Here is a calculation regarding observers placing bets in the many-worlds scenario.

Let us assume that an observer has some current rank n in the human race (there have been n-1 humans born before him/her).

Let us assume a many futures scenario with the total population sizes N>=n weighted by the function W(N>=n) given by

W(N>=n) = n^d d / N^(1+d)

where we can let d approach zero. As I understand it this scale-free distribution is the broadest normalized function that we can use (as opposed to a uniform distribution over all N>=n, for example, which cannot be normalized).

The observer bets that the total population size N that he shall subjectively experience will be less than or equal to m * n (given that versions of himself will live long enough to see the end of the human race).

The fraction of the future versions of the observer that will be correct is given by

P(N <= m * n) = Integral (N=n to m * n) n^d d / N^(1+d) dN = 1 – 1/m^d

For any value of m, if we let d->0 then P(N <= m * n) -> 0.

Thus the fraction of future versions of the observer who observe N <= n * m is vanishingly small. Almost all future versions of the observer lose their money.

Thus the doomsday argument doesn’t work if there will be many actually occurring futures.

This is in contrast to the Doomsday argument in the standard scenario of one actually occurring future. In that case the Doomsday Argument says that the fraction of observers who correctly predict that N <= n * m, P(N <= n * m) = 1 - 1/m. (To derive this result consider an observer's fractional position f along the human race. The probability that f is larger or equal to some ratio r, P(f>=r), is given by P(f>=r) = 1 - r. Substitute f=n/N and r = 1/m so that we obtain P(N <= m * n) = 1 - 1/m).

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    Many worlds does not exist it is not real so there is not a real answer for this, this means you can create thought experiments like yours to define if a theory is real or not and in this case the many worlds is just an incorrect theory
    – Fuel
    Jun 10 '18 at 3:21

You have given no reason why we should assume that the prior over universe sizes should have a long tail.

If you want to use a probabilistic model you should have some reason for why you use this particular model. Do you have observations from our current universe which would give you reason to assume this particular form? If not, why not model it using an exponential model, or any other model whatsoever?

By using a distribution with a long tail you are precisely assuming what you set out to 'show' in the first place.

  • I am assuming that a priori we are completely ignorant about the total number of humans,N, who will ever live. If we know nothing about N then we should represent our knowledge (or more accurately our complete ignorance) with a probability distribution that contains no scale information whatsoever. The limiting probability distribution P(N) = 1/N uniquely contains no scale information. I am arguing that we can go one further in our assumption of complete prior ignorance by dropping the implicit assumption of one future. To do this we assume a scale-free weighting of futures given by W(N)=1/N. Jun 13 '18 at 15:33

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