Generalizing Foundations of Decision Theory

This post is more about ar­tic­u­lat­ing mo­ti­va­tions than about pre­sent­ing any­thing new, but I think read­ers may learn some­thing about the foun­da­tions of clas­si­cal (ev­i­den­tial) de­ci­sion the­ory as they stand.

The Project

Most peo­ple in­ter­ested in de­ci­sion the­ory know about the VNM the­o­rem and the Dutch Book ar­gu­ment, and not much more. The VNM the­o­rem shows that if we have to make de­ci­sions over gam­bles which fol­low the laws of prob­a­bil­ity, and our prefer­ences obey four plau­si­ble pos­tu­lates of ra­tio­nal­ity (the VNM ax­ioms), then our prefer­ences over gam­bles can be rep­re­sented as an ex­pected util­ity func­tion. On the other hand, the Dutch Book ar­gu­ment as­sumes that we make de­ci­sions by ex­pected util­ity, but per­haps with a non-prob­a­bil­is­tic be­lief func­tion. It then proves that any vi­o­la­tion of prob­a­bil­ity the­ory im­plies a will­ing­ness to take sure-loss gam­bles. (Re­v­erse Dutch Book ar­gu­ments show that in­deed, fol­low­ing the laws of prob­a­bil­ity elimi­nates these sure-loss bets.)

So we can more or less ar­gue for ex­pected util­ity the­ory start­ing from prob­a­bil­ity the­ory, and ar­gue for prob­a­bil­ity the­ory start­ing from ex­pected util­ity the­ory; but clearly, this is not enough to provide good rea­son to en­dorse Bayesian de­ci­sion the­ory over­all. Sub­se­quent in­ves­ti­ga­tions which I will sum­ma­rize have at­tempted to ad­dress this gap.

But first, why care?

  • Log­i­cal In­duc­tion can be seen as re­sult­ing from a small tweak to the Dutch Book setup, re­lax­ing it enough that it could ap­ply to math­e­mat­i­cal un­cer­tainty. Although we were ini­tially op­ti­mistic that Log­i­cal In­duc­tion would al­low sig­nifi­cant progress in de­ci­sion the­ory, it has proven difficult to get a satis­fy­ing log­i­cal-in­duc­tion DT. Per­haps it would be use­ful to in­stead un­der­stand the ar­gu­ment for DT as a whole, and try to re­lax the foun­da­tions of DT in “the same way” we re­laxed the foun­da­tions of prob­a­bil­ity the­ory.

  • It seems likely to me that such a re-ex­am­i­na­tion of the foun­da­tions would au­to­mat­i­cally provide jus­tifi­ca­tion for re­flec­tively con­sis­tent de­ci­sion the­o­ries like UDT. Hope­fully I can make my in­tu­itions clear as I de­scribe things.

  • Fur­ther­more, the foun­da­tions of DT seem like they aren’t that solid. Per­haps we’ve put blin­ders on by not in­ves­ti­gat­ing these ar­gu­ments for DT in full. Even with­out the kind of mod­ifi­ca­tion to the as­sump­tions which I’m propos­ing, we may find sig­nifi­cant gen­er­al­iza­tions of DT are given just by drop­ping un­jus­tified ax­ioms in the ex­ist­ing foun­da­tions. We can already see one such gen­er­al­iza­tion, the use of in­finites­i­mal prob­a­bil­ity, by study­ing the his­tory; I’ll ex­plain this more.

Longer History

Jus­tify­ing Prob­a­bil­ity Theory

Be­fore go­ing into the at­tempts to jus­tify Bayesian de­ci­sion the­ory in its en­tirety, it’s worth men­tion­ing Cox’s the­o­rem, which is an­other way of jus­tify­ing prob­a­bil­ity alone. Un­like the Dutch Book ar­gu­ment, it doesn’t rely on a con­nec­tion be­tween be­liefs and de­ci­sions; in­stead, Cox makes a se­ries of plau­si­ble as­sump­tions about the na­ture of sub­jec­tive be­lief, and con­cludes that any ap­proach must ei­ther vi­o­late those as­sump­tions or be es­sen­tially equiv­a­lent to prob­a­bil­ity the­ory.

There has been some con­tro­versy about holes in Cox’s ar­gu­ment. Like other holes in the foun­da­tions which I will dis­cuss later, it seems one con­clu­sion we can draw by drop­ping un­jus­tified as­sump­tions is that there is no good rea­son to rule out in­finites­i­mal prob­a­bil­ities. I haven’t un­der­stood the is­sues with Cox’s the­o­rem yet, though, so I won’t re­mark on this fur­ther.

This is an opinionated sum­mary of the foun­da­tions of de­ci­sion the­ory, so I’ll re­mark on the rel­a­tive qual­ity of the jus­tifi­ca­tions pro­vided by the Dutch Book vs Cox. The Dutch Book ar­gu­ment pro­vides what could be called con­se­quen­tial­ist con­straints on ra­tio­nal­ity: if you don’t fol­low them, some­thing bad hap­pens. I’ll treat this as the “high­est tier” of ar­gu­ment. Cox’s ar­gu­ment re­lies on more de­on­tolog­i­cal con­straints: if you don’t fol­low them, it seems in­tu­itively as if you’ve done some­thing wrong. I’ll take this to be the sec­ond tier of jus­tifi­ca­tion.

Jus­tify­ing De­ci­sion Theory

VNM

Be­fore we move on to at­tempts to jus­tify de­ci­sion the­ory in full, let’s look at the VNM ax­ioms in a lit­tle de­tail.

The set-up is that we’ve got a set of out­comes , and we con­sider lot­ter­ies over out­comes which as­so­ci­ate a prob­a­bil­ity with each out­come (such that and ). We have a prefer­ence re­la­tion over out­comes, , which must obey the fol­low­ing prop­er­ties:

  1. (Com­plete­ness.) For any two lot­ter­ies , ei­ther , or , or nei­ther, writ­ten . (” or ” will be ab­bre­vi­ated as “” as usual.)

  2. (Tran­si­tivity.) If and , then .

  3. (Con­ti­nu­ity.) If , then there ex­ists such that a gam­ble as­sign­ing prob­a­bil­ity to and to satis­fies .

  4. (In­de­pen­dence.) If , then for any and , we have .

Tran­si­tivity is of­ten con­sid­ered to be jus­tified by the money-pump ar­gu­ment. Sup­pose that you vi­o­late tran­si­tivity for some ; that is, and , but . Then you’ll be will­ing to trade away for and then for (per­haps in ex­change for a triv­ial amount of money). But, then, you’ll have ; and since , you’ll gladly pay (a non-triv­ial amount) to switch back to . I can keep send­ing you through this loop to get more money out of you un­til you’re broke.

The money-pump ar­gu­ment seems similar in na­ture to the Dutch Book ar­gu­ment; both re­quire a slightly un­nat­u­ral setup (mak­ing the as­sump­tion that util­ity is always ex­change­able with money), but re­sult­ing in strong con­se­quen­tial­ist jus­tifi­ca­tions for ra­tio­nal­ity ax­ioms. So, I place the money-pump ar­gu­ment (and thus tran­si­tivity) in my “first tier” along with Dutch Book.

Com­plete­ness is less clear. Ac­cord­ing to the SEP, “most de­ci­sion the­o­rists sug­gest that ra­tio­nal­ity re­quires that prefer­ences be co­her­ently ex­tendible. This means that even if your prefer­ences are not com­plete, it should be pos­si­ble to com­plete them with­out vi­o­lat­ing any of the con­di­tions that are ra­tio­nally re­quired, in par­tic­u­lar Tran­si­tivity.” So, I sug­gest we place this in a third tier, the so-called struc­tural ax­ioms: those which are not re­ally jus­tified at all, ex­cept that as­sum­ing them al­lows us to prove our re­sults.

“Struc­tural ax­ioms” are a some­what cu­ri­ous arte­fact found in al­most all of the ax­iom-sets which we will look at. Th­ese ax­ioms usu­ally have some­thing to do with re­quiring that the do­main is rich enough for the in­tended proof to go through. Com­plete­ness is not usu­ally referred to as struc­tural, but if we agree with the quo­ta­tion above, I think we have to re­gard it as such.

I take the ax­iom of in­de­pen­dence to be tier two: an in­tu­itively strong ra­tio­nal­ity prin­ci­ple, but not one that’s en­forced by nasty things that hap­pen if we vi­o­late it. It sur­prises me that I’ve only seen this kind of jus­tifi­ca­tion for one of the four VNM ax­ioms. Ac­tu­ally, I sus­pect that in­de­pen­dence could be jus­tified in a tier-one way; it’s just that I haven’t seen it. (Devel­op­ing a frame­work in which an ar­gu­ment for in­de­pen­dence can be made just as well as the money-pump and dutch-book ar­gu­ments is part of my goal.)

I think many peo­ple would put con­ti­nu­ity at tier two, a strong in­tu­itive prin­ci­ple. I don’t see why, per­son­ally. For me, it seems like an as­sump­tion which only makes sense if we already have the in­tu­ition that ex­pected util­ity is go­ing to be the right way of do­ing things. This puts it in tier 3 for me; an­other struc­tural ax­iom. (The analogs of con­ti­nu­ity in the rest of the de­ci­sion the­o­ries I’ll men­tion come off as very struc­tural.)

Savage

Leonard Sav­age took on the task of pro­vid­ing si­mul­ta­neous jus­tifi­ca­tion of the en­tire Bayesian de­ci­sion the­ory, ground­ing sub­jec­tive prob­a­bil­ity and ex­pected util­ity in one set of ax­ioms. I won’t de­scribe the en­tire frame­work, as it’s fairly com­pli­cated; see the SEP sec­tion. I will note sev­eral fea­tures of it, though:

  • Sav­age makes the some­what pe­cu­liar move of sep­a­rat­ing the ob­jects of be­lief (“states”) and ob­jects of de­sire (“out­comes”). How we go about sep­a­rat­ing parts of the world into one or the other seems quite un­clear.

  • He re­places the gam­bles from VNM with “acts”: an act is a func­tion from states to out­comes (he’s prac­ti­cally beg­ging us to make ter­rible puns about his “sav­age acts”). Just as the VNM the­o­rem re­quires us to as­sume that the agent has prefer­ences on all lot­ter­ies, Sav­age’s the­o­rem re­quires the agent to have prefer­ences over all acts; that is, all func­tions from states to out­comes. Some of these may be quite ab­surd.

  • As the pa­per Ac­tu­al­ist Ra­tion­al­ity com­plains, Sav­age’s jus­tifi­ca­tion for his ax­ioms is quite de­on­tolog­i­cal; he is pri­mar­ily say­ing that if you no­ticed any vi­o­la­tion of the ax­ioms in your­self, you would feel there’s some­thing wrong with your think­ing and you would want to cor­rect it some­how. This doesn’t mean we can’t put some of his ax­ioms in tier 1; af­ter all, he’s got a tran­si­tivity ax­iom like ev­ery­one else. How­ever, on Sav­age’s ac­count, it’s all what I’d call tier-two jus­tifi­ca­tion.

  • Sav­age cer­tainly has what I’d call tier-three ax­ioms, as well. The SEP ar­ti­cle iden­ti­fies P5 and P6 as such. His ax­iom P6 re­quires that there ex­ist world-states which are suffi­ciently im­prob­a­ble so as to make even the worst pos­si­ble con­se­quences neg­ligible. Surely it can’t be a “re­quire­ment of ra­tio­nal­ity” that the state-space be com­plex enough to con­tain neg­ligible pos­si­bil­ities; this is just some­thing he needs to prove his the­o­rem. P6 is Sav­age’s ana­log of the con­ti­nu­ity ax­iom.

  • Sav­age chooses not to define prob­a­bil­ities on a sigma-alge­bra. I haven’t seen any de­ci­sion-the­o­rist who prefers to use sigma-alge­bras yet. Similarly, he only de­rives finite ad­di­tivity, not countable ad­di­tivity; this also seems com­mon among de­ci­sion the­o­rists.

  • Sav­age’s rep­re­sen­ta­tion the­o­rem shows that if his ax­ioms are fol­lowed, there ex­ists a unique prob­a­bil­ity dis­tri­bu­tion and a util­ity func­tion which is unique up to a lin­ear trans­for­ma­tion, such that the prefer­ence re­la­tion on acts is also the or­der­ing with re­spect to ex­pected util­ity.

Jeffrey-Bolker Axioms

In con­trast to Sav­age, Jeffrey’s de­ci­sion the­ory makes the ob­jects of be­lief and the ob­jects of de­sire the same. Both be­lief and de­sire are func­tions of log­i­cal propo­si­tions.

The most com­mon ax­iom­a­ti­za­tion is Bolker’s. We as­sume that there is a boolean field, with a prefer­ence re­la­tion , fol­low­ing these ax­ioms:

  1. is tran­si­tive and com­plete. is defined on all el­e­ments of the field ex­cept . (Jeffrey does not wish to re­quire prefer­ences over propo­si­tions which the agent be­lieves to be im­pos­si­ble, in con­trast to Sav­age.)

  2. The boolean field is com­plete and atom­less. More speci­fi­cally:

    • An up­per bound of a (pos­si­bly in­finite) set of propo­si­tions is a propo­si­tion im­plied by ev­ery propo­si­tion in that set. The supre­mum of is an up­per bound which im­plies ev­ery up­per bound. Define lower bound and in­fi­mum analo­gously. A com­plete Boolean alge­bra is one in which ev­ery set of propo­si­tions has a supre­mum and an in­fi­mum.

    • An atom is a propo­si­tion other than which is im­plied by it­self and , but by no other propo­si­tions. An atom­less Boolean alge­bra has no atoms.

  3. (Law of Aver­ag­ing.) If ,

    • If , then

    • If , then

  4. (Im­par­tial­ity.) If and , then if for some where and not , then for ev­ery such .

  5. (Con­ti­nu­ity.) Sup­pose that is the supre­mum (in­fi­mum) of a set of propo­si­tions , and . Then there ex­ists such that if is im­plied by (or where is the in­fi­mum, im­plies ), then .

The cen­tral ax­iom to Jeffrey’s de­ci­sion the­ory is the law of av­er­ag­ing. This can be seen as a kind of con­se­quen­tial­ism. If I vi­o­late this ax­iom, I would ei­ther value some gam­ble less than both its pos­si­ble out­comes and , or value it more. In the first case, we could charge an agent for switch­ing from the gam­ble to ; this would worsen the agent’s situ­a­tion, since one of or was true already, , and the agent has just lost money. In the other case, we can set up a proper money pump: charge the agent to keep switch­ing to the gam­ble , which it will hap­pily do whichever of or come out true.

So, I ten­ta­tively put ax­iom 3 in my first tier (pend­ing bet­ter for­mal­iza­tion of that ar­gu­ment).

I’ve already dealt with ax­iom 1, since it’s just the first two ax­ioms of VNM rol­led into one: I count tran­si­tivity as tier one, and com­plete­ness as tier two.

Ax­ioms two and five are clearly struc­tural, so I place them in my third tier. Bolker is es­sen­tially set­ting things up so that there will be an iso­mor­phism to the real num­bers when he de­rives the ex­is­tence of a prob­a­bil­ity and util­ity dis­tri­bu­tion from the ax­ioms.

Ax­iom 4 has to be con­sid­ered struc­tural in the sense I’m us­ing here, as well. Jeffrey ad­mits that there is no in­tu­itive mo­ti­va­tion for it un­less you already think of propo­si­tions as hav­ing some kind of mea­sure which de­ter­mines their rel­a­tive con­tri­bu­tion to ex­pected util­ity. If you do have such an in­tu­ition, ax­iom 4 is just say­ing that propo­si­tions whose weight is equal in one con­text must have equal weight in all con­texts. (Sav­age needs a similar ax­iom which says that prob­a­bil­ities do not change in differ­ent con­texts.)

Un­like Sav­age’s, Bolker’s rep­re­sen­ta­tion the­o­rem does not give us a unique prob­a­bil­ity dis­tri­bu­tion. In­stead, we can trade be­tween util­ity and prob­a­bil­ity via a cer­tain for­mula. Prob­a­bil­ity zero events are not dis­t­in­guish­able from events which cause the util­ities of all sub-events to be con­stant.

Jeffrey-Do­mo­tor Axioms

Zoltan Do­mo­tor pro­vides an al­ter­na­tive set of ax­ioms for Jeffrey’s de­ci­sion the­ory. Do­mo­tor points out that Bolker’s ax­ioms are suffi­cient, but not nec­es­sary, for his rep­re­sen­ta­tion the­o­rem. He sets out to con­struct a nec­es­sary and suffi­cient ax­iom­a­ti­za­tion. This ne­ces­si­tates deal­ing with finite and in­com­plete boolean fields. The re­sult is a rep­re­sen­ta­tion the­o­rem which al­lows non­stan­dard re­als; we can have in­finites­i­mal prob­a­bil­ities, and in­finites­i­mal or in­finite util­ities. So, we have a sec­ond point of ev­i­dence in fa­vor of that.

Although look­ing for nec­es­sary and suffi­cient con­di­tions seems promis­ing as a way of elimi­nat­ing struc­tural as­sump­tions like com­plete­ness and atom­less­ness, it ends up mak­ing all ax­ioms struc­tural. In fact, Do­mo­tor gives es­sen­tially one sig­nifi­cant ax­iom: his ax­iom J2. J2 is to­tally in­scrutable with­out a care­ful read­ing of the no­ta­tion in­tro­duced in his pa­per; it would be pointless to re­pro­duce it here. The ax­iom is cho­sen to ex­actly state the con­di­tions for the ex­is­tence of a prob­a­bil­ity and util­ity func­tion, and can’t be jus­tified in any other way—at least not with­out pro­vid­ing a full jus­tifi­ca­tion for Jeffrey’s de­ci­sion the­ory by other means!

Another con­se­quence of Do­mo­tor’s ax­iom­a­ti­za­tion is that the rep­re­sen­ta­tion be­comes wildly non-unique. This has to be true for a rep­re­sen­ta­tion the­o­rem deal­ing with finite situ­a­tions, since there is a lot of wig­gle room in what prob­a­bil­ities and util­ities rep­re­sent prefer­ences over finite do­mains. It gets even worse with the ad­di­tion of in­finites­i­mals, though; the choice of non­stan­dard-real field con­fronts us as well.

Con­di­tional Prob­a­bil­ity as Primitive

Hajek

In What Con­di­tional Prob­a­bil­ities Could Not Be, Alan Ha­jek ar­gues that con­di­tional prob­a­bil­ity can­not pos­si­bly be defined by Bayes’ fa­mous for­mula, due pri­mar­ily to its in­ad­e­quacy when con­di­tion­ing on events of prob­a­bil­ity zero. He also takes is­sue with other pro­posed defi­ni­tions, ar­gu­ing that con­di­tional prob­a­bil­ity should in­stead be taken as prim­i­tive.

The most pop­u­lar way of do­ing this are Pop­per’s ax­ioms of con­di­tional prob­a­bil­ity. In Learn­ing the Im­pos­si­ble (Vann McGee, 1994), it’s shown that con­di­tional prob­a­bil­ity func­tions fol­low­ing Pop­per’s ax­ioms and non­stan­dard-real prob­a­bil­ity func­tions with con­di­tion­als defined ac­cord­ing to Bayes’ the­o­rem are in­ter-trans­lat­able. Ha­jek doesn’t like the in­finites­i­mal ap­proach be­cause of the re­sult­ing non-unique­ness of rep­re­sen­ta­tion; but, for those who don’t see this as a prob­lem but who put some stock in Ha­jek’s other ar­gu­ments, this would be an­other point in fa­vor of in­finites­i­mal prob­a­bil­ity.

Richard Bradley

In A unified Bayesian de­ci­sion the­ory, Richard Bradley shows that Sav­age’s and Jeffrey’s de­ci­sion the­o­ries can be seen as spe­cial cases of a more gen­eral de­ci­sion the­ory which takes con­di­tional prob­a­bil­ities as a ba­sic el­e­ment. Bradley’s the­ory groups all the “struc­tural” as­sump­tions to­gether, as ax­ioms which pos­tu­late a rich set of “neu­tral” propo­si­tions (es­sen­tially, pos­tu­lat­ing a suffi­ciently rich set of coin-flips to mea­sure the prob­a­bil­ities of other propo­si­tions against). He needs to speci­fi­cally make an archimedean as­sump­tion to rule out non­stan­dard num­bers, which could eas­ily be dropped. He man­ages to de­rive a unique prob­a­bil­ity dis­tri­bu­tion in his rep­re­sen­ta­tion the­o­rem, as well.

OK, So What?

In gen­eral, I have hope that most of the tier-two ax­ioms could be­come tier-one; that is, it seems pos­si­ble to cre­ate a gen­er­al­iza­tion of dutch-book/​money-pump ar­gu­ments which cov­ers most of what de­ci­sion the­o­rists con­sider to be prin­ci­ples of ra­tio­nal­ity. I have an in­com­plete at­tempt which I’ll de­velop for a fu­ture post. I don’t ex­pect tier-three ax­ioms to be jus­tifi­able in this way.

With such a for­mal­ism in hand, the next step would be to try to de­rive a rep­re­sen­ta­tion the­o­rem: how can we un­der­stand the prefer­ences of an agent which doesn’t fall into these gen­er­al­ized traps? I’m not sure what gen­er­al­iza­tions to ex­pect be­yond in­finites­i­mal prob­a­bil­ity. It’s not even clear that such an agent’s prefer­ences will always be rep­re­sentable as a prob­a­bil­ity func­tion and util­ity func­tion pair; some more com­pli­cated struc­ture may be im­pli­cated (in which case it will likely be difficult to find!). This would tell us some­thing new about what agents look like in gen­eral.

The gen­er­al­ized dutch-book would likely dis­al­low prefer­ence func­tions which put agents in situ­a­tions they’ll pre­dictably re­gret. This sounds like a tem­po­ral con­sis­tency con­straint; so, it might also jus­tify up­date­less­ness au­to­mat­i­cally or with a lit­tle mod­ifi­ca­tion. That would cer­tainly be in­ter­est­ing.

And, as I said be­fore, if we have this kind of foun­da­tion we can at­tempt to “do the same thing we did with log­i­cal in­duc­tion” to get a de­ci­sion the­ory which is ap­pro­pri­ate for situ­a­tions of log­i­cal un­cer­tainty as well.