Problems and Solutions in Infinite Ethics

Sum­mary: The uni­verse may very well be in­finite, and hence con­tain an in­finite amount of hap­piness and sad­ness. This causes sev­eral prob­lems for al­tru­ists; for ex­am­ple: we can plau­si­bly only af­fect a finite sub­set of the uni­verse, and an in­finite quan­tity of hap­piness is un­changed by the ad­di­tion or sub­trac­tion of a finite amount of hap­piness. This would im­ply that all forms of al­tru­ism are equally in­effec­tive.

Like ev­ery­thing in life, the canon­i­cal refer­ence in philos­o­phy about this prob­lem was writ­ten by Nick Bostrom. How­ever, I found that an area of eco­nomics known as “sus­tain­able de­vel­op­ment” has ac­tu­ally made much fur­ther progress on this sub­ject than the philos­o­phy world. In this post I go over some of what I con­sider to be the most in­ter­est­ing re­sults.

NB: This as­sumes a lot of math­e­mat­i­cal liter­acy and fa­mil­iar­ity with the sub­ject mat­ter, and hence isn’t tar­geted to a gen­eral au­di­ence. Most peo­ple will prob­a­bly pre­fer to read my other posts:

1. Sum­mary of the most in­ter­est­ing results

  1. There’s no eth­i­cal sys­tem which in­cor­po­rates all the things we might want.

  2. Even if we have pretty min­i­mal re­quire­ments, satis­fac­tory eth­i­cal sys­tems might ex­ist but we can’t prove their ex­is­tence, much less ac­tu­ally con­struct them

  3. Dis­counted util­i­tar­i­anism, whereby we value peo­ple less just be­cause they are fur­ther away in time, is ac­tu­ally a pretty rea­son­able thing de­spite philoso­phers con­sid­er­ing it ridicu­lous.

    1. (I con­sider this to be the first rea­son­able ar­gu­ment for lo­ca­vorism I’ve ever heard)

2. Definitions

In gen­eral, we con­sider a pop­u­la­tion to con­sist of an in­finite util­ity vec­tor (u0,u1,…) where ui is the ag­gre­gate util­ity of the gen­er­a­tion al­ive at time i. Utility is a bounded real num­ber (the fact that economists as­sume util­ity to be bounded con­fused me for a long time!). Our goal is to find a prefer­ence or­der­ing over the set of all util­ity vec­tors which is in some sense “rea­son­able”. While philoso­phers have un­der­stood for a long time that find­ing such an or­der­ing is difficult, I will pre­sent sev­eral the­o­rems which show that it is in fact im­pos­si­ble.

Due to a lack of la­tex sup­port I’m go­ing to give English-lan­guage defi­ni­tions and re­sults in­stead of math-ey ones; in­ter­ested peo­ple should look at the pa­pers them­selves any­way.

    3. Im­pos­si­bil­ity Results

    3.0 Spe­cific defs

    • Strong Pareto: if you can make a gen­er­a­tion bet­ter off, and none worse off, you should.

    • Weak Pareto: if you can make ev­ery gen­er­a­tion bet­ter off, you should.

    • In­ter­gen­er­a­tional equity: util­ity vec­tors are un­changed in value by any per­mu­ta­tion of their com­po­nents.

      • There is an im­por­tant dis­tinc­tion here be­tween al­low­ing a finite num­ber of el­e­ments to be per­muted and an in­finite num­ber; I will re­fer to the former as “finite in­ter­gen­er­a­tional equity” and the lat­ter as just “in­ter­gen­er­a­tional equity”

    • Eth­i­cal re­la­tion: one which obeys both weak Pareto and finite in­ter­gen­er­a­tional equity

    • So­cial welfare func­tion: an or­der-pre­serv­ing func­tion from the set of pop­u­la­tions (util­ity vec­tors) to the real numbers

    3.1 Di­a­mond-Basu-Mi­tra Im­pos­si­bil­ity Re­sult1

    1. There is no so­cial welfare func­tion which obeys Strong Pareto and finite in­ter­gen­er­a­tional equity. This means that any sort of util­i­tar­i­anism won’t work, un­less we look out­side the real num­bers.

    3.2 Zame’s im­pos­si­bil­ity re­sult2

    1. If an or­der­ing obeys finite in­ter­gen­er­a­tional equity over [0,1]N, then al­most always we can’t tell which of two pop­u­la­tions is bet­ter

      1. (i.e. the set of pop­u­la­tions {X,Y: nei­ther X<Y nor X>Y} has outer mea­sure one)

    2. The ex­is­tence of an eth­i­cal prefer­ence re­la­tion on [0,1]N is in­de­pen­dent of ZF plus the ax­iom of choice

    4. Pos­si­bil­ity Results

    We’ve just shown that it’s im­pos­si­ble to con­struct or even prove the ex­is­tence of any use­ful eth­i­cal sys­tem. But not all hope is lost!

    The im­por­tant idea here is that of a “sub­re­la­tion”: < is a sub­re­la­tion to <’ if x<y im­plies x<’y.

    Our ar­gu­ments will work like this:

    Sup­pose we could ex­tend util­i­tar­i­anism to the in­finite case. (We don’t, of course, know that we can ex­tend util­i­tar­i­anism to the in­finite case. But sup­pose we could.) Then A, B and C must fol­low.

    Tech­ni­cally: sup­pose util­i­tar­i­anism is a sub­re­la­tion of <. Then < must have prop­er­ties A, B and C.

    Every­thing in this sec­tion comes from (3). This is a great re­view of the liter­a­ture.

    4.1 Definition

    • Utili­tar­i­anism: we ex­tend the stan­dard to­tal util­i­tar­i­anism or­der­ing to in­finite pop­u­la­tions in the fol­low­ing way: sup­pose there is some time T af­ter which ev­ery gen­er­a­tion in X is at least as well off as ev­ery gen­er­a­tion in Y, and that the to­tal util­ity in X be­fore T is at least as good as the to­tal util­ity in Y be­fore T. Then X is at least as good as Y.

      • Note that this is not a com­plete or­der­ing! In fact, as per Zame’s re­sult above, the set of pop­u­la­tions it can mean­ingfully speak about has mea­sure zero.

    • Par­tial trans­la­tion scale in­var­i­ance: sup­pose af­ter some time T, X and Y be­come the same. Then we can add any ar­bi­trary util­ity vec­tor A to both X and Y with­out chang­ing the or­der­ing. (I.e. X > Y ó X+A > Y+A)

    4.2 Theorem

    1. Utili­tar­i­anism is a sub­re­la­tion of > if and only if > satis­fies strong Pareto, finite in­ter­gen­er­a­tional equity and par­tial trans­la­tion scale in­var­i­ance.

      1. This means that if we want to ex­tend util­i­tar­i­anism to the in­finite case, we can’t use a so­cial welfare func­tion, as per the above Basu-Mi­tra result

    4.3 Definition

    • Over­tak­ing util­i­tar­i­anism: sup­pose there is some point T af­ter which the to­tal util­ity of the first N gen­er­a­tions in X is always greater than the to­tal util­ity of the first N gen­er­a­tions in Y (given N > T). Then X is bet­ter than Y.

      • Note that util­i­tar­i­anism is a sub­re­la­tion of over­tak­ing utilitarianism

    • Weak limit­ing prefer­ence: sup­pose that for any time T, X trun­cated at time T is bet­ter than Y trun­cated at time T. Then X is bet­ter than Y.

    4.4 Theorem

    1. Over­tak­ing util­i­tar­i­anism is a sub­re­la­tion of < if and only if < satis­fies strong Pareto, finite in­ter­gen­er­a­tional equity, par­tial trans­la­tion scale in­var­i­ance, and weak limit­ing preference

    4.5 Definition

    • Dis­counted util­i­tar­i­anism: the util­ity of a pop­u­la­tion is the sum of its com­po­nents, dis­counted by how far away in time they are

    • Separa­bil­ity:

      • Separable pre­sent: if you can im­prove the first T gen­er­a­tions with­out af­fect­ing the rest, you should

      • Separable fu­ture: if you can im­prove ev­ery­thing af­ter the first T gen­er­a­tions with­out af­fect­ing the rest, you should

    • Sta­tion­ar­ity: prefer­ences are time invariant

    • Weak sen­si­tivity: for any util­ity vec­tor, we can mod­ify its first gen­er­a­tion some­how to make it bet­ter or worse

    4.6 Theorem

    1. The only con­tin­u­ous, mono­tonic re­la­tion which obeys weak sen­si­tivity, sta­tion­ary, and sep­a­ra­bil­ity is dis­counted utilitarianism

    4.7 Definition

    • Dic­ta­tor­ship of the pre­sent: there’s some time T af­ter which chang­ing the util­ity of gen­er­a­tions doesn’t matter

    4.8 Theorem

    1. Dis­counted util­i­tar­i­anism re­sults in a dic­ta­tor­ship of the pre­sent. (Re­mem­ber that each gen­er­a­tion’s util­ity is as­sumed to be bounded!)

    4.9 Definition

    • Sus­tain­able prefer­ence: a con­tin­u­ous or­der­ing which doesn’t have a dic­ta­tor­ship of the pre­sent but fol­lows strong Pareto and sep­a­ra­bil­ity.

    4.10 Theorem

    1. The only or­der­ing which is sus­tain­able is to take dis­counted util­i­tar­i­anism and add an “asymp­totic” part which en­sures that in­finitely long changes in util­ity mat­ter. (Of course, finite changes in util­ity still won’t mat­ter.)

    5. Conclusion

    I hope I’ve con­vinced you that there’s a “there” there: in­finite ethics is some­thing that peo­ple can make progress on, and it seems that most of the progress is be­ing made in the field of sus­tain­able de­vel­op­ment.

    Fun fact: the au­thor of the last the­o­rem (the one which defined “sus­tain­able”) was one of the lead economists on the Ky­oto pro­to­col. Who says in­finite ethics is im­prac­ti­cal?

    6. References

    1. Basu, Kaushik, and Ta­pan Mi­tra. “Ag­gre­gat­ing in­finite util­ity streams with in­ter­gen­er­a­tional equity: the im­pos­si­bil­ity of be­ing Pare­tian.” Econo­met­rica 71.5 (2003): 1557-1563. http://​​​​gasheim/​​zB%26M2003.pdf

    2. Zame, William R. “Can in­ter­gen­er­a­tional equity be op­er­a­tional­ized?.” (2007). https://​​​​bit­stream/​​1807/​​9745/​​1/​​1204.pdf

    3. Asheim, Geir B. “In­ter­gen­er­a­tional equity.” Annu. Rev. Econ. 2.1 (2010): 197-222.http://​​​​gasheim/​​A-ARE10.pdf