Confused about the doomsday argument, please help

That wikipedia article is pretty bad. Read this and this instead. It looks like your confusion is caused by there being multiple versions of the doomsday argument; see the second question in the linked FAQ.

Assuming we're using Leslie's version:

As far as I can understand, after receiving this information you're certain to be among the first 10% of humans ever born, because it's true in every possible universe where you receive such information.

Yes, that's right: it's only before learning your birth rank that you assign a 10% probability of being in the first 10% of humans ever born.

Also learning your index doesn't seem to tell you very much about the date of the doomsday: it doesn't change the relative probabilities of doomsday dates that are consistent with your existence.

But it does: finding you're the 50th human is much more likely under the hypothesis that there will be 1000 humans than under the hypothesis that there will be 1000000 humans.

My understanding of the doomsday argument, perhaps mistaken, is that one assumes that population growth is exponential in the long term, in which case most observers live just before the end.

This is doubtless a frequentist approach, which perhaps isn't allowed, but I asked myself, how should an observer update on the information that civilization has existed up until that time, if observers collectively wanted to minimize their total error.

Suppose that a civilization lasts T years, and I assume that observations are uniformly distributed over the T years. (This would mean, for example, that the civilization at each time point in the interval (0,T) makes a collective, single vote.) Given only the information that civilization has lasted x years (clearly, 0<=x<=T), the observer would guess that civilization will actually last some multiple of x years: cx. Should they choose a c that is low (close to 1) or high, etc?

The optimal value of c can be calculated as exactly sqrt(2).

By taking the minimum of the function that measures the total error, the integral of the error (error=|T-cx|) integrated from T to 0.

You would get a different c if you assume that growth is exponential and weight by the number of observers. It would be closer to 1.

Also, implicit is the assumption that T is uniformly distributed over an unknown range. Instead, T might be normally distributed with an unknown mean or extremely tight-tailed. These would also affect c, but by moving c further from 1 I think.

I've been meaning to post about the Doomsday Argument for awhile. I have a strong sense that it's wrong, but I've had a hell of a time trying to put my finger on how it fails. The best that I can come up with is as follows: Aumann's agreement theorem says that two rational agents cannot disagree, in the long run. In particular, two rational agents presented with the same evidence should update their probability distribution in the same direction. Suppose I learn that I am the 50th human, and I am led to conclude that it is far more likely that only 1000 humans should ever live, than 100,000. But suppose I go tell Bob that I'm the 50th human; it would be senseless for him to come to the same conclusion that I have. Formally, it looks something like this:

P(1000 humans|I am human #50)>P(1000 humans)

but

P(1000 humans|Skatche is human #50)=P(1000 humans)

where the right hand sides are the prior probabilities. The same information has been conveyed in each case, yet very different conclusions have been reached. Since this cannot be, I conclude that the Doomsday Argument is mistaken. This could perhaps be adapted as an argument against anthropic reasoning more generally.