Summary: Surreal Decisions

This post sum­marises a pa­per by Eddy Chen and Daniel Ru­bio on us­ing Sur­real num­bers to re­solve prob­lems of In­finite Ethics. Fu­ture posts will ar­gue that sur­re­als are the cor­rect ap­proach to this prob­lem be­fore ex­tend­ing upon this work. How­ever, this post merely aims to sum­marise this pa­per.


The prob­lem of In­finite Paral­y­sis is best de­scribed as fol­lows. Sup­pose that there are in­finite peo­ple and that they are happy so that there is in­finite util­ity. I then come along and punch 100 peo­ple de­stroy­ing 100 util­ity. Since there was in­finite util­ity at the start and in­finity minus 100 is still in­finity, so ar­guably I’ve done noth­ing wrong. How­ever, this seems to be a re­duc­tio ad ab­sur­dum if I’ve ever seen one.

One ap­proach men­tioned by Bostrom is to use hy­per­re­als to rep­re­sent in­finite se­quences of util­ity. In par­tic­u­lar, he sets the ith in­dex of the hy­per­real rep­re­sent­ing util­ity to the sum of the first i num­bers.

Un­for­tu­nately, there is not a unique defi­ni­tion of the hy­per­re­als—they re­quire what’s called a non-prin­ci­ple ul­tra­filter to be defined in or­der to de­ter­mine the or­der­ing. Our choice of this seems es­sen­tially ar­bi­trary and there­fore hard to jus­tify prin­ci­pally. Ad­di­tion­ally, sum­ma­tion re­quires a preferred lo­ca­tion around which to sum, which can be hard to jus­tify philo­soph­i­cally.

Sur­re­als Decisions

Chen and Ru­bio out­line a sur­real de­ci­sion the­ory by adapt­ing the the Von Neu­mann-Mor­gen­stern ax­ioms. They then use it to analyse Pas­cal’s Wager to demon­strate that the val­idity of the ar­gu­ment de­pends on par­tic­u­lar in­finite val­ues as­signed in the prob­lem and the var­i­ous deities that ex­ist.

They note that Ex­pected Utility The­ory with stan­dard in­fini­ties (car­di­nal num­bers) seems to pro­duce ab­surd re­sults. In par­tic­u­lar, it is in­differ­ent be­tween the fol­low­ing, when most peo­ple would pre­fer them in order

  • In­finity or some­thing: In­finite util­ity if heads, 10000 util­ity if tails

  • In­finity or noth­ing: In­finite util­ity if heads, 0 if tails

  • In­finity or bust: In­finite util­ity if heads, −10000 if tails

They provide an­other ex­am­ple, where they ar­gue that the or­der­ing is ob­vi­ous, but that this is un­defined for Ex­pected Utility The­ory:

  • Bi­ased pos­i­tive in­finity: 910 times plus in­finity, 110 mi­nis infinity

  • Fair in­finity: 5050 chance of plus in­finity or nega­tive infinity

  • Bi­ased nega­tive in­finity: 910 times minus in­finity, 110 plus infinity

They then ar­gue that sur­real num­bers can cor­rectly solve these prob­lems. At this stage I’ll note that the “ob­vi­ous” solu­tion re­quires an ad­di­tional as­sump­tion that the in­fini­ties in the above prob­lem all have the same mag­ni­tude. Without this as­sump­tion, the an­swer re­ally is un­defined.

Sur­real Numbers

They then out­line what sur­real num­bers are and how they are con­structed. For our pur­poses, all that mat­ters is that they have the fol­low­ing two prop­er­ties:

  • x+1 ≠ x

  • Stan­dard ar­ith­metic works the same, ie. ad­di­tion and mul­ti­pli­ca­tion are com­mu­ta­tive, as­so­ci­a­tivity, dis­tribu­tivity, ad­di­tive in­verses, mul­ti­plica­tive in­verses for non-zero val­ues, ect

Sur­real Von Neu­man-Mor­ganstern Ax­ioms:

Just like the finite ver­sion, this the­o­rem states that a set of prefer­ences can be rep­re­sented by a util­ity func­tion un­der cer­tain as­sump­tions such that the or­der­ing always prefers the op­tion with higher util­ity.

The pa­per lists the fol­low­ing four as­sump­tions (link to image if the text be­low is bro­ken):

  • Com­plete­ness: ∀x, y ∈ X, ei­ther x ≼ y or y ≼ x

  • Tran­si­tivity: ∀x, y, z ∈ X, if x ≼ y and y ≼ z, then x ≼ z

  • Con­ti­nu­ity*: ∀x, y, z ∈ X, if x ≼ y ≼ z, then there ex­ists a sur­real p ∈ *[0, 1] such that px + (1 − p)z ∼ y.

  • 4. In­de­pen­dence*: ∀x, y, z ∈ X, ∀p ∈ *(0, 1], x ≼ y if and only if px + (1 − p)z ≼ py + (1 − p)z.

(Here *[0,1] means the range 0 to 1 in the sur­real num­bers)

They then con­sider the po­ten­tial for prob­a­bil­ity the­ory to be ex­tend­ing to the sur­re­als by con­sid­er­ing the com­pat­i­bil­ity of the Kol­mogorov ax­ioms. Most are com­pat­i­ble, but the countable ad­di­tivity of events can’t be main­tained if we in­sist that prob­a­bil­ities re­main nor­mal­ised. They sug­gest that an al­ter­na­tive for­mu­la­tion of countable ad­di­tivity might be able to work around this limi­ta­tion, but they aren’t too con­cerned about this as finite ad­di­tivity is suffi­cient for this pa­per.

Pas­cal’s Wager:

One re­sponse to Pas­cal’s Wager is the Mixed Strat­egy ap­proach. Us­ing typ­i­cal in­fini­ties, any finite chance of an in­finity with no chance of nega­tive in­finity causes the ex­pected value to be in­finite. Since re­gard­less of our de­ci­sion at this point in time, we will still have a non-zero chance of us even­tu­ally be­com­ing Chris­tian, we should there­fore be in­differ­ent be­tween all ac­tions. This ar­gu­ment seems ab­surd and this can be see to be the case once we use Sur­real Num­bers as a smaller chance of be­com­ing Chris­tian leads to smaller in­finite value.

Another re­sponse is the Many Gods re­sponse. This re­sponse ar­gues that the origi­nal Pas­cal’s Wager as­sumes with­out rea­son that there is only one pos­si­ble God, when there might be mul­ti­ple pos­si­ble Gods offer­ing differ­ent lev­els of plus or minus in­finite util­ity. This pa­per is able to make this ar­gu­ment more pre­cise than it is nor­mally made thanks to sur­real num­bers. They then con­clude that Pas­cal’s Wager doesn’t de­liver what it promises: it says that you should be­lieve in God re­gard­less of the ev­i­dence, when in fact it de­pends on the like­li­hood of each de­ity ex­ist­ing and your ex­pec­ta­tion of their pun­ish­ments.

Rele­vance to In­finite Ethics:

This pa­per doesn’t dis­cuss in­finite ethics, but the ap­pli­ca­tion is rather triv­ial. In the Sur­re­als, X+1 is differ­ent from X, so we don’t run into in­finite paral­y­sis. This will be dis­cussed in more de­tail in fu­ture posts.