Faustian bargains and discounting
I was reading TV Tropes on Hell, and it occurred to me: If your discounting was sufficiently hyperbolic, or indeed plain exponential with a low enough time preference, it would in some sense be rational to take a literal Faustian bargain. The integral to infinite time of some constant amount of torture per unit time, discounted exponentially or hyperbolically, is finite; enough worldly power and pleasure would outweight it.
But this clashes rather strongly with my intuition. Notice that the argument doesn’t depend on hyperbolic discounting; no preference pumping is involved. It works just fine with exponentials and a high decay constant. Or, if the worldly pleasures were strong enough, a low decay constant, that is, a high time preference, such as (I assume) most LWers have. For example, would you take eternal torture for a guarantee of living until the heat-death of the universe, 10^130 years from now, with all the refinements of Fun Theory along the way? Intuition says no, infinity being infinity, but then again intuition is notoriously bad at dealing with very large and very small numbers. If I calculate the thing in time-discounted utilons, it seems to me that my decay constant has to be very tiny indeed for me to care about what happens at the end of *10^130* years.
So should I discard my intuition, and take such a bargain if Mephistopheles should suddenly turn up? (Noting that in 10^130 years, I might learn a thing or two about getting out of such difficulties...) Or alternatively, should I stop discounting future utilons?
The integral of 1/t is log(t), thus hyperbolically discounted constant amount of torture over infinite time is still infinite. Hence your intuition about the Faustian bargain is consistent with hyperbolic discounting (but not with exponential discounting).
Ah. Very good point. I hadn’t actually checked that hyperbolic discounting would have the property of integrating to a constant. I assumed that since it falls faster than exponential, the integral would be smaller—but it only falls faster initially.
I always envied Faust; the temptation of being able to instantly realize knowledge in all disciplines is something that I would find very difficult to resist, regardless of the consequences. (As Bender says to the Robot Devil, “Hmm. I forgot you could tempt me with things I want.”) I think that this is because I’ve worked so hard to gain what little knowledge I have that the promise of knowing so much more with little additional effort has just become more alluring over time.
More to the point of what you were talking about: I’m not sure why hyperbolic discounting is emphasized over exponential here. A quick look at Wikipedia confirms that exponential decays faster. (In fact, unless I’m misunderstanding hyperbolic discounting, the integral doesn’t converge, you need to use exponential discounting.) I wouldn’t discount exponentially, so I personally would not take the offer if thinking clearly. But if Omega caught me in an excitable moment, maybe after reading an Iain Banks novel, I would be sorely tempted.
Didn’t check my math assumptions, yes. In fact hyperbolic discounting agrees with my intuition, while exponential doesn’t. I’m not sure if this demonstrates that exponential discounting is irrational (at least in sufficiently contrived scenarios!) or that I’m just a hyperbolic discounter at heart.
Well, according to experiments we all are, right? So maybe you could turn this around and say that the reason humans don’t like Faustian bargains is that we discount hyperbolically—and that a hypothetical race of Vulcans whose utility function is exponential would find the bargain intuitively appealing.
If your intuition is that 10^130 years of pleasure upfront don’t cancel out an infinity of pain afterwards, then your utility function just doesn’t discount in the way that you’re describing. Doesn’t preclude that some people’s might (to the extent that humans have utility functions.)
Eliezer made a related point in Against discount rates. I suspect the solution may be to treat the discount/interest rate as a consequence of the economy’s existence, rather than as a hardcoded constant in your utility function.
I’m not sure I understand this. You are suggesting that I should discount the infinite torture by some constant for all t, rather than exponentially, because there’s no economy at that time? This would give a non-finite integral again, thus agreeing with my intuition.
Yes, I agree.
To me it more depends on how you estimate the probability of being able to get out of the deal given enough time to try to find loopholes or other ways to break the deal.
Part of the “discounting future utilons” comes from the fact that we don’t know the future for sure, and the further the future is, the less we know about it. That applies to (some?) smokers (who don’t know for sure they’ll have cancer) or some criminals (who hope they’ll escape justice). That “oh, I’ll see when we’ll get closer to it” mindset is to a point reasonable—there are many problems you can solve later. But humans tend to not multiply right—a high chance of cancer or long prison term should still outweigh the temporary gain. Or a high chance of infinity of torture.
I observe that if we take this as straight Christian theology, you cannot plan to repent, and true repentance—not just wishing to avoid the consequences! - is the only way out. For other varieties of Omega, who knows?
As others have pointed out, a type of discounting is not “correct”. If you find yourself disagreeing with the conclusions of a certain type of discounting then you probably don’t discount that way.
Exponential discounting with a discount rate that has a positive lower bound for all time makes perfect sense if you assume that there is always an opportunity cost to immediate consumption. Indeed this has been true for all of human history… and prehistory, ever since the first protoman discovered that sharpening rocks and weaving baskets instead of hunting and gathering now now leads to much better hunting and gathering later. But there are reasonable assumptions (e.g. “the universe is finite”) that would prevent it from being true forever. On the other hand, I think most of those assumptions would also preclude the possibility of infinite torture. Is that what’s happening here? Your intuition is effectively telling you “infinite torture is impossible”, so you’re willing to look at the other side of the bargain?
Why the hell would you want to doom the vast majority of future-you’s to an eternity of torture?
Faustian bargain is paying a finite amount of pleasure with an infinite amount of pain. Contrary to the Pascal bargain (wager) where he is purchasing the infinite amount of pleasure with a finite amount of pain.
In fact, Pascal tries to switch between the infinite amount of pain and the infinite amount of pleasure with some finite effort. What is the essence of the Christianity.
A Christian would reply that it’s explained by God doing a token gesture of respect for Man’s free will, carrying out almost all of the (unfathomable) work of absolving a soul’s sins, but stopping His hand just in time to require the sinner to acknowledge their salvation.