Logarithms and Total Utilitarianism

Epistemic sta­tus: I might be rein­vent­ing the wheel here

A com­mon cause for re­jec­tion of to­tal util­i­tar­i­anism is that it im­plies the so-called Repug­nant Con­clu­sion, of which a lot has been writ­ten el­se­where. I will ar­gue that while this im­pli­ca­tion is solid in the­ory, it does not ap­ply in our cur­rent known uni­verse. My view is similar to the one ex­pressed here, but I try to give more de­tails.

The Repug­nant Con­clu­sion IRL

The great­est rele­vance of the RC in prac­tice arises in situ­a­tions of scarce re­sources and Malthu­sian pop­u­la­tion traps¹: We com­pare pop­u­la­tion A, where there are few peo­ple with each one hav­ing plen­tiful re­sources, and pop­u­la­tion Z, which has grown from A un­til the av­er­age per­son lives in near-sub­sis­tence con­di­tions.

Let’s for­mal­ize this a bit: sup­pose each per­son re­quires 1 unit of re­sources for liv­ing, so that the util­ity of a per­son liv­ing on 1 re­sources is ex­actly 0: a com­pletely neu­tral life. Fur­ther­more, sup­pose util­ity is lin­ear w.r.t. re­sources: dou­bling re­sources means dou­bling util­ity and 10 re­sources cor­re­spond to 1 util­ity. If there are 100 re­sources in the world, pop­u­la­tion A might con­tain 10 peo­ple with 10 re­sources each and to­tal util­ity 10; pop­u­la­tion Z might con­tain 99 peo­ple with 10099 re­sources each and to­tal util­ity also 10.

So in this model, we are in­differ­ent be­tween A and Z even as ev­ery­one in Z is barely sub­sist­ing, and this would be the Repug­nant Con­clu­sion². But this con­clu­sion de­pends cru­cially on the re­la­tion­ship be­tween re­sources and util­ity which we have as­sumed to be lin­ear. What if our as­sump­tion is wrong? What is this re­la­tion­ship in the ac­tual world? Note that this is an em­piri­cal ques­tion³.

It is well known that self-re­ported hap­piness varies log­a­r­ith­mi­cally with in­come⁴, both be­tween coun­tries and for in­di­vi­d­u­als within each coun­try, so it seems rea­son­able to as­sume that the util­ity-re­sources re­la­tion is log­a­r­ith­mic: ex­po­nen­tial in­creases in re­sources bring lin­ear in­creases in util­ity.

Back to our model, as­sum­ing log util­ity, how do we now com­pare A and Z? If util­ity per per­son is where are the re­sources available to that per­son, then to­tal util­ity is . As­sum­ing equal­ity in the pop­u­la­tion (see the Equal­ity sec­tion), if are to­tal re­sources and is pop­u­la­tion size, each per­son has re­sources and so we have

We can plot to­tal util­ity (ver­ti­cal axis) as a func­tion of N (hori­zon­tal axis) for

Here we can see two ex­tremes of cero util­ity: at where there are no per­sons and at where each per­son lives with 1 re­sources, at sub­sis­tence level. In the mid­dle there is a sweet spot, and the max­i­mum M lies at around 37 peo­ple⁵.

Now we can an­swer our ques­tion! Pop­u­la­tion A, where is bet­ter than pop­u­la­tion Z where , but M is a su­pe­rior al­ter­na­tive to both.

So I have shown that there is a pop­u­la­tion M greater and bet­ter than A where ev­ery­one is worse off, how is that differ­ent from the RC? Well, the differ­ence is that this does not hap­pen for ev­ery pop­u­la­tion, but only for those where av­er­age well be­ing is rel­a­tively high. Fur­ther­more, the av­er­age in­di­vi­d­ual in M is far above sub­sis­tence.


Equality

In my model I as­sumed an equal dis­tri­bu­tion of re­sources over the pop­u­la­tion, mainly to sim­plify the calcu­la­tions, but also be­cause un­der the log re­la­tion­ship and if the pop­u­la­tion is held con­stant, to­tal util­i­tar­i­anism en­dorses equal­ity. I will try to give an in­tu­ition for this and then a for­mal proof.

This graph rep­re­sents in­di­vi­d­ual util­ity (ver­ti­cal axis) vs in­di­vi­d­ual re­sources (hori­zon­tal axis). If there are two peo­ple, A and B, each hav­ing 2.5 and 7.5 re­sources re­spec­tively, we can re­al­lo­cate re­sources so that both now are at point M, with 5 each. Note that the in­crease in util­ity for A is 3, while the de­crease for B is a bit less than 2, so to­tal util­ity in­creases by more than 1.

This hap­pens no mat­ter where in the graph are A and B due to the prop­er­ties of the log func­tion. As long as there is a differ­ence in wealth you can in­crease to­tal util­ity by re­dis­tribut­ing re­sources equally.

For a for­mal proof, see ⁶.


Implications

The main con­clu­sion I get from this is that al­though to­tal util­i­tar­i­anism is far from perfect, it might give good re­sults in prac­tice. The Repug­nant Con­clu­sion is not dead, how­ever. We can cer­tainly imag­ine some sen­tient aliens, AIs or an­i­mals whose util­ity func­tion is such that greater, worse-av­er­age-util­ity pop­u­la­tions end up be­ing bet­ter. But in this case, should we re­ally call it re­pug­nant? Could our in­tu­ition be fine-tuned for think­ing about hu­mans, and thus not ap­pli­ca­ble to those hy­po­thet­i­cal be­ings?

I don’t know to what ex­tent have oth­ers ex­plored the con­nec­tion be­tween to­tal util­i­tar­i­anism and equal­ity, but I was sur­prised when I re­al­ized that the former could im­ply the lat­ter. Of course, even if to­tal util­ity is all that mat­ters, it might not be pos­si­ble to reshuffle it among in­di­vi­d­u­als with com­plete liberty, which is the case in my model.


Footnotes

1: One might con­sider other ways of con­trol­ling in­di­vi­d­ual util­ity in a pop­u­la­tion be­sides re­sources (e.g. mind de­sign, tor­ture...) but these seem less rele­vant to me.

2: Ac­tu­ally, in the origi­nal for­mu­la­tion Z is shown to be bet­ter than A, not just equally good.

3: As long as util­ity is well defined, that is. Here I will use self-re­ported hap­piness as a proxy for util­ity.

4: See the charts here

5: We can find the ex­act max­i­mum for any R with a bit of calcu­lus:

A nice prop­erty of this is that the ra­tio that max­i­mizes is con­stant for al (the ex­act con­stant ob­tained here is just due to the ar­bi­trary choice of base 10 for the log­a­r­ithms)


6: For a pop­u­la­tion of in­di­vi­d­u­als the dis­tri­bu­tion of re­sources which max­i­mizes to­tal util­ity is that where for all . The proof goes by in­duc­tion on .

This is ob­vi­ous in the case . For the in­duc­tion step, we can sep­a­rate a pop­u­la­tion of into two sets of and 1 in­di­vi­d­u­als re­spec­tively so that to­tal util­ity is . Sup­pose we al­lo­cate re­sources to the group of , and to the last per­son. By hy­poth­e­sis, each of the peo­ple must re­ceive re­sources to max­i­mize their to­tal util­ity so

Now we have to de­cide how much should be.

Solv­ing for :

There­fore, for each of the first in­di­vi­d­u­als and for the last one