Boltzmann brain decision theory

Sup­pose I told you that you had many Boltz­mann brain copies of you. Is it then your duty to be as happy as pos­si­ble, so that these copies were also happy?

(Now some peo­ple might ar­gue that you can’t re­ally make your­self happy through men­tal effort; but you can cer­tainly make your­self sad, so avoid­ing do­ing that also counts.)

So, I told you that some pro­por­tion of your Boltz­mann brain copies were happy and some were sad, it seems that the best thing you could do, to in­crease the pro­por­tion of happy ones, is to be happy all the time—af­ter all, who knows when in your life a Boltz­mann brain might “hap­pen”?

But that rea­son­ing is wrong, a stan­dard er­ror of ev­i­den­tial de­ci­sion the­o­ries. Be­ing happy doesn’t make your Boltz­mann brain copies happy; in­stead, it en­sures that among all the ex­ist­ing Boltz­mann brains, only the happy ones may be copies of you.

This is similar to the XOR black­mail prob­lem in func­tional de­ci­sion the­ory. If you pay Omega when they send the black­mail let­ter (“You cur­rently have a ter­mite prob­lem in your house iff you won’t sent me £1000”), you’re not pro­tect­ing your house; in­stead, you’re de­ter­min­ing whether you live in a world where Omega will send the let­ter.

On the other hand, if there were long-lived iden­ti­cal copies of you scat­tered around the place, and you cared about their hap­piness in a to­tal util­i­tar­ian style way, then it seems there is a strong ar­gu­ment you should make your­self happy. So, some­where be­tween in­stan­ta­neous Boltz­mann brains and perfect copies, your de­ci­sion pro­cess changes. What forms the bound­aries be­tween the cat­e­gories?

Du­ra­tion and causality

If a Boltz­mann brain only has time to make a sin­gle de­ci­sion, then im­me­di­ately van­ishes, then that de­ci­sion is ir­rele­vant. So we have to have long-lived Boltz­mann brains, where long-lived means a sec­ond or more.

Similarly, the de­ci­sion has to be causally con­nected to the Boltz­mann brain’s sub­se­quent ex­pe­rience. It makes no sense if you de­cide to be happy, and then your brain gets im­me­di­ately flooded with pain im­me­di­ately af­ter—or the con­verse. Your de­ci­sion only mat­ters if your view of causal­ity is some­what ac­cu­rate. There­fore you re­quire long-lived Boltz­mann brains who re­spect causal­ity.

In a pre­vi­ous post, I showed that the ev­i­dence seems to sug­gest that Boltz­mann brains caused by quan­tum fluc­tu­a­tions are gen­er­ally very short-lived (this seems a re­al­is­tic re­sult) and that they don’t obey causal­ity (I’m more un­cer­tain about this).

In con­trast, for Boltz­mann brains cre­ated by nu­cle­ation in an ex­pand­ing uni­verse, most ob­server mo­ments be­long to Boltz­mann brains in Boltz­mann simu­la­tions: ex­cep­tion­ally long lived, with causal­ity. They are, how­ever, much—much! - less prob­a­ble than quan­tum fluc­tu­a­tions.

De­ci­sion the­ory: causal

As­sume, for the mo­ment, that you are an un­bound­ely ra­tio­nal agent (con­grat­u­la­tions, btw, on win­ning all the Clay in­sti­tute prizes, on crack­ing all pub­lic-key en­cryp­tion, on reg­is­ter­ing patents on all imag­in­able in­ven­tions, and for solv­ing friendly AI).

You have de­cent es­ti­mates as to how many Boltz­mann brains are long-lived with causal­ity, how many use your de­ci­sion the­ory, and how many are ex­act copies of you.

If you are us­ing a causal de­ci­sion the­ory, then only your ex­act copies mat­ter—the ones where you are un­sure of whether you are them or you are “your­self”. Let be the prob­a­bil­ity that you are a Boltz­mann brain at this very mo­ment, let be an ac­tion and de­com­pose your prefer­ences into , where is some util­ity func­tion and is hap­piness. By an abuse of no­ta­tion, I’ll write for the ex­pected given that ac­tion is taken by the “real” you, for ex­pected given that ac­tion is taken by a Boltz­mann brain copy of you, and similarly for .

Then the ex­pected util­ity for ac­tion is:

If we re­strict our at­ten­tion to medium-long du­ra­tion Boltz­mann brains, say ten sec­onds or less (though re­mem­ber that Boltz­mann simu­la­tions are an is­sue!), and as­sume that is rea­son­ably defined over the real world, we can ne­glect (since all ac­tions the Boltz­mann brain takes will have lit­tle to no im­pact on ), and use the ex­pres­sion:

This for­mula seems pretty in­tu­itive: you trade off the small in­crease in hap­piness in your Boltz­mann brain (), with the prob­a­bil­ity of be­ing a Boltz­mann brain (), and the util­ity and hap­piness you can ex­pect from your nor­mal life.

De­ci­sion the­ory: functional

If you’re us­ing a more sen­si­ble func­tional de­ci­sion the­ory, and are a to­tal util­i­tar­ian al­tru­ist where hap­piness is con­cerned, the ex­pres­sion is some­what differ­ent. Let be the set of Boltz­mann brains (not nec­es­sar­ily copies of you) that will take de­ci­sion iff you do. For any given , let be the fact that takes ac­tion , let be the prob­a­bil­ity of ex­ist­ing (not the prob­a­bil­ity of you be­ing ), and let be the hap­piness of .

Then the ex­pected util­ity for ac­tion a is:

Note that need not be the util­ity of b at all—you are al­tru­is­tic for hap­piness, not for gen­eral goals. As be­fore, if con­tains only medium-long du­ra­tion Boltz­mann brains (or if the ac­tions of these agents are in­de­pen­dent of ), we can sim­plify to:

Be­cause of the sum­ma­tion, the hap­piness of the Boltz­mann brains can come to dom­i­nate your de­ci­sions, even if you your­self are pretty cer­tain not to be a Boltz­mann brain.

Vari­a­tions of this, for differ­ent lev­els and types of al­tru­ism, should be clear now.

Bounded rationality

But nei­ther of us is un­bound­edly ra­tio­nal (a shame, re­ally). What should we do, in the real world, if we think that Boltz­mann brains are worth wor­ry­ing about? As­sume that your al­tru­ism and prob­a­bil­ities point to­wards Boltz­mann brain hap­piness be­ing the dom­i­nant con­sid­er­a­tion.

A key point of FDT/​UDT is that your de­ci­sions only make a differ­ence when they make some­thing hap­pen differ­ently. That sounds en­tirely tau­tolog­i­cal, but let’s think about the mo­ment in which an un­bounded ra­tio­nal agent might be tak­ing a differ­ent de­ci­sion in or­der to make Boltz­mann brains happy. When it is do­ing this, it is ap­ply­ing FDT, and con­sid­er­ing the hap­piness of the Boltz­mann brains, and then de­cid­ing to be happy.

And the same is true for you. Apart from per­sonal hap­piness, you should take ac­tions to make your­self hap­pier only when you are us­ing al­tru­is­tic UDT and think­ing about Boltz­mann brain prob­lems. So right now might be a good time.

This might feel per­verse—is that re­ally the only time the fact is rele­vant? Is there noth­ing else you could do—like make your­self into a con­sis­tent UDT agent in the first place?

But re­mem­ber the point in the in­tro­duc­tion—naively mak­ing your­self happy means that your Boltz­mann brain copies will be happy: but this isn’t ac­tu­ally in­creas­ing the hap­piness across all Boltz­mann brains, just chang­ing which ones are copies of you. Similarly, none of the fol­low­ing will change any­thing about other brains in the uni­verse:

  • Be­com­ing more of FDT/​UDT agent.

  • Be­com­ing more happy in gen­eral.

  • Be­com­ing more al­tru­is­tic in gen­eral.

They won’t change any­thing, be­cause they don’t have any acausal im­pact on Boltz­mann brain copies. More sur­pris­ingly, nei­ther will the fol­low­ing:

  • Make it eas­ier for you to make your­self happy when needed.

That will not make any differ­ence; some Boltz­mann brains will find it easy to make them­selves happy, oth­ers will find it hard. But the ac­tion a that Boltz­mann brains should take in these situ­a­tions is some­thing like “make your­self happy, as best you can”. Chang­ing the “as best you can” for you doesn’t change it for Boltz­mann brains.