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This is a very technical paper, which requires some (or a lot) of familiarity with Bostrom/Kulczycki's "patched" Simulation Argument (www.simulation-argument.com/patch.pdf). I'm choosing to publish it here after experiencing Analysis's depressing version of peer review (they rejected a shorter, more-professional version of the paper based on one very positive review, and one negative review, from a superficial reading of the paper, that is almost certainly written by Kulczycki or Bostrom themself).

The positive review (of the earlier shorter, more-professional version of the paper) does a better job of summarizing the contribution than I did, so with the permission of the reviewer I'm including an excerpt here:

Bostrom (2003) argued that at least one of the following three claims is true: (1) the fraction of civilizations that reach a 'post-human' stage is approximately zero; (2) the fraction of post-human civilizations interested in running 'significant numbers' of simulations of their own ancestors is approximately zero; (3) the fraction of observers with human-type experiences that are simulated is approximately one.

The informal argument for this three-part disjunction is that, given what we know about the physical limits of computation, a post-human civilization would be so technologically advanced that it could run 'hugely many' simulations of observers very easily, should it choose to do so, so that the falsity of (1) and (2) implies the truth of (3). However, this informal argument falls short of a formal proof.

Bostrom himself saw that his attempt at a formal proof in the (2003) paper was sloppy, and he attempted to put it right in Bostrom and Kulczycki (2011). The take-home message of Sections 1 and 2 of the manuscript under review is that these (2011) reformulations of the argument are still rather sloppy. For example, the author points out (p. 6) that the main text of B&K inaccurately describes the mathematical argument in the appendix: the appendix uses an assumption much more favourable to B&K's desired conclusion than the assumption stated in the main text. Moreover, B&K's use of vague terms such as 'significant number' and 'astronomically large factor' creates a misleading impression. The author shows, amusingly, that the 'significant number' must be almost 1 million times greater than the 'astronomically large factor' for their argument to work (p. 9).

In Section 3, the author provides a new formulation of the simulation argument that is easily the most rigorous I have seen. This formulation deserves to be the reference point for future discussions of the argument's epistemological consequences."