Toy model piece #4: partial preferences, re-re-visited

Two failed attempts

I ini­tially defined par­tial prefer­ences in terms of fore­ground vari­ables and back­ground vari­ables .

Then a par­tial prefer­ence would be defined by and in , such that, for any , the world de­scribed by would be bet­ter than the world de­scribed by . The idea be­ing that, ev­ery­thing else be­ing equal (ie the same ), a world with was bet­ter than a world with . The other as­sump­tion is that, within men­tal mod­els, hu­man prefer­ences can be phrased as one or many bi­nary com­par­i­sons. So if we have a par­tial prefer­ence like : “I pre­fer a choco­late ice-cream to get­ting kicked in the groin”, then and are oth­er­wise iden­ti­cal wor­lds with a choco­late ice-cream and a groin-kick, re­spec­tively.

Note that in this for­mal­ism, there are two sub­sets of the set of wor­lds, and , and map be­tween them (which just sends to ).

In a later post, I re­al­ised that such a for­mal­ism can’t cap­ture seem­ingly sim­ple prefer­ences, such as : ” peo­ple is bet­ter than peo­ple”. The prob­lem is that that prefer­ences like that don’t talk about just two sub­sets of wor­lds, but many more.

Thus a par­tial prefer­ence was defined as a pre­order. Now, a pre­order is cer­tainly rich enough to in­clude prefer­ences like , but its al­lows for far too many differ­ent types of struc­tures, need­ing a com­pli­cated en­ergy-min­imi­sa­tion pro­ce­dure to turn a pre­order into a util­ity func­tion.

This post pre­sents an­other for­mal­ism for par­tial prefer­ences, that keeps the ini­tial in­tu­ition but can cap­ture prefer­ences like .

The formalism

Let be the (finite) set of all wor­lds, seen as uni­verses with their whole his­tory.

Let be a sub­set of , and let be an in­jec­tive (one-to-one) map from to . Define , the image of , and as the in­verse.

Then the prefer­ence is de­ter­mined by:

  • For all , .

If and are dis­joint, this just re­pro­duces the origi­nal defi­ni­tion, with and .

But it also al­lows prefer­ences like , defin­ing as some­thing like “the same world as , but with one less per­son”. In that case, maps some parts of to it­self[1].

Then for any el­e­ment , we can con­struct its up­wards and down­wards chain:

  • .

Th­ese chains end when they cy­cle: so there is an and an so that (equiv­a­lently, ).

If they don’t cy­cle, the up­wards chain ends when there is an which is not an el­e­ment of (hence is not defined on in), and the down­ward chain ends when there is an which is not in (and hence is not defined on it).

So, for ex­am­ple, for , all the chains con­tain two el­e­ments only: and . For , there are no cy­cles, and the lower chain ends when the pop­u­la­tion hits zero, while the up­per chain ends when the pop­u­la­tion hits some max­i­mal value.

Utilities differ­ence be­tween clearly com­pa­rable worlds

Since the wor­lds of de­com­pose ei­ther into chains or cy­cles via , there is not need for the full ma­chin­ery for util­ities con­structed in this post.

One thing we can define un­am­bigu­ously, is the rel­a­tive util­ity be­tween two el­e­ments of the same chain/​cy­cle:

  • If and are in the same cy­cle, then .

  • Other­wise, if and are in the same chain, then .

Cur­rently, lets nor­mal­ise these rel­a­tive util­ities to , by nor­mal­is­ing each chain in­di­vi­d­u­ally; note that if ev­ery world in the chain is reach­able, this is the same as the mean-max nor­mal­i­sa­tion on each chain:

  • If and are in the same cy­cle, then .

  • Other­wise, if and are in the same chain with to­tal el­e­ments in the chain, then .

We we could try and ex­tend to a global util­ity func­tion which com­pares differ­ent chains and com­pares val­ues in chains with val­ues out­side of . But as we shall see in the next post, this doesn’t work when com­bin­ing differ­ent par­tial prefer­ences.

In­ter­pre­ta­tion of

The in­ter­pre­ta­tion of is some­thing like “this is the key differ­ence in fea­tures that causes the differ­ence in world-rank­ings”. So, for , the switches out a choco­late ice-cream and sub­sti­tutes a groin-kick. While for , the sim­ply re­moves one per­son from the world.

This means that, lo­cally, we can ex­press in the same for­mal­ism as in the first post. Here the are the back­ground vari­ables, while is a dis­crete vari­able that op­er­ates on.

We can­not nec­es­sar­ily ex­press this product globally. Con­sider, for , a situ­a­tion where is an idyl­lic village, is an Earth­bound hu­man pop­u­la­tion, and a star-span­ning civ­i­liza­tion with ex­ten­sive use of hu­man up­loads.

And if de­notes the num­ber of peo­ple in each world, it’s clear that hits a low max­i­mum for (thou­sands?), can rise much higher for (trillions?), and even higher for (need to use sci­en­tific no­ta­tion). So though makes sense, is non­sense. So there is no global de­com­po­si­tion of these wor­lds as .


  1. Note that there is a similar­ity with CP-nets, if we con­sider this as ex­press­ing a prefer­ence over pop­u­la­tion size while keep­ing other vari­ables con­stant. ↩︎