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.