# Matt_Simpson

Karma: 2,887
• See also Kreps, Notes on the The­ory of Choice. Note that one of these two re­stric­tions are re­quired in or­der to speci­fi­cally pre­vent in­finite ex­pected util­ity. So if a lot­tery spits out in­finite ex­pected util­ity, you broke some­thing in the VNM ax­ioms.

For any­one who’s in­ter­ested, a quick and dirty ex­pla­na­tion is that the prefer­ence re­la­tion is prim­i­tive, and we’re try­ing to come up with an in­dex (a util­ity func­tion) that re­pro­duces the prefer­ence re­la­tion. In the case of cer­tainty, we want a func­tion U:O->R where O is the out­come space and R is the real num­bers such that U(o1) > U(o2) if and only if o1 is preferred to o2. In the case of un­cer­tainty, U is defined on the set of prob­a­bil­ity dis­tri­bu­tions over O, i.e. U:M(O) → R. With the VNM ax­ioms, we get U(L) = E_L[u(o)] where L is some lot­tery (i.e. a prob­a­bil­ity dis­tri­bu­tion over O). U is strictly pro­hibited from tak­ing the value of in­finity in these defi­ni­tion. Now you prob­a­bly could ex­tend them a lit­tle bit to al­low for such in­fini­ties (at the cost of VNM util­ity per­haps), but you would need ev­ery lot­tery with in­finite ex­pected value to be tied for the best lot­tery ac­cord­ing to the prefer­ence re­la­tion.

• Funny, I read your post and my ini­tial re­ac­tion was that this ev­i­dence cuts against PUA. (Now I’m not sure whether it sup­ports PUA or not, but I lean to­wards sup­port).

PUA would pre­dict that this phrase

...while I de­vote my­self to wor­shiping the ground she walks on.

is unattrac­tive.

• If the basilisk is cor­rect* it seems any in­di­rect ap­proach is doomed, but I don’t see how it pre­vents a di­rect ap­proach. But that has it’s own set of prob­a­bly-in­sur­mountable prob­lems, I’d wa­ger.

* I re­main highly un­cer­tain about that, but it’s not some­thing I can claim to have a good grasp on or to have thought a lot about.

• I think I un­der­stand X, and it seems like a le­gi­t­i­mate prob­lem, but the com­ment I think you’re refer­ring to here seems to con­tain (nearly) all of X and not just half of it. So I’m con­fused and think I don’t com­pletely un­der­stand X.

Edit: I think I found the miss­ing part of X. Ouch.

• Poli­tics is the mind kil­ler for a va­ri­ety of rea­sons be­sides ridicu­lously strong pri­ors that are never swayed by ev­i­dence. Strong pri­ors isn’t even the en­tirety of the phe­nom­ena to be ex­plained (though it is a big part), let alone a fun­da­men­tal ex­pla­na­tion.

Also, I re­ally like Noah’s post (and was about to post it in the cur­rent open thread be­fore I found your post). Not only did Noah at­tach a word to a pretty com­monly oc­cur­ring phe­nomenon, the word seems to have a great set of con­no­ta­tions at­tached to it, given some goals about im­prov­ing dis­course.

• What do you mean by ‘con­tent’ here? The ba­sic nar­ra­tive each model tells about the econ­omy?

I think I agree with you. The big differ­ence be­tween the mod­els I learned in un­der­grad and the mod­els I learned in grad school was that in un­der­grad, ev­ery­thing was static. In grad school, the mod­els were dy­namic—i.e. a se­quence of equil­ibria over time in­stead of just one.

• FWIW I’m a grad stu­dent in econ, and in my ex­pe­rience the un­der­grad and grad­u­ate macro are com­pletely differ­ent. I re­call Greg Mankiw shar­ing a similar sen­ti­ment on his blog at some point, but can’t be both­ered to look it up.

• That was like, half the point of my post. I ob­vi­ously suck at ex­plain­ing my­self.

I think the com­bi­na­tion of me skim­ming and think­ing in terms of the un­der­ly­ing prefer­ence re­la­tion in­stead of in­terthe­o­retic weights caused me to miss it, but yeah, It’s clear you already said that.

Thanks for throw­ing your brain into the pile.

No prob­lem :) Here are some more thoughts:

It seems cor­rect to al­low the prob­a­bil­ity dis­tri­bu­tion over eth­i­cal the­o­ries to de­pend on the out­come—there are facts about the world which would change my prob­a­bil­ity dis­tri­bu­tion over eth­i­cal the­o­ries, e.g. facts about the brain or hu­man psy­chol­ogy. Not all meta-eth­i­cal the­o­ries would al­low this, but some do.

I’m nearly cer­tain that if you use prefer­ence re­la­tion over sets frame­work, you’ll re­cover a ver­sion of each eth­i­cal the­ory’s util­ity func­tion, and this even hap­pens if you al­low the true eth­i­cal the­ory to be cor­re­lated with the out­come of the lot­tery by us­ing a con­di­tional dis­tri­bu­tion P(m|o) in­stead of P(m). Im­plic­itly, this will define your k_m’s and c_m’s, given a ver­sion of each m’s util­ity func­tion, U_m(.).

It seems straight­for­ward to add un­cer­tainty over meta-prefer­ences into the mix, though now we’ll need meta-meta-prefer­ences over M2xM1xO. In gen­eral, you can always add un­cer­tainty over meta^n-prefer­ences, and the stan­dard VNM ax­ioms should get you what you want, but in the limit the space be­comes in­finite-di­men­sional and thus in­finite, so the usual VNM proof doesn’t ap­ply to the in­finite tower of un­cer­tainty.

It seems in­cor­rect to have M be a finite set in the first place since com­pet­ing eth­i­cal the­o­ries will say some­thing like “1 hu­man life = X dog lives”, and X could be any real num­ber. This means, once again, we blow up the VNM proof. On the other hand, I’m not sure this is any differ­ent than com­plain­ing that O is finite, in which case if you’re go­ing to sim­plify and as­sume O is finite, you may as well do the same for M.

• This strikes me as the wrong ap­proach. I think that you prob­a­bly need to go down to the level of meta-prefer­ences and ap­ply VNM-type rea­son­ing to this struc­ture rather than work­ing with the higher-level con­struct of util­ity func­tions. What do I mean by that? Well, let M de­note the model space and O de­note the out­come space. What I’m talk­ing about is a prefer­ence re­la­tion > on the space MxO. If we sim­ply as­sume such a > is given (satis­fy­ing the con­straint that (m1, o1) > (m1, o2) iff o1 >_m1 o2 where >_m1 is model m1′s prefer­ence re­la­tion) , then the VNM ax­ioms ap­plied to (>, MxO) and the dis­tri­bu­tion on M are prob­a­bly suffi­cient to give a util­ity func­tion, and it should have some in­ter­est­ing re­la­tion­ship with the util­ity func­tions of each com­pet­ing eth­i­cal model. (I don’t ac­tu­ally know this, it just seems in­tu­itively plau­si­ble. Feel free to do the ac­tual math and prove me wrong.)

On the other hand, we’d like to al­low the set of >_m’s to de­ter­mine > (along with P(m)), but I’m not op­ti­mistic. It seems like this should only hap­pen when the util­ity func­tions as­so­ci­ated with each >_m, U_m(o), are fully unique rather than unique up to af­fine trans­for­ma­tion. Ba­si­cally, we need our meta-prefer­ences over the rel­a­tive bad­ness of do­ing the wrong thing un­der com­pet­ing eth­i­cal the­o­ries to play some role in de­ter­min­ing >, and that in­for­ma­tion sim­ply isn’t pre­sent in the >_m’s.

(Even though my com­ment is a crit­i­cism, I still liked the post—it was good enough to get me think­ing at least)

Edit: clar­ity and fix­ing _′s

• One in­ter­est­ing fact from Chap­ter 4 (on weather pre­dic­tions) that seems worth men­tion­ing: Weather fore­cast­ers are also very good at man­u­ally and in­tu­itively (i.e. with­out some rigor­ous math­e­mat­i­cal method) fix­ing the pre­dic­tions of their mod­els. E.g. they might know that model A always pre­dicts rain a hun­dred miles or so too far west from the Rocky Moun­tains. So to fix this, they take the com­puter out­put and man­u­ally re­draw the lines (de­mark­ing level sets of pre­cip­i­ta­tion) about a hun­dred miles east, and this sig­nifi­cantly im­proves their fore­casts.

Also: the na­tional weather ser­vice gives the most ac­cu­rate weather pre­dic­tions. Every­one else will ex­ag­ger­ate to a greater or lesser de­gree in or­der to avoid get­ting flak from con­sumers about, e.g., rain on their wed­ding day (be­cause not-rain or their not-wed­ding day is far less of a prob­lem).

• I just started a re­search pro­ject with my ad­viser de­vel­op­ing new pos­te­rior sam­pling al­gorithms for dy­namic lin­ear mod­els (lin­ear gaus­sian dis­crete time state space mod­els). Right now I’m in the pro­cess of writ­ing up the re­sults of some simu­la­tions test­ing a cou­ple known al­gorithms, and am about to start some simu­la­tions test­ing some AFAIK un­known al­gorithms. There’s a cou­ple in­ter­est­ing di­ver­gent threads com­ing off this pro­ject, but I haven’t re­ally got­ten into those yet.

• Off the cuff: it’s prob­a­bly a ran­dom walk.

Edit: It’s now pretty clear to me that’s false, but plot­ting the er­godic means of sev­eral “chains” seems like a good way to figure it out.

Edit 2: In ret­ro­spect, I should have pre­dicted that. If any­one is in­ter­ested, I can post some R code so you can see what hap­pens.

• The book, Wicked is based on Wizard of Oz and has some re­lated themes IIRC. (I re­ally didn’t like the mu­si­cal based on the book though. But I might just dis­like mu­si­cals in gen­eral; FWIW I also didn’t like the only other mu­si­cal I’ve seen in per­son—Rent.)