with “an in­finite num­ber of im­mor­tal peo­ple,” we’re talk­ing about a point far away from the ori­gin, where both and are pos­i­tive and in­finite. But from the den­sity plot, it’s clear that we could be talk­ing about many differ­ent points with differ­ent val­ues, and we have to spec­ify which di­rec­tion we go to get there. For ex­am­ple, if we fol­low the red dot­ted line out from the ori­gin, we will find that choice 1 gets bet­ter and bet­ter the more we in­crease and . If we fol­low the blue dot­ted line, choice 2 is bet­ter for large and . Fi­nally, if we fol­low the green dot­ted line out to in­finity, we find that for all and , mean­ing that our choices are equally good (or bad).

Density plot of differential utility

Figure 3: Utility vs. pop­u­la­tion and life span. The best choice de­pends on what path we fol­low to take the limit.

Prac­ti­cally, these lines rep­re­sent situ­a­tions in which we choose a finite num­ber of peo­ple and a finite life span, and then mon­i­tor what hap­pens to as we in­crease them both at a con­stant (but not nec­es­sar­ily equal) rate. We find that the an­swer de­pends on the differ­ence in rates. To for­mal­ize this re­sult, we can set in the equa­tion, where has units of [peo­ple/​day], and is al­lowed to be frac­tional. That way, ev­ery time we dou­ble , we mul­ti­ply by , and we move to­wards in­finity that way. Put an­other way, we con­strain our­selves to a lin­ear re­la­tion­ship be­tween and (we could take a non-lin­ear path out to in­finity if we cared to). We then have

The limit of this func­tion as is if , if , and if , which cov­ers all of the pos­si­ble lin­ear paths to in­finite pop­u­la­tion and in­finite lifes­pan.

Ex­pected Per­sonal Utility

Another in­ter­est­ing ques­tion to ask is “If I were one of the peo­ple in the prob­lem, what would I ex­pect my differ­en­tial util­ity to be?” We can an­swer this ques­tion us­ing the same method as above. I define my util­ity per day in each sphere as and in units of [util­ity/​day]. The ex­pected num­ber of days that I will be in the start­ing sphere for a finite pop­u­la­tion is , and the num­ber of days in the sec­ond sphere for a finite pop­u­la­tion and life span is . My ex­pected util­ity is there­fore given by

for choice 1, and

for choice 2. The differ­en­tial util­ity is the differ­ence be­tween these, or

If we just plug in in­fini­ties, the an­swer is un­defined. But if we use the same method as above, and define , then we can write

and we have well defined limits for all slopes . If I choose , I find that my lifes­pan grows faster than the pop­u­la­tion grows, and I’m bet­ter off start­ing in the bad sphere, ex­pect­ing to be trans­fered be­fore half my life is over. If I choose , the pop­u­la­tion grows too quickly, and I would pre­fer to start in the good sphere, since on av­er­age I’ll die be­fore I live more than half my life in the bad sphere. And as be­fore, if , I’m in­differ­ent to the two plans, since on av­er­age I will live half my life in each sphere ei­ther way.

The Answer

If we want to make a de­ci­sion based on ad­di­tive util­ity, the in­finite prob­lem is ill posed; it has no unique solu­tion un­less we take on ad­di­tional as­sump­tions. In par­tic­u­lar, we in­tro­duced an ad­di­tional pa­ram­e­ter , mak­ing clear three well defined solu­tions that span the space of pos­si­ble solu­tions. In do­ing so, we solved a similar well posed prob­lem, but not the origi­nal one.

So why bother, if we didn’t solve the prob­lem? It’s worth work­ing through the math be­cause tt gave us an in­tu­ition for how the sys­tem works, in­clud­ing a quan­ti­ta­tive un­der­stand­ing of cross­ing points, in­ter­cepts, and cur­va­tures of the util­ity curves. While the prob­lem state­ment cor­re­sponds to an im­pos­si­ble situ­a­tion, the finite prob­lem is quite pos­si­ble, and tak­ing lin­ear limits could very well cor­re­spond to a real phys­i­cal pro­cess. If we no­tice a para­dox and then stop think­ing, we lose an op­por­tu­nity to gain a bet­ter un­der­stand­ing of the de­ci­sion pro­cess.

We could also con­sider us­ing a differ­ent paradigm of de­ci­sion the­ory that might deal with in­finite quan­tities bet­ter. With that said, I think that ad­di­tive util­i­tar­i­anism rep­re­sents well the in­tu­itive para­dox that pre­sents each choice as bet­ter than the other. Our graphs with one vari­able fixed showed that these are both in­tu­itively valid view­points when taken alone, and their rec­on­cili­a­tion is the challeng­ing part.

If you’re in­ter­ested in similar prob­lems, the St. Peters­burg Para­dox also in­volves di­verg­ing util­lities, and it has has been “dis­pel­led” here.

Author’s notes: [last edit Oct.15, 2013] The sub­ject of un­bounded util­ities is not new here at Less Wrong. Stu­art_Arm­strong has a well writ­ten anal­y­sis of the Heaven and Hell prob­lem, among oth­ers. PhilGoetz has a use­ful note on in­fini­ties. And on a re­lated topic, in­finite set athe­ism abounds. I have been un­able to find on LW ex­am­ples of care­ful math­e­mat­i­cal treat­ments of ap­par­ent para­doxes that re­sult from mi­suse of in­finites in util­i­tar­ian calcu­la­tions, and I hope this ar­ti­cle serves as a con­crete ex­am­ple of how easy it is to defeat (some) such prob­lems. If there are other ex­am­ples at LW, please post them in the com­ments: I’d love to read them. It is in­ter­est­ing to note that Pas­cal’s Wager, a fa­mous re­lated prob­lem, is rather more difficult to solve. It in­volves not only in­fini­ties, but also a hy­poth­e­sis space whose car­di­nal­ity and par­tic­u­lar mem­bers are not ob­vi­ous.

Thanks to Me­stroyer for bring­ing this prob­lem to my at­ten­tion, Man­fred for helping me work through it, and Vin­cen­tYu for point­ing out the is­sues sur­round­ing ad­di­tive util­i­tar­i­anism as a de­ci­sion paradigm.