A summary of Savage’s foundations for probability and utility.

Edit: I think the P2c I wrote origi­nally may have been a bit too weak; fixed that. Nev­er­mind, recheck­ing, that wasn’t needed.

More ed­its (now con­soli­dated): Edited non­triv­ial­ity note. Edited to­tal­ity note. Added in the defi­ni­tion of nu­mer­i­cal prob­a­bil­ity in terms of qual­i­ta­tive prob­a­bil­ity (though not the proof that it works). Also slight clar­ifi­ca­tions on im­pli­ca­tions of P6′ and P6‴ on par­ti­tions into equiv­a­lent and al­most-equiv­a­lent parts, re­spec­tively.

One very late edit, June 2: Even though we don’t get countable ad­di­tivity, we still want a σ-alge­bra rather than just an alge­bra (this is needed for some of the proofs in the “par­ti­tion con­di­tions” sec­tion that I don’t go into here). Also noted nonempti­ness of gam­bles.

The idea that ra­tio­nal agents act in a man­ner iso­mor­phic to ex­pected-util­ity max­i­miz­ers is of­ten used here, typ­i­cally jus­tified with the Von Neu­mann-Mor­gen­stern the­o­rem. (The last of Von Neu­mann and Mor­gen­stern’s ax­ioms, the in­de­pen­dence ax­iom, can be grounded in a Dutch book ar­gu­ment.) But the Von Neu­mann-Mor­gen­stern the­o­rem as­sumes that the agent already mea­sures its be­liefs with (finitely ad­di­tive) prob­a­bil­ities. This in turn is of­ten jus­tified with Cox’s the­o­rem (valid so long as we as­sume a “large world”, which is im­plied by e.g. the ex­is­tence of a fair coin). But Cox’s the­o­rem as­sumes as an ax­iom that the plau­si­bil­ity of a state­ment is taken to be a real num­ber, a very large as­sump­tion! I have also seen this jus­tified here with Dutch book ar­gu­ments, but these all seem to as­sume that we are already us­ing some no­tion of ex­pected util­ity max­i­miza­tion (which is not only some­what cir­cu­lar, but also a con­sid­er­ably stronger as­sump­tion than that plau­si­bil­ities are mea­sured with real num­bers).

There is a way of ground­ing both (finitely ad­di­tive) prob­a­bil­ity and util­ity si­mul­ta­neously, how­ever, as de­tailed by Leonard Sav­age in his Foun­da­tions of Statis­tics (1954). In this ar­ti­cle I will state the ax­ioms and defi­ni­tions he gives, give a sum­mary of their log­i­cal struc­ture, and sug­gest a slight mod­ifi­ca­tion (which is equiv­a­lent math­e­mat­i­cally but slightly more philo­soph­i­cally satis­fy­ing). I would also like to ask the ques­tion: To what ex­tent can these ax­ioms be grounded in Dutch book ar­gu­ments or other more ba­sic prin­ci­ples? I warn the reader that I have not worked through all the proofs my­self and I sug­gest sim­ply find­ing a copy of the book if you want more de­tail.

Peter Fish­burn later showed in Utility The­ory for De­ci­sion Mak­ing (1970) that the ax­ioms set forth here ac­tu­ally im­ply that util­ity is bounded.

(Note: The ver­sions of the ax­ioms and defi­ni­tions in the end pa­pers are for­mu­lated slightly differ­ently from the ones in the text of the book, and in the 1954 ver­sion have an er­ror. I’ll be us­ing the ones from the text, though in some cases I’ll re­for­mu­late them slightly.)

Prim­i­tive no­tions; prefer­ence given a set of states

We will use the fol­low­ing prim­i­tive no­tions. Firstly, there is a set S of “states of the world”; the ex­act cur­rent state of the world is un­known to the agent. Se­condly, there is a set F of “con­se­quences”—things that can hap­pen as a re­sult of the agent’s ac­tions. Ac­tions or acts will be in­ter­preted as func­tions f:S→F, as two ac­tions which have the same con­se­quences re­gard­less of the state of the world are in­dis­t­in­guish­able and hence con­sid­ered equal. While the agent may be un­cer­tain as to the ex­act re­sults of its ac­tions, this can be folded into his un­cer­tainty about the state of the world. Fi­nally, we in­tro­duces as prim­i­tive a re­la­tion ≤ on the set of ac­tions, in­ter­preted as “is not preferred to”. I.e., f≤g means that given a choice be­tween ac­tions f and g, the agent will ei­ther pre­fer g or be in­differ­ent. As usual, sets of states will be referred to as “events”, and for the usual rea­sons we may want to re­strict the set of ad­mis­si­ble events to a boolean σ-sub­alge­bra of ℘(S), though I don’t know if that’s re­ally nec­es­sary here (Sav­age doesn’t seem to do so, though he does dis­cuss it some).

In any case, we then have the fol­low­ing ax­iom:

P1. The re­la­tion ≤ is a to­tal pre­order.

The in­tu­ition here for tran­si­tivity is pretty clear. For to­tal­ity, if the agent is pre­sented with a choice of two acts, it must choose one of them! Or be in­differ­ent. Per­haps we could in­stead use a par­tial pre­order (or or­der?), though this would give us two differ­ent in­dis­t­in­guish­able fla­vors of in­differ­ence, which seems prob­le­matic. But this could be use­ful if we wanted in­tran­si­tive in­differ­ence. So long as in­differ­ence is tran­si­tive, though, we can col­lapse this into a to­tal pre­order.

As usual we can then define f≥g, f<g (mean­ing “it is false that g≤f”), and g>f. I will use f≡g to mean “f≤g and g≤f”, i.e., the agent is in­differ­ent be­tween f and g. (Sav­age uses an equals sign with a dot over it.)

Note that though ≤ is defined in terms of how the agent chooses when pre­sented with two op­tions, Sav­age later notes that there is a con­struc­tion of W. Allen Wal­lis that al­lows one to ad­duce the agent’s prefer­ence or­der­ing among a finite set of more than two op­tions (mod­ulo in­differ­ence): Sim­ply tell the agent to rank the op­tions given, and that af­ter­ward, two of them will be cho­sen uniformly at ran­dom, and it will get whichever one it ranked higher.

The sec­ond ax­iom states that if two ac­tions have the same con­se­quences in some situ­a­tion, just what that equal con­se­quence is does not af­fect their rel­a­tive or­der­ing:

P2. Sup­pose f≤g, and B is a set of states such f and g agree on B. If f’ and g’ are an­other pair of acts which, out­side of B, agree with f and g re­spec­tively, and on B, agree with each other, then f’≤g’.

In other words, to de­cide be­tween two ac­tions, only the cases where they ac­tu­ally have differ­ent con­se­quences mat­ter.

With this ax­iom, we can now define:

D1. We say “f≤g given B” to mean that if f’ and g’ are ac­tions such that f’ agrees with f on B, g’ agrees with g on B, and f’ and g’ agree with each other out­side of B, then f’≤g’.

Due to ax­iom P2, this is well-defined.

Here is where I would like to sug­gest a small mod­ifi­ca­tion to this setup. The no­tion of “f≤g given B” is im­plic­itly taken to be how the agent makes de­ci­sions if it knows that B ob­tains. How­ever it seems to me that we should ac­tu­ally take “f≤g given B”, rather than f≤g, to be the prim­i­tive no­tion, ex­plic­itly in­ter­peted as “the agent does not pre­fer f to g if it knows that B ob­tains”. The agent always has some state of prior knowl­edge and this way we have ex­plic­itly speci­fied de­ci­sions un­der a given state of knowl­edge—the acts we are con­cerned with—as the ba­sis of our the­ory. Rather than defin­ing f≤g given B in terms of ≤, we can define f≤g to mean “f≤g given S” and then add ad­di­tional ax­ioms gov­ern­ing the re­la­tion be­tween “≤ given B” for vary­ing B, which in Sav­age’s setup are the­o­rems or part of the defi­ni­tion D1.

(Speci­fi­cally, I would mod­ify P1 and P2 to talk about “≤ given B” rather than ≤, and add the fol­low­ing the­o­rems as ax­ioms:

P2a. If f and g agree on B, then f≡g given B.

P2b. If B⊆C, f≤g given C, and f and g agree out­side B, then f≤g given B.

P2c. If B and C are dis­joint, and f≤g given B and given C, then f≤g given B∪C.

This is a lit­tle un­wieldy and per­haps there is an eas­ier way—these might not be min­i­mal. But they do seem to be suffi­cient.)

In any case, re­gard­less which way we do it, we’ve now es­tab­lished the no­tion of prefer­ence given that a set of states ob­tains, as well as prefer­ence with­out ad­di­tional knowl­edge, so hence­forth I’ll freely use both as Sav­age does with­out wor­ry­ing about which makes a bet­ter foun­da­tion, since they are equiv­a­lent.

Order­ing on preferences

The next defi­ni­tion is sim­ply to note that we can sen­si­bly talk about f≤b, b≤f, b≤c where here b and c are con­se­quences rather than ac­tions, sim­ply by in­ter­pret­ing con­se­quences as con­stant func­tions. (So the agent does have a prefer­ence or­der­ing on con­se­quences, it’s just in­duced from its or­der­ing on ac­tions. We do it this way since it’s its choices be­tween ac­tions we can ac­tu­ally see.)

How­ever, the third ax­iom reifies this in­duced or­der­ing some­what, by de­mand­ing that it be in­var­i­ant un­der gain­ing new in­for­ma­tion.

P3′. If b and c are con­se­quences and b≤c, then b≤c given any B.

Thus the fact that the agent may change prefer­ences given new in­for­ma­tion, just re­flects its un­cer­tainty about the re­sults of their ac­tions, rather than ac­tu­ally prefer­ring differ­ent con­se­quences in differ­ent states (any such prefer­ences can be done away with by sim­ply ex­pand­ing the set of con­se­quences).

Really this is not strong enough, but to state the ac­tual P3 we will first need a defi­ni­tion:

D3. An event B is said to be null if f≤g given B for any ac­tions f and g.

Null sets will cor­re­spond to sets of prob­a­bil­ity 0, once nu­mer­i­cal prob­a­bil­ity is in­tro­duced. Prob­a­bil­ity here is to be ad­duced from the agent’s prefer­ences, so we can­not dis­t­in­guish be­tween “the agent is cer­tain that B will not hap­pen” and “if B ob­tains, the agent doesn’t care what hap­pens”.

Now we can state the ac­tual P3:

P3. If b and c are con­se­quences and B is not null, then b≤c given B if and only if b≤c.

P3′, by con­trast, al­lowed some col­laps­ing of prefer­ence on gain­ing new in­for­ma­tion; here we have dis­al­lowed that ex­cept in the case where the new in­for­ma­tion is enough to col­lapse all prefer­ences en­tirely (a sort of “end of the world” or “fatal er­ror” sce­nario).

Qual­i­ta­tive probability

We’ve in­tro­duced above the idea of “prob­a­bil­ity 0” (and hence im­plic­itly prob­a­bil­ity 1; ob­serve that “¬B is null” is equiv­a­lent to “for any f and g, f≤g given B if and only if f≤g”). Now we want to ex­pand this to prob­a­bil­ity more gen­er­ally. But we will not ini­tially get num­bers out of it; rather we will first just get an­other to­tal pre­order­ing, A≤B, “A is at most as prob­a­ble as B”.

How can we de­ter­mine which of two events the agent thinks is more prob­a­ble? Have it bet on them, of course! First, we need a non­triv­ial­ity ax­iom so it has some things to bet on.

P5. There ex­ist con­se­quences b and c such that b>c.

(I don’t know what the re­sults would be if in­stead we used the weaker non­triv­ial­ity ax­iom “there ex­ist ac­tions f and g such that f<g”, i.e., “S is not null”. That we even­tu­ally get that ex­pected util­ity for com­par­ing all acts sug­gests that this should work, but I haven’t checked.)

So let us now con­sider a class of ac­tions which I will call “wa­gers”. (Sav­age doesn’t have any spe­cial term for these.) Define “the wa­ger on A for b over c” to mean the ac­tion that, on A, re­turns b, and oth­er­wise, re­turns c. Denote this by wA,b,c. Then we pos­tu­late:

P4. Let b>b’ be a pair of con­se­quences, and c>c’ an­other such pair. Then for any events A and B, wA,b,b’≤wB,b,b’ if and only if wA,c,c’≤wB,c,c’.

That is to say, if the agent is given the choice be­tween bet­ting on event A and bet­ting on event B, and the prize and booby prize are the same re­gard­less of which it bets on, then it shouldn’t just mat­ter just what the prize and booby prize are—it should just bet on whichever it thinks is more prob­a­ble. Hence we can define:

D4. For events A and B, we say “A is at most as prob­a­ble as B”, de­noted A≤B, if wA,b,b’≤wB,b,b’, where b>b’ is a pair of con­se­quences.

By P4, this is well-defined. We can then show that the re­la­tion on events ≤ is a to­tal pre­order, so we can use the usual no­ta­tion when talk­ing about it (again, ≡ will de­note equiv­alence).

In fact, ≤ is not only a to­tal pre­order, but a qual­i­ta­tive prob­a­bil­ity:

  1. ≤ is a to­tal preorder

  2. ∅≤A for any event A

  3. ∅<S

  4. Given events B, C, and D with D dis­joint from B and C, then B≤C if and only if B∪D≤C∪D.

(There is no con­di­tion cor­re­spond­ing to countable ad­di­tivity; as men­tioned above, we sim­ply won’t get countable ad­di­tivity out of this.) Note also that un­der this, A≡∅ if and only if A is null in the ear­lier sense. Also, we can define “A≤B given C” by com­par­ing the wa­gers given C; this is equiv­a­lent to the con­di­tion that A∩C≤B∩C. This re­la­tion is too a qual­i­ta­tive prob­a­bil­ity.

Par­ti­tion con­di­tions and nu­mer­i­cal probability

In or­der to get real num­bers to ap­pear, we are of course go­ing to have to make some sort of Archimedean as­sump­tion. In this sec­tion I dis­cuss what some of these look like and then ul­ti­mately state P6, the one Sav­age goes with.

First, defi­ni­tions. We will be con­sid­er­ing finitely-ad­di­tive prob­a­bil­ity mea­sures on the set of states, i.e. a func­tion P from the set of events to the in­ter­val [0,1] such that P(S)=1, and for dis­joint B and C, P(B∪C)=P(B)+P(C). We will say “P agrees with ≤” if for ev­ery A and B, A≤B if and only if P(A)≤P(B); and we will say “P al­most agrees with ≤” if for ev­ery A and B, A≤B im­plies P(A)≤P(B). (I.e., in the lat­ter case, nu­mer­i­cal prob­a­bil­ity is al­lowed to col­lapse some dis­tinc­tions be­tween events that the agent might not ac­tu­ally be in­differ­ent be­tween.)

We’ll be con­sid­er­ing here par­ti­tions of the set of states S. We’ll say a par­ti­tion of S is “uniform” if the parts are all equiv­a­lent. More gen­er­ally we’ll say it is “al­most uniform” if, for any r, the union of any r parts is at most as prob­a­ble as the union of any r+1 parts. (This is us­ing ≤, re­mem­ber; we don’t have nu­mer­i­cal prob­a­bil­ities yet!) (Note that any uniform par­ti­tion is al­most uniform.) Then it turns out that the fol­low­ing are equiv­a­lent:

  1. There ex­ist al­most-uniform par­ti­tions of S into ar­bi­trar­ily large num­bers of parts.

  2. For any B>∅, there ex­ists a par­ti­tion of S with each part less prob­a­ble than B.

  3. There ex­ists a (nec­es­sar­ily unique) finitely ad­di­tive prob­a­bil­ity mea­sure P that al­most agrees with ≤, which has the prop­erty that for any B and any 0≤λ≤1, there is a C⊆B such that P(C)=λP(B).

(Definitely not go­ing into the proof of this here. How­ever, the ac­tual defi­ni­tion of the nu­mer­i­cal prob­a­bil­ity P(A) is not so com­pli­cated: Let k(A,n) de­note the largest r such that there ex­ists an al­most-uniform par­ti­tion of S into n parts, for which there is some union of r parts, C, such that C≤A. Then the se­quence k(A,n)/​n always con­verges, and we can define P(A) to be its limit.)

So we could use this as our 6th ax­iom:

P6‴. For any B>∅, there ex­ists a par­ti­tion of S with each part less prob­a­ble than B.

Sav­age notes that other au­thors have as­sumed the stronger

P6″. There ex­ist uniform par­ti­tions of S into ar­bi­trar­ily large num­bers of parts.

since there’s an ob­vi­ous jus­tifi­ca­tion for this: the ex­is­tence of a fair coin! If a fair coin ex­ists, then we can gen­er­ate a uniform par­ti­tion of S into 2n parts sim­ply by flip­ping it n times and con­sid­er­ing the re­sult. We’ll ac­tu­ally end up as­sum­ing some­thing even stronger than this.

So P6‴ does get us nu­mer­i­cal prob­a­bil­ities, but they don’t nec­es­sar­ily re­flect all of the qual­i­ta­tive prob­a­bil­ity; P6‴ is only strong enough to force al­most agree­ment. Though it is stronger than that when ∅ is in­volved—it does turn out that P(B)=0 if and only if B≡∅. (And hence also P(B)=1 if and only if B≡S.) But more gen­er­ally it turns out that P(B)=P(C) if and only if B and C are “al­most equiv­a­lent”, which I will de­note B≈C (Sav­age uses a sym­bol I haven’t seen el­se­where), which is defined to mean that for any E>∅ dis­joint from B, B∪E≥C, and for any E>∅ dis­joint from C, C∪E≥B.

(It’s not ob­vi­ous to me that ≈ is in gen­eral an equiv­alence re­la­tion, but it cer­tainly is in the pres­ence of P6‴; Sav­age seems to use this im­plic­itly. Note also that an­other con­se­quence of P6‴ is that for any n there ex­ists a par­ti­tion of S into n al­most-equiv­a­lent parts; such a par­ti­tion is nec­es­sar­ily al­most-uniform.)

How­ever the fol­low­ing stronger ver­sion of P6‴ gets rid of this dis­tinc­tion:

P6′. For any B>C, there ex­ists a par­ti­tion of S, each part D of which satis­fies C∪D<B.

(Ob­serve that P6‴ is just P6′ for C=∅.) Un­der P6′, al­most equiv­alence is equiv­alence, and so nu­mer­i­cal prob­a­bil­ity agrees with qual­i­ta­tive prob­a­bil­ity, and we fi­nally have what we wanted. (So by ear­lier, P6′ im­plies P6″, not just P6‴. In­deed by above it im­plies the ex­is­tence of uniform par­ti­tions into n parts for any n, not just ar­bi­trar­ily large n.)

In ac­tu­al­ity, Sav­age as­sumes an even stronger ax­iom, which is needed to get util­ity and not just prob­a­bil­ity:

P6. For any acts g<h, and any con­se­quence b, there is a par­ti­tion of S such that if g is mod­ified on any one part to be con­stantly b there, we would still have g<h; and if h is mod­ified on any one part to be con­stantly b there, we would also still have g<h.

Ap­ply­ing P6 to wa­gers yields the weaker P6′.

We can now also get con­di­tional prob­a­bil­ity—if P6′ holds, it also holds for the pre­order­ings “≤ given C” for non-null C, and hence we can define P(B|C) to be the prob­a­bil­ity of B un­der the quan­ti­ta­tive prob­a­bil­ity we get cor­re­spond­ing to the qual­i­ta­tive prob­a­bilty “≤ given C”. Us­ing the unique­ness of agree­ing prob­a­bil­ity mea­sures, it’s easy to check that in­deed, P(B|C)=P(B∩C)/​P(C).

Utility for finite gambles

Now that we have nu­mer­i­cal prob­a­bil­ity, we can talk about finite gam­bles. If we have con­se­quences b1, …, bn, and prob­a­bil­ities λ1, …, λn sum­ming to 1, we can con­sider the gam­ble ∑λibi, rep­re­sented by any ac­tion which yields b1 with prob­a­bil­ity λ1, b2 with prob­a­bil­ity λ2, etc. (And with prob­a­bil­ity 0 does any­thing; we don’t care about events with prob­a­bil­ity 0.) Note that by above such an ac­tion nec­es­sar­ily ex­ists. It can be proven that any two ac­tions rep­re­sent­ing the same gam­ble are equiv­a­lent, and hence we can talk about com­par­ing gam­bles. We can also sen­si­bly talk about mix­ing gam­bles—tak­ing ∑λifi where the fi are finite gam­bles, and the λi are prob­a­bil­ities sum­ming to 1 - in the ob­vi­ous fash­ion.

With these defi­ni­tions, it turns out that Von Neu­mann and Mor­gen­stern’s in­de­pen­dence con­di­tion holds, and, us­ing ax­iom P6, Sav­age shows that the con­ti­nu­ity (i.e. Archimedean) con­di­tion also holds, and hence there is in­deed a util­ity func­tion, a func­tion U:F→R such that for any two finite gam­bles rep­re­sented by f and g re­spec­tively, f≤g if and only if the ex­pected util­ity of the first gam­ble is less than or equal to that of the sec­ond. Fur­ther­more, any two such util­ity func­tions are re­lated via an in­creas­ing af­fine trans­for­ma­tion.

We can also take ex­pected value know­ing that a given event C ob­tains, since we have nu­mer­i­cal prob­a­bil­ity; and in­deed this agrees with the prefer­ence or­der­ing on gam­bles given C.

Ex­pected util­ity in gen­eral and bound­ed­ness of utility

Fi­nally, Sav­age shows that if we as­sume one more ax­iom, P7, then we have that for any es­sen­tially bounded ac­tions f and g, we have f≤g if and only if the ex­pected util­ity of f is at most that of g. (It is pos­si­ble to define in­te­gra­tion with re­spect to a finitely ad­di­tive mea­sure similarly to how one does with re­spect to a countably ad­di­tive mea­sure; the re­sult is lin­ear and mono­tonic but doesn’t satisfy con­ver­gence prop­er­ties.) Similarly with re­spect to a given event C.

The ax­iom P7 is:

P7. If f and g are acts and B is an event such that f≤g(s) given B for ev­ery s∈B, then f≤g given B. Similarly, if f(s)≤g given B for ev­ery s in B, then f≤g given B.

So this is just an­other var­i­ant on the “sure-thing prin­ci­ple” that I ear­lier la­beled P2c.

Now in fact it turns out as men­tioned above that P7, when taken to­gether with the rest, im­plies that util­ity is bounded, and hence that we do in­deed have that for any f and g, f≤g if and only if the ex­pected util­ity of f is at most that of g! This is due to Peter Fish­burn and post­dates the first edi­tion of Foun­da­tions of Statis­tics, so in there Sav­age sim­ply notes that it would be nice if this worked for f and g not nec­es­sar­ily es­sen­tially bounded (so long as their ex­pected val­ues ex­ist, and al­low­ing them to be ±∞), but that he can’t prove this, and then adds a foot­note giv­ing a refer­ence for bounded util­ity. (Though he does prove us­ing P7 that if you have two acts f and g such that f,g≤b for all con­se­quences b, then f≡g; similarly if f,g≥b for all b. Ac­tu­ally, this is a key lemma in prov­ing that util­ity is bounded; Fish­burn’s proof works by show­ing that if util­ity were un­bounded, you could con­struct two ac­tions that con­tra­dict this.)

Of course, if you re­ally don’t like the con­clu­sion that util­ity is bounded, you could throw out ax­iom 7! It’s pretty in­tu­itive, but it’s not clear that ig­nor­ing it could ac­tu­ally get you Dutch-booked. After all, the first 6 ax­ioms are enough to han­dle finite gam­bles, 7 is only needed for more gen­eral situ­a­tions. So long as your Dutch bookie is limited to finite gam­bles, you don’t need this.

Ques­tions on fur­ther justification

So now that I’ve laid all this out, here’s the ques­tion I origi­nally meant to ask: To what ex­tent can these ax­ioms be grounded in more ba­sic prin­ci­ples, e.g. Dutch book ar­gu­ments? It seems to me that most of these are too ba­sic for that to ap­ply—Dutch book ar­gu­ments need more work­ing in the back­ground. Still, it seems to me ax­ioms P2, P3, and P4 might plau­si­bly be grounded this way, though I have not yet at­tempted to figure out how. P7 pre­sum­ably can’t, for the rea­sons noted in the pre­vi­ous sec­tion. P1 I as­sume is too ba­sic. P5 ob­vi­ously can’t (if the agent doesn’t care about any­thing, that’s its own prob­lem).

P6 is an Archimedean con­di­tion. Typ­i­cally I’ve seen those (speci­fi­cally Von Neu­mann and Mor­gen­stern’s con­ti­nu­ity con­di­tion) jus­tified on this site with the idea that in­finites­i­mals will never be rele­vant in any prac­ti­cal situ­a­tion—if c has only in­finites­i­mally more util­ity than b, the only case when the dis­tinc­tion would be rele­vant is if the prob­a­bil­ities of ac­com­plish­ing them were ex­actly equal, which is not re­al­is­tic. I’m guess­ing in­finites­i­mal prob­a­bil­ities can prob­a­bly be done away with in a similar man­ner?

Or are these not good ax­ioms in the first place? You all are more fa­mil­iar with these sorts of things than me. Ideas?