As a result, the effective discount falls off as 2^{-Kolmogorov complexity of t} which is only slightly faster than 1/t.
It is about 1/t x 1/log t x 1/log log t etc. for most values of t (taking base 2 logarithms). There are exceptions for very regular values of t.
Incidentally, I’ve been thinking about a similar weighting approach towards anthropic reasoning, and it seems to avoid a strong form of the Doomsday Argument (one where we bet heavily against our civilisation expanding). Imagine listing all the observers (or observer moments) in order of appearance since the Big Bang (use cosmological proper time). Then assign a prior probability 2^-K(n) to being the nth observer (or moment) in that sequence.
Now let’s test this distribution against my listed hypotheses above:
1. No other civilisations exist or have existed in the universe apart from us.
Fit to observations: Not too bad. After including the various log terms in 2^-K(n), the probability of me having an observer rank n between 60 billion and 120 billion (we don’t know it more precisely than that) seems to be about 1/log (60 billion) x 1/log (36) or roughly 1⁄200.
Still, the hypothesis seems a bit dodgy. How could there be exactly one civilisation over such a large amount of space and time? Perhaps the evolution of intelligence is just extraordinarily unlikely, a rare fluke that only happened once. But then the fact that the “fluke” actually happened at all makes this hypothesis a poor fit. A better hypothesis is that the chance of intelligence evolving is high enough to ensure that it will appear many times in the universe: Earth-now is just the first time it has happened. If observer moments were weighted uniformly, we would rule that out (we’d be very unlikely to be first), but with the 2^-K(n) weighting, there is rather high probability of being a smaller n, and so being in the first civilisation. So this hypothesis does actually work. One drawback is that living 13.8 billion years after the Big Bang, and with only 5% of stars still to form, we may simply be too late to be the first among many. If there were going to be many civilisations, we’d expect a lot of them to have already arrived.
Predictions for Future of Humanity: No doomsday prediction at all; the probability of my n falling in the range 60-120 billion is the same sum over 2^-K(n) regardless of how many people arrive after me. This looks promising.
2. A few have existed apart from us, but none have expanded (yet)
Fit to observations: Pretty good e.g. if the average number of observers per civilisation is less than 1 trilllion. In this case, I can’t know what my n is (since I don’t know exactly how many civilisations existed before human beings, or how many observers they each had). What I can infer is that my relative rank within my own civilisation will look like it fell at random between 1 and the average population of a civilisation. If that average population is less than 1 trillion, there will be a probability of > 1 in 20 of seeing a relative rank like my current one.
Predictions for Future of Humanity: There must be a fairly low probability of expanding, since other civilisations before us didn’t expand. If there were 100 of them, our own estimated probability of expanding would be less than 0.01 and so on. But notice that we can’t infer anything in particular about whether our own civilisation will expand: if it does expand (against the odds) then there will be a very large number of observer moments after us, but these will fall further down the tail of the Kolmogorov distribution. The probability of my having a rank n where it is (at a number before the expansion) doesn’t change. So I shouldn’t bet against expansion at odds much different from 100:1.
3. A few have existed, and a few have expanded, but we can’t see them (yet)
Fit to observations: Poor. Since some civilisations have already expanded, my own n must be very high (e.g. up in the trillions of trillions). But then most values of n which are that high and near to my own rank will correspond to observers inside one of the expanded civilisations. Since I don’t know my own n, I can’t expect it to just happen to fall inside one of the small civilisations. My observations look very unlikely under this model.
Predictions for Future of Humanity: Similar to 2
4. Lots have existed, but none have expanded (very strong future filter)
Fit to observations: Mixed. It can be made to fit if the average number of observers per civilisation is less than 1 trilllion; this is for reasons simlar to 2. While that gives a reasonable degree of fit, the prior likelihood of such a strong filter seems low.
Predictions for Future of Humanity: Very pessimistic, because of the strong universal filter.
5. Lots have existed, and a few have expanded (still a strong future filter), but we can’t see the expanded ones (yet)
Fit to observations: Poor. Things could still fit if the average population of a civilisation is less than a trillion. But that requires that the small, unexpanded, civilisations massively outnumber the big, expanded ones: so much so that most of the population is in the small ones. This requires an extremely strong future filter. Again, the prior likelihood of this strength of filter seems very low.
Predictions for Future of Humanity: Extremely pessimistic, because of the strong universal filter.
6. Lots have existed, and lots have expanded, so the uinverse is full of expanded civilisations; we don’t see that, but that’s because we are in a zoo or simulation of some sort.
Fit to observations: Poor: even worse than in case 5. Most values of n close to my own (enormous) value of n will be in one of the expanded civilisations. The most likely case seems to be that I’m in a simulation; but still there is no reason at all to suppose the simulation would look like this.
Predictions for Future of Humanity: Uncertain. A significant risk is that someone switches our simulation off, before we get a chance to expand and consume unavailable amounts of simulation resources (e.g. by running our own simulations in turn). This switch-off risk is rather hard to estimate. Most simulations will eventually get switched off, but the Kolmogorov weighting may put us into one of the earlier simulations, one which is running when lots of resources are still available, and doesn’t get turned off for a long time.
It is about 1/t x 1/log t x 1/log log t etc. for most values of t (taking base 2 logarithms). There are exceptions for very regular values of t.
Incidentally, I’ve been thinking about a similar weighting approach towards anthropic reasoning, and it seems to avoid a strong form of the Doomsday Argument (one where we bet heavily against our civilisation expanding). Imagine listing all the observers (or observer moments) in order of appearance since the Big Bang (use cosmological proper time). Then assign a prior probability 2^-K(n) to being the nth observer (or moment) in that sequence.
Now let’s test this distribution against my listed hypotheses above:
1. No other civilisations exist or have existed in the universe apart from us.
Fit to observations: Not too bad. After including the various log terms in 2^-K(n), the probability of me having an observer rank n between 60 billion and 120 billion (we don’t know it more precisely than that) seems to be about 1/log (60 billion) x 1/log (36) or roughly 1⁄200.
Still, the hypothesis seems a bit dodgy. How could there be exactly one civilisation over such a large amount of space and time? Perhaps the evolution of intelligence is just extraordinarily unlikely, a rare fluke that only happened once. But then the fact that the “fluke” actually happened at all makes this hypothesis a poor fit. A better hypothesis is that the chance of intelligence evolving is high enough to ensure that it will appear many times in the universe: Earth-now is just the first time it has happened. If observer moments were weighted uniformly, we would rule that out (we’d be very unlikely to be first), but with the 2^-K(n) weighting, there is rather high probability of being a smaller n, and so being in the first civilisation. So this hypothesis does actually work. One drawback is that living 13.8 billion years after the Big Bang, and with only 5% of stars still to form, we may simply be too late to be the first among many. If there were going to be many civilisations, we’d expect a lot of them to have already arrived.
Predictions for Future of Humanity: No doomsday prediction at all; the probability of my n falling in the range 60-120 billion is the same sum over 2^-K(n) regardless of how many people arrive after me. This looks promising.
2. A few have existed apart from us, but none have expanded (yet)
Fit to observations: Pretty good e.g. if the average number of observers per civilisation is less than 1 trilllion. In this case, I can’t know what my n is (since I don’t know exactly how many civilisations existed before human beings, or how many observers they each had). What I can infer is that my relative rank within my own civilisation will look like it fell at random between 1 and the average population of a civilisation. If that average population is less than 1 trillion, there will be a probability of > 1 in 20 of seeing a relative rank like my current one.
Predictions for Future of Humanity: There must be a fairly low probability of expanding, since other civilisations before us didn’t expand. If there were 100 of them, our own estimated probability of expanding would be less than 0.01 and so on. But notice that we can’t infer anything in particular about whether our own civilisation will expand: if it does expand (against the odds) then there will be a very large number of observer moments after us, but these will fall further down the tail of the Kolmogorov distribution. The probability of my having a rank n where it is (at a number before the expansion) doesn’t change. So I shouldn’t bet against expansion at odds much different from 100:1.
3. A few have existed, and a few have expanded, but we can’t see them (yet)
Fit to observations: Poor. Since some civilisations have already expanded, my own n must be very high (e.g. up in the trillions of trillions). But then most values of n which are that high and near to my own rank will correspond to observers inside one of the expanded civilisations. Since I don’t know my own n, I can’t expect it to just happen to fall inside one of the small civilisations. My observations look very unlikely under this model.
Predictions for Future of Humanity: Similar to 2
4. Lots have existed, but none have expanded (very strong future filter)
Fit to observations: Mixed. It can be made to fit if the average number of observers per civilisation is less than 1 trilllion; this is for reasons simlar to 2. While that gives a reasonable degree of fit, the prior likelihood of such a strong filter seems low.
Predictions for Future of Humanity: Very pessimistic, because of the strong universal filter.
5. Lots have existed, and a few have expanded (still a strong future filter), but we can’t see the expanded ones (yet)
Fit to observations: Poor. Things could still fit if the average population of a civilisation is less than a trillion. But that requires that the small, unexpanded, civilisations massively outnumber the big, expanded ones: so much so that most of the population is in the small ones. This requires an extremely strong future filter. Again, the prior likelihood of this strength of filter seems very low.
Predictions for Future of Humanity: Extremely pessimistic, because of the strong universal filter.
6. Lots have existed, and lots have expanded, so the uinverse is full of expanded civilisations; we don’t see that, but that’s because we are in a zoo or simulation of some sort.
Fit to observations: Poor: even worse than in case 5. Most values of n close to my own (enormous) value of n will be in one of the expanded civilisations. The most likely case seems to be that I’m in a simulation; but still there is no reason at all to suppose the simulation would look like this.
Predictions for Future of Humanity: Uncertain. A significant risk is that someone switches our simulation off, before we get a chance to expand and consume unavailable amounts of simulation resources (e.g. by running our own simulations in turn). This switch-off risk is rather hard to estimate. Most simulations will eventually get switched off, but the Kolmogorov weighting may put us into one of the earlier simulations, one which is running when lots of resources are still available, and doesn’t get turned off for a long time.