Partial preferences and models

Note: work­ing on a re­search agenda, hence the large amount of small in­di­vi­d­ual posts, to have things to link to in the main doc­u­ments.

EDIT: This model is cur­rently ob­so­lete, see here for the most cur­rent ver­sion.

I’ve talked about par­tial prefer­ences and par­tial mod­els be­fore. I haven’t been par­tic­u­larly con­sis­tent in ter­minol­ogy so far (“proto-prefer­ences”, “model frag­ments”), but from now on I’ll stick with “par­tial”.

Definitions

So what are par­tial mod­els, and par­tial prefer­ences?

As­sume that ev­ery world is de­scribed by the val­ues of differ­ent vari­ables, .

A par­tial model is given by two sets, and , along with an ad­di­tion map . Thus for and , is an el­e­ment of .

We’ll want to have ‘rea­son­able’ prop­er­ties; for the mo­ment I’m imag­in­ing and as man­i­folds and as lo­cal home­o­mor­phism. If you don’t un­der­stand that ter­minol­ogy, it just means that is well be­haved and that as you move and around, you move in ev­ery di­rec­tion in .

A par­tial prefer­ence given the par­tial model above are two val­ues , along with the value judge­ment that:

  • for all , de­scribes a bet­ter world than .

We can gen­er­al­ise to non-lin­ear sub­spaces, but this ver­sion works well for many cir­cum­stances.

Interpretations

The are the fore­ground vari­ables that we care about in our par­tial model. The are the ‘back­ground vari­ables’ that are not rele­vant to the par­tial model at the mo­ment.

So, for ex­am­ple, when I con­tem­plate whether to walk or run back home, then the GDP of Swe­den, the dis­tance Voy­ager 2 is from Earth, the ac­tual value of the cos­molog­i­cal con­stant, the num­ber of deaths from malaria, and so on, are not ac­tu­ally rele­vant to that model. They are grouped un­der the (ir­rele­vant) back­ground vari­ables cat­e­gory.

No­tice that these vari­ables are only ir­rele­vant if they are in a ‘rea­son­able range’. If the GDP of Swe­den had sud­denly hit zero, if Voy­ager 2 was about to crash into my head, if the cos­molog­i­cal con­stant sud­denly jumped, or if malaria deaths reached of the pop­u­la­tion, then this would af­fect my walk­ing/​run­ning speed.

So the set also en­codes back­ground ex­pec­ta­tions about the world. Be­ing able to say that cer­tain val­ues are in an ‘ir­rele­vant’ range is a key part of sym­bol ground­ing and the frame prob­lem: it al­lows us to sep­a­rate and as be­ing, in a sense, com­ple­men­tary or or­thog­o­nal to each other. Note that hu­man defi­ni­tions of are im­plicit, in­com­plete, and of­ten wrong. But that doesn’t mat­ter; whether I be­lieve that wor­ld­wide deaths from malaria are in the thou­sands or in the mil­lions, that’s equally ir­rele­vant for my cur­rent de­ci­sion.

In com­par­i­son, the and the val­ues are much sim­pler, and are about the fac­tors I’m cur­rently con­tem­plat­ing: one of them in­volves run­ning, the other walk­ing. The vari­ables of could be fu­ture health, cur­rent tired­ness, how peo­ple might look at me as I run, how run­ning would make me feel, and how I cur­rently feel about run­ning. Or it could just be a sin­gle vari­able, like the mon­ster be­hind me with the teeth, or the whether I will be home on time to meet a friend.

So the par­tial prefer­ence is say­ing that, hold­ing the rest of the val­ues of the world con­stant, when look­ing at these is­sues, I cur­rently pre­fer to run or to walk.

Re-in­vent­ing the wheel

This whole con­struc­tion feels like re-in­vent­ing the wheel: surely some­one has de­signed some­thing like par­tial mod­els be­fore? What are the search terms I’m miss­ing?